[ { "id": "Statistics_0000", "subject": "Statistics", "topic": "Experimental design", "subtopic": "Concepts", "level": "3", "keywords": [ "Study", "Experiment", "Observation", "statistics", "introduction", "concepts" ], "problem_v1": "Determine whether the follow descriptions correspond to an observational study or an experiment. Write \"EXPERIMENT\" for experiment and \"OBSERVATION\" for observational study. (without quotations)\n(a) Classifying different stages of a child's language development. answer: [ANS]\n(b) The effectiveness of lecture teaching is tested with a sample of students who has completed numerous lecture style courses. answer: [ANS]\n(c) Studying how patients respond when given a placebo. answer: [ANS]", "answer_v1": [ "OBSERVATION", "OBSERVATION", "EXPERIMENT" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "EXPERIMENT", "OBSERVATION" ], [ "EXPERIMENT", "OBSERVATION" ], [ "EXPERIMENT", "OBSERVATION" ] ], "problem_v2": "Determine whether the follow descriptions correspond to an observational study or an experiment. Write \"EXPERIMENT\" for experiment and \"OBSERVATION\" for observational study. (without quotations)\n(a) Cartons on milk are opened, and the volumes of the contents are measured. answer: [ANS]\n(b) A new study examines how efficiently althletes burn calories by limiting their diets. answer: [ANS]\n(c) A new antibiotic is tested in effectiveness by recording how the drug works on patients that already take the drug. answer: [ANS]", "answer_v2": [ "EXPERIMENT", "EXPERIMENT", "OBSERVATION" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "EXPERIMENT", "OBSERVATION" ], [ "EXPERIMENT", "OBSERVATION" ], [ "EXPERIMENT", "OBSERVATION" ] ], "problem_v3": "Determine whether the follow descriptions correspond to an observational study or an experiment. Write \"EXPERIMENT\" for experiment and \"OBSERVATION\" for observational study. (without quotations)\n(a) A new antibiotic is tested in effectiveness by recording how the drug works on patients that already take the drug. answer: [ANS]\n(b) Classifying different stages of a child's language development. answer: [ANS]\n(c) The effectiveness of lecture teaching is tested with a sample of students who has completed numerous lecture style courses. answer: [ANS]", "answer_v3": [ "OBSERVATION", "OBSERVATION", "OBSERVATION" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "EXPERIMENT", "OBSERVATION" ], [ "EXPERIMENT", "OBSERVATION" ], [ "EXPERIMENT", "OBSERVATION" ] ] }, { "id": "Statistics_0001", "subject": "Statistics", "topic": "Experimental design", "subtopic": "Concepts", "level": "2", "keywords": [ "statistic", "producing data", "experiments" ], "problem_v1": "For each problem, select the best response.\n(a) A marketing class designs two videos advertising an expensive Mercedes sports car. They test the videos by asking fellow students to view both (in random order) and say which makes them more likely to buy the car. Mercedes should be reluctant to agree that the video favored in this study will sell more cars because [ANS] A. the study used matched pairs design instead of a completely randomized design. B. this is an observational study, not an experiment. C. results from students may not generalize to the older and richer customers who might buy a Mercedes. D. None of the above.\n(b) What electrical changes occur in muscles as they get tired? Student subjects hold their arms above their shoulders until they have to drop them. Meanwhile, the electrical activity in their arm muscles is measured. This is [ANS] A. an experiment with no control group. B. an observational study. C. a randomized comparative experiment. D. None of the above.\n(c) Can changing diet reduce high blood pressure? Vegetarian diets and low-salt diets are both promising. Men with high blood pressure are assigned at random to one of four diets: (1) normal diet with unrestricted salt; (2) vegetarian with unrestricted salt; (3) normal with restricted salt; and (4) vegetarian with restricted salt. This experiment has [ANS] A. four factors, the four diets being compared. B. two factors, normal/vegetarian diet and unrestricted/restricted salt. C. one factor, the choice of diet. D. None of the above.", "answer_v1": [ "C", "A", "B" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "For each problem, select the best response.\n(a) The Nurses' Health Study has interviewed a sample of more than 100,000 female registered nurses every two years since 1976. The study finds that \" light-to-moderate drinkers had a significantly lower risk of death\" than either nondrinkers or heavy drinkers. The Nursers' Health Study is [ANS] A. an observational study. B. an experiment. C. Can't tell without more information.\n(b) A marketing class designs two videos advertising an expensive Mercedes sports car. They test the videos by asking fellow students to view both (in random order) and say which makes them more likely to buy the car. Mercedes should be reluctant to agree that the video favored in this study will sell more cars because [ANS] A. this is an observational study, not an experiment. B. the study used matched pairs design instead of a completely randomized design. C. results from students may not generalize to the older and richer customers who might buy a Mercedes. D. None of the above.\n(c) What electrical changes occur in muscles as they get tired? Student subjects hold their arms above their shoulders until they have to drop them. Meanwhile, the electrical activity in their arm muscles is measured. This is [ANS] A. a randomized comparative experiment. B. an observational study. C. an experiment with no control group. D. None of the above.", "answer_v2": [ "A", "C", "C" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "For each problem, select the best response.\n(a) What electrical changes occur in muscles as they get tired? Student subjects hold their arms above their shoulders until they have to drop them. Meanwhile, the electrical activity in their arm muscles is measured. This is [ANS] A. a randomized comparative experiment. B. an experiment with no control group. C. an observational study. D. None of the above.\n(b) Can changing diet reduce high blood pressure? Vegetarian diets and low-salt diets are both promising. Men with high blood pressure are assigned at random to one of four diets: (1) normal diet with unrestricted salt; (2) vegetarian with unrestricted salt; (3) normal with restricted salt; and (4) vegetarian with restricted salt. This experiment has [ANS] A. one factor, the choice of diet. B. four factors, the four diets being compared. C. two factors, normal/vegetarian diet and unrestricted/restricted salt. D. None of the above.\n(c) The Nurses' Health Study has interviewed a sample of more than 100,000 female registered nurses every two years since 1976. The study finds that \" light-to-moderate drinkers had a significantly lower risk of death\" than either nondrinkers or heavy drinkers. The Nursers' Health Study is [ANS] A. an experiment. B. an observational study. C. Can't tell without more information.", "answer_v3": [ "B", "C", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C" ] ] }, { "id": "Statistics_0002", "subject": "Statistics", "topic": "Experimental design", "subtopic": "Concepts", "level": "2", "keywords": [ "statistic", "producing data", "experiments" ], "problem_v1": "Researchers wish to determine if a new experimental medication will reduce the symptoms of allergy sufferers without the side-effect of drowsiness. To investigate this question, the researchers give the new medication to 50 adult volunteers who suffer from allergies. Forty-four of these volunteers report a significant reduction in their allergy symptoms without any drowsiness.\n(a) This is an example of [ANS] A. the establishing of a causal relationship. B. a double blind experiment. C. a randomized study. D. a block design. E. None of the above.\n(b) The experimental units are [ANS] A. the six volunteers who did not report a significant reduction in their allergy symptoms without any drowsiness. B. the researchers. C. the 44 volunteers who reported a significant reduction in their allergy symptoms without any drowsiness. D. the 50 adult volunteers.\n(c) This study could be improved by [ANS] A. using a control group. B. including people who do not suffer from allergies in the study in order to represent a more diverse population. C. repeating the study with only the 44 volunteers who reported a significant reduction in their allergy symptoms without any drowsiness, and giving them a higher dosage this time. D. all of the above.", "answer_v1": [ "E", "D", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Researchers wish to determine if a new experimental medication will reduce the symptoms of allergy sufferers without the side-effect of drowsiness. To investigate this question, the researchers give the new medication to 50 adult volunteers who suffer from allergies. Forty-four of these volunteers report a significant reduction in their allergy symptoms without any drowsiness.\n(a) This study could be improved by [ANS] A. repeating the study with only the 44 volunteers who reported a significant reduction in their allergy symptoms without any drowsiness, and giving them a higher dosage this time. B. including people who do not suffer from allergies in the study in order to represent a more diverse population. C. using a control group. D. all of the above.\n(b) This is an example of [ANS] A. a block design. B. a randomized study. C. the establishing of a causal relationship. D. a double blind experiment. E. None of the above.\n(c) The experimental units are [ANS] A. the 50 adult volunteers. B. the 44 volunteers who reported a significant reduction in their allergy symptoms without any drowsiness. C. the six volunteers who did not report a significant reduction in their allergy symptoms without any drowsiness. D. the researchers.", "answer_v2": [ "C", "E", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Researchers wish to determine if a new experimental medication will reduce the symptoms of allergy sufferers without the side-effect of drowsiness. To investigate this question, the researchers give the new medication to 50 adult volunteers who suffer from allergies. Forty-four of these volunteers report a significant reduction in their allergy symptoms without any drowsiness.\n(a) This study could be improved by [ANS] A. using a control group. B. including people who do not suffer from allergies in the study in order to represent a more diverse population. C. repeating the study with only the 44 volunteers who reported a significant reduction in their allergy symptoms without any drowsiness, and giving them a higher dosage this time. D. all of the above.\n(b) This is an example of [ANS] A. a double blind experiment. B. a block design. C. a randomized study. D. the establishing of a causal relationship. E. None of the above.\n(c) The experimental units are [ANS] A. the 50 adult volunteers. B. the 44 volunteers who reported a significant reduction in their allergy symptoms without any drowsiness. C. the researchers. D. the six volunteers who did not report a significant reduction in their allergy symptoms without any drowsiness.", "answer_v3": [ "A", "E", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0003", "subject": "Statistics", "topic": "Experimental design", "subtopic": "Concepts", "level": "2", "keywords": [ "statistic", "producing data", "experiments" ], "problem_v1": "A market research company wishes to find out whether College of Idaho students prefer brand A or brand B of instant coffee. A random sample of students is selected, and each one is asked to try brand A first and then brand B (or vice versa, with the order determined at random). They then indicate which brand they prefer.\n(a) The response variable here is [ANS] A. which brand they prefer. B. whether brand A or brand B is tried first. C. the identity of the student. D. None of the above.\n(b) This is an example of [ANS] A. block design. B. an observational study, not an experiment. C. a matched-pairs experiment D. stratified sampling design.", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "A market research company wishes to find out whether College of Idaho students prefer brand A or brand B of instant coffee. A random sample of students is selected, and each one is asked to try brand A first and then brand B (or vice versa, with the order determined at random). They then indicate which brand they prefer.\n(a) This is an example of [ANS] A. block design. B. an observational study, not an experiment. C. stratified sampling design. D. a matched-pairs experiment\n(b) The response variable here is [ANS] A. the identity of the student. B. whether brand A or brand B is tried first. C. which brand they prefer. D. None of the above.", "answer_v2": [ "D", "C" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "A market research company wishes to find out whether College of Idaho students prefer brand A or brand B of instant coffee. A random sample of students is selected, and each one is asked to try brand A first and then brand B (or vice versa, with the order determined at random). They then indicate which brand they prefer.\n(a) This is an example of [ANS] A. block design. B. a matched-pairs experiment C. stratified sampling design. D. an observational study, not an experiment.\n(b) The response variable here is [ANS] A. which brand they prefer. B. whether brand A or brand B is tried first. C. the identity of the student. D. None of the above.", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0004", "subject": "Statistics", "topic": "Experimental design", "subtopic": "Concepts", "level": "2", "keywords": [ "statistic", "producing data", "experiments" ], "problem_v1": "A study of human development showed two types of movies to a group of children. Crackers were available in a bowl, and the investigators compared the number of crackers eaten by the children while watching the different kinds of movies. One kind was shown at 8 A.M. and another at 11 A.M. It was found that during the movie shown at 11 A.M., more crackers were eaten than during the movie shown at 8 A.M. The investigators concluded that the different types of movies had an effect on appetite.\n(a) This is an example of a [ANS] A. block design. B. simple random sample. C. matched pairs design. D. None of the above.\n(b) The response variable in this experiment is [ANS] A. the bowls. B. the different kinds of movies. C. the time the movie was shown. D. the number of crackers eaten.\n(c) A lurking variable in this experiment is [ANS] A. the time the movie was shown. B. the bowls. C. the different kinds of movies. D. the number of crackers eaten.", "answer_v1": [ "C", "D", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "A study of human development showed two types of movies to a group of children. Crackers were available in a bowl, and the investigators compared the number of crackers eaten by the children while watching the different kinds of movies. One kind was shown at 8 A.M. and another at 11 A.M. It was found that during the movie shown at 11 A.M., more crackers were eaten than during the movie shown at 8 A.M. The investigators concluded that the different types of movies had an effect on appetite.\n(a) A lurking variable in this experiment is [ANS] A. the different kinds of movies. B. the bowls. C. the number of crackers eaten. D. the time the movie was shown.\n(b) This is an example of a [ANS] A. matched pairs design. B. simple random sample. C. block design. D. None of the above.\n(c) The response variable in this experiment is [ANS] A. the number of crackers eaten. B. the time the movie was shown. C. the bowls. D. the different kinds of movies.", "answer_v2": [ "D", "A", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "A study of human development showed two types of movies to a group of children. Crackers were available in a bowl, and the investigators compared the number of crackers eaten by the children while watching the different kinds of movies. One kind was shown at 8 A.M. and another at 11 A.M. It was found that during the movie shown at 11 A.M., more crackers were eaten than during the movie shown at 8 A.M. The investigators concluded that the different types of movies had an effect on appetite.\n(a) A lurking variable in this experiment is [ANS] A. the time the movie was shown. B. the number of crackers eaten. C. the different kinds of movies. D. the bowls.\n(b) This is an example of a [ANS] A. block design. B. matched pairs design. C. simple random sample. D. None of the above.\n(c) The response variable in this experiment is [ANS] A. the number of crackers eaten. B. the time the movie was shown. C. the different kinds of movies. D. the bowls.", "answer_v3": [ "A", "B", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0005", "subject": "Statistics", "topic": "Experimental design", "subtopic": "Concepts", "level": "2", "keywords": [ "statistic", "producing data", "experiments" ], "problem_v1": "One hundred volunteers who suffer from severe depression are available for a study. Fifty are selected at random and are given a new drug that is thought to be particularly effective in treating severe depression. The other fifty are given an existing drug for treating severe depression. A psychiatrist evaluates the symptoms of all volunteers after four weeks in order to determine if there has been substantial improvement in the severity of the depression.\n(a) Suppose the volunteers were first divided into men and women, and then half of the men were randomly assigned to the new drug and half of the women were assigned to the new drug. The remaining volunteers received the other drug. This would be an example of [ANS] A. a matched pairs design. B. confounding. The effects of gender will be mixed up with the effects of the drugs. C. replication. D. a block design.\n(b) The study would be double blind if [ANS] A. neither the volunteers nor the psychiatrist knew which treatment any person had received. B. neither drug had any identifying marks on it. C. all volunteers were not allowed to see the psychiatrist nor the psychiatrist allowed to see the volunteers during the session in which the psychiatrist evaluated the severity of the depression. D. All of the above.\n(c) The factor in this study is [ANS] A. which treatment the volunteers receive. B. the use of a psychiatrist to evaluate the severity of depression. C. the use of randomization and the fact that this was a comparative study. D. the extent to which the depression was reduced.", "answer_v1": [ "D", "A", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "One hundred volunteers who suffer from severe depression are available for a study. Fifty are selected at random and are given a new drug that is thought to be particularly effective in treating severe depression. The other fifty are given an existing drug for treating severe depression. A psychiatrist evaluates the symptoms of all volunteers after four weeks in order to determine if there has been substantial improvement in the severity of the depression.\n(a) The factor in this study is [ANS] A. the use of randomization and the fact that this was a comparative study. B. the use of a psychiatrist to evaluate the severity of depression. C. the extent to which the depression was reduced. D. which treatment the volunteers receive.\n(b) Suppose the volunteers were first divided into men and women, and then half of the men were randomly assigned to the new drug and half of the women were assigned to the new drug. The remaining volunteers received the other drug. This would be an example of [ANS] A. a matched pairs design. B. a block design. C. replication. D. confounding. The effects of gender will be mixed up with the effects of the drugs.\n(c) The study would be double blind if [ANS] A. all volunteers were not allowed to see the psychiatrist nor the psychiatrist allowed to see the volunteers during the session in which the psychiatrist evaluated the severity of the depression. B. neither drug had any identifying marks on it. C. neither the volunteers nor the psychiatrist knew which treatment any person had received. D. All of the above.", "answer_v2": [ "D", "B", "C" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "One hundred volunteers who suffer from severe depression are available for a study. Fifty are selected at random and are given a new drug that is thought to be particularly effective in treating severe depression. The other fifty are given an existing drug for treating severe depression. A psychiatrist evaluates the symptoms of all volunteers after four weeks in order to determine if there has been substantial improvement in the severity of the depression.\n(a) The factor in this study is [ANS] A. which treatment the volunteers receive. B. the extent to which the depression was reduced. C. the use of randomization and the fact that this was a comparative study. D. the use of a psychiatrist to evaluate the severity of depression.\n(b) Suppose the volunteers were first divided into men and women, and then half of the men were randomly assigned to the new drug and half of the women were assigned to the new drug. The remaining volunteers received the other drug. This would be an example of [ANS] A. confounding. The effects of gender will be mixed up with the effects of the drugs. B. a block design. C. replication. D. a matched pairs design.\n(c) The study would be double blind if [ANS] A. all volunteers were not allowed to see the psychiatrist nor the psychiatrist allowed to see the volunteers during the session in which the psychiatrist evaluated the severity of the depression. B. neither the volunteers nor the psychiatrist knew which treatment any person had received. C. neither drug had any identifying marks on it. D. All of the above.", "answer_v3": [ "A", "B", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0006", "subject": "Statistics", "topic": "Experimental design", "subtopic": "Concepts", "level": "2", "keywords": [ "statistic", "producing data", "experiments" ], "problem_v1": "For each problem, select the best response.\n(a) Can pleasant aromas help a student learn better? Two researchers believed that the presence of a floral scent could improve a person's learning ability in certain situations. They had twenty-two people work through a pencil and paper maze six times, three times while wearing a floral-scented mask and three times while wearing an unscented mask. The three trials for each mask closely followed one another. Testers measured the length of time it took the subjects to complete each of the six trials. They reported that, on average, the subjects wearing the floral-scented mask completed the maze more quickly than those wearing the unscented mask, though the difference was not statistically significant. This study is [ANS] A. an experiment, but not a double-blind experiment. B. a double-blind experiment. C. an observational study, but not an experiment. D. a convenience sample.\n(b) A study of the effects of running on personality involved 231 male runners who each ran about 20 miles a week. The runners were given the Cattell Sixteen Personality Factors Questionnaire, a 187-item multiple-choice test often used by psychologists. A news report (The New York Times, Feb. 15, 1988) stated, The researchers found statistically significant personality differences between the runners and the 30-year-old male population as a whole. A headline on the article said, Research has shown that running can alter one's moods. This study was [ANS] A. a randomized, double-blind experiment. B. an experiment, but not a double-blind experiment. C. an observational study, but not an experiment. D. a double-blind experiment, but not a randomized experiment.", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "For each problem, select the best response.\n(a) A study of the effects of running on personality involved 231 male runners who each ran about 20 miles a week. The runners were given the Cattell Sixteen Personality Factors Questionnaire, a 187-item multiple-choice test often used by psychologists. A news report (The New York Times, Feb. 15, 1988) stated, The researchers found statistically significant personality differences between the runners and the 30-year-old male population as a whole. A headline on the article said, Research has shown that running can alter one's moods. This study was [ANS] A. a randomized, double-blind experiment. B. an experiment, but not a double-blind experiment. C. a double-blind experiment, but not a randomized experiment. D. an observational study, but not an experiment.\n(b) Can pleasant aromas help a student learn better? Two researchers believed that the presence of a floral scent could improve a person's learning ability in certain situations. They had twenty-two people work through a pencil and paper maze six times, three times while wearing a floral-scented mask and three times while wearing an unscented mask. The three trials for each mask closely followed one another. Testers measured the length of time it took the subjects to complete each of the six trials. They reported that, on average, the subjects wearing the floral-scented mask completed the maze more quickly than those wearing the unscented mask, though the difference was not statistically significant. This study is [ANS] A. an observational study, but not an experiment. B. a double-blind experiment. C. a convenience sample. D. an experiment, but not a double-blind experiment.", "answer_v2": [ "D", "D" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "For each problem, select the best response.\n(a) A study of the effects of running on personality involved 231 male runners who each ran about 20 miles a week. The runners were given the Cattell Sixteen Personality Factors Questionnaire, a 187-item multiple-choice test often used by psychologists. A news report (The New York Times, Feb. 15, 1988) stated, The researchers found statistically significant personality differences between the runners and the 30-year-old male population as a whole. A headline on the article said, Research has shown that running can alter one's moods. This study was [ANS] A. a randomized, double-blind experiment. B. an observational study, but not an experiment. C. a double-blind experiment, but not a randomized experiment. D. an experiment, but not a double-blind experiment.\n(b) Can pleasant aromas help a student learn better? Two researchers believed that the presence of a floral scent could improve a person's learning ability in certain situations. They had twenty-two people work through a pencil and paper maze six times, three times while wearing a floral-scented mask and three times while wearing an unscented mask. The three trials for each mask closely followed one another. Testers measured the length of time it took the subjects to complete each of the six trials. They reported that, on average, the subjects wearing the floral-scented mask completed the maze more quickly than those wearing the unscented mask, though the difference was not statistically significant. This study is [ANS] A. an experiment, but not a double-blind experiment. B. a convenience sample. C. an observational study, but not an experiment. D. a double-blind experiment.", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0007", "subject": "Statistics", "topic": "Experimental design", "subtopic": "Concepts", "level": "2", "keywords": [ "statistic", "producing data", "experiments" ], "problem_v1": "A marketing experiment compares four different types of packaging for blank computer CDs. Each type of packaging can be presented in three different colors. Each combination of package type with a particular color is shown to 45 potential customers, who rate the overall attractiveness on a scale of 1 to 5.\n(a) The factors are [ANS] A. the three different colors. B. type of packaging and color. C. the rating scale and package combination. D. the potential customers.\n(b) The experimental units in this experiment are [ANS] A. the measure of attractiveness B. type of packaging and color. C. the three different colors. D. the potential customers.", "answer_v1": [ "B", "D" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "A marketing experiment compares five different types of packaging for blank computer CDs. Each type of packaging can be presented in four different colors. Each combination of package type with a particular color is shown to 40 potential customers, who rate the overall attractiveness on a scale of 1 to 8.\n(a) The experimental units in this experiment are [ANS] A. the measure of attractiveness B. the potential customers. C. the four different colors. D. type of packaging and color.\n(b) The factors are [ANS] A. the rating scale and package combination. B. type of packaging and color. C. the four different colors. D. the potential customers.", "answer_v2": [ "B", "B" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "A marketing experiment compares five different types of packaging for blank computer CDs. Each type of packaging can be presented in four different colors. Each combination of package type with a particular color is shown to 45 potential customers, who rate the overall attractiveness on a scale of 1 to 5.\n(a) The experimental units in this experiment are [ANS] A. type of packaging and color. B. the potential customers. C. the four different colors. D. the measure of attractiveness\n(b) The factors are [ANS] A. the rating scale and package combination. B. the four different colors. C. type of packaging and color. D. the potential customers.", "answer_v3": [ "B", "C" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0008", "subject": "Statistics", "topic": "Experimental design", "subtopic": "Concepts", "level": "2", "keywords": [ "statistic", "producing data", "experiments" ], "problem_v1": "A group of college students believes that herbal tea has remarkable restorative powers. To test their theory, they make weekly visits to a local nursing home, visiting with residents, talking with them, and serving them herbal tea. After several months, many of the residents are more cheerful and healthy.\n(a) Which of the following may be correctly concluded from this study? [ANS] A. The results of the study are not convincing because only a local nursing home was used and only for a few months. B. There is some evidence that herbal tea may improve one's emotional state. The results would be completely convincing if a scientist had conducted the study rather than a group of college students. C. Herbal tea does improve one's emotional state, at least for the residents of nursing homes. D. The results of the study are not convincing because the effect of herbal tea is confounded with several other factors.\n(b) The lurking variable in this experiment is [ANS] A. the emotional state of the residents. B. the fact that this is a local nursing home. C. herbal tea. D. visits of the college students.\n(c) The explanatory variable in this experiment is [ANS] A. herbal tea. B. visits of the college students. C. the fact that this is a local nursing home. D. the emotional state of the residents.", "answer_v1": [ "D", "D", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "A group of college students believes that herbal tea has remarkable restorative powers. To test their theory, they make weekly visits to a local nursing home, visiting with residents, talking with them, and serving them herbal tea. After several months, many of the residents are more cheerful and healthy.\n(a) The explanatory variable in this experiment is [ANS] A. the fact that this is a local nursing home. B. visits of the college students. C. the emotional state of the residents. D. herbal tea.\n(b) Which of the following may be correctly concluded from this study? [ANS] A. The results of the study are not convincing because only a local nursing home was used and only for a few months. B. The results of the study are not convincing because the effect of herbal tea is confounded with several other factors. C. Herbal tea does improve one's emotional state, at least for the residents of nursing homes. D. There is some evidence that herbal tea may improve one's emotional state. The results would be completely convincing if a scientist had conducted the study rather than a group of college students.\n(c) The lurking variable in this experiment is [ANS] A. visits of the college students. B. herbal tea. C. the emotional state of the residents. D. the fact that this is a local nursing home.", "answer_v2": [ "D", "B", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "A group of college students believes that herbal tea has remarkable restorative powers. To test their theory, they make weekly visits to a local nursing home, visiting with residents, talking with them, and serving them herbal tea. After several months, many of the residents are more cheerful and healthy.\n(a) The explanatory variable in this experiment is [ANS] A. herbal tea. B. the emotional state of the residents. C. the fact that this is a local nursing home. D. visits of the college students.\n(b) Which of the following may be correctly concluded from this study? [ANS] A. There is some evidence that herbal tea may improve one's emotional state. The results would be completely convincing if a scientist had conducted the study rather than a group of college students. B. The results of the study are not convincing because the effect of herbal tea is confounded with several other factors. C. Herbal tea does improve one's emotional state, at least for the residents of nursing homes. D. The results of the study are not convincing because only a local nursing home was used and only for a few months.\n(c) The lurking variable in this experiment is [ANS] A. visits of the college students. B. herbal tea. C. the fact that this is a local nursing home. D. the emotional state of the residents.", "answer_v3": [ "A", "B", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0009", "subject": "Statistics", "topic": "Experimental design", "subtopic": "Concepts", "level": "3", "keywords": [ "Study", "Experiment", "Observation" ], "problem_v1": "Determine whether the follow descriptions correspond to an observational study (OS), a controlled experiment (CE), or a double blind experiment (DBE). Write the abbreviations above as your answers. (without the parenthesis)\n(a) A group of 500 adult males entered a study in which half would be fed a high fiber diet while the other half would be fed a minimal fiber diet in order to determine the effect of fiber on cholesterol. answer: [ANS]\n(b) A therapist compiles information on his/her patients who have survived child abuse to explain the effects of child abuse on adult survivors. answer: [ANS]\n(c) A new study examines the effects of regular exercise on child obesity by taking a survey of 500 children on their weight and exercise habits. answer: [ANS]", "answer_v1": [ "CE", "OS", "OS" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "CE", "OS", "DBE" ], [ "CE", "OS", "DBE" ], [ "CE", "OS", "DBE" ] ], "problem_v2": "Determine whether the follow descriptions correspond to an observational study (OS), a controlled experiment (CE), or a double blind experiment (DBE). Write the abbreviations above as your answers. (without the parenthesis)\n(a) A nurse gives a group of patients either the drug being studied or a placebo; however neither the nurse nor the patient knows which they are receiving. Only the person who prepared the doses knows. answer: [ANS]\n(b) The effects of fish oil on inflammation were studied by administering either a dose of 2000 mcg of fish oil each day or a placebo each day to two different groups of arthritis patients. answer: [ANS]\n(c) A doctor gives half of his/her patients in a study a treatment and the other half a placebo, but the patients do not know which they received. answer: [ANS]", "answer_v2": [ "DBE", "CE", "CE" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "CE", "OS", "DBE" ], [ "CE", "OS", "DBE" ], [ "CE", "OS", "DBE" ] ], "problem_v3": "Determine whether the follow descriptions correspond to an observational study (OS), a controlled experiment (CE), or a double blind experiment (DBE). Write the abbreviations above as your answers. (without the parenthesis)\n(a) A doctor gives half of his/her patients in a study a treatment and the other half a placebo, but the patients do not know which they received. answer: [ANS]\n(b) A group of 500 adult males entered a study in which half would be fed a high fiber diet while the other half would be fed a minimal fiber diet in order to determine the effect of fiber on cholesterol. answer: [ANS]\n(c) A therapist compiles information on his/her patients who have survived child abuse to explain the effects of child abuse on adult survivors. answer: [ANS]", "answer_v3": [ "CE", "CE", "OS" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "CE", "OS", "DBE" ], [ "CE", "OS", "DBE" ], [ "CE", "OS", "DBE" ] ] }, { "id": "Statistics_0010", "subject": "Statistics", "topic": "Experimental design", "subtopic": "Concepts", "level": "4", "keywords": [ "Experimental design", "factors", "treatments" ], "problem_v1": "A study investigated the effect of the length and the repetition of TV advertisements on students' desire to eat at a Sub-U-Like sandwich franchise. Sixty students watched a 50-minute television program that showed at least one commercial for Sub-U-Like during advertisement breaks. Some students saw a 30-second commerical, others a 90-second commerical. The same commerical was shown one, three, or five times during the program. After the viewing, each student was asked to rate their craving for a Sub-U-Like sandwich on a scale of 0 to 10.\n(a) What kind of study is this? [ANS] A. An experiment because the study investigator controlled the amount of exposure to advertisements the participating students received. B. An experiment because the study investigator compared the degree of craving between different amounts of exposure to advertisements in the study. C. An observational study because the study investigator observed the students\u2019 ratings on their craving for the Sub-U-Like sandwich. D. An observational study because there was no control group.\n(b) What are the subjects in the \"Sub-U-Like\" study? [ANS] A. The 60 students B. The effect of the length and repetition of the TV commercials C. One, three, or five commercials during the 50-minute television program D. The 50-minute television program E. The craving for a Sub-U-Like sandwich on a scale of 0 to 10\n(c) What is/are the response variables in the \"Sub-U-Like\" study? [ANS] A. The 60 students B. One, three, or five commercials during the 50-minute television program C. Craving for Sub-U-Like on a scale of 0 to 10 D. The length and repetition of the TV advertisements E. The 50-minute television program\n(d) How many treatments are there in the \"Sub-U-Like\" study? Enter your answer as a number (e.g. e.g. 1), not as text. Answer: [ANS]", "answer_v1": [ "A", "A", "C", "6" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "NV" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [] ], "problem_v2": "A study investigated the effect of the length and the repetition of TV advertisements on students' desire to eat at a Sub-U-Like sandwich franchise. Sixty students watched a 50-minute television program that showed at least one commercial for Sub-U-Like during advertisement breaks. Some students saw a 30-second commerical, others a 90-second commerical. The same commerical was shown one, three, or five times during the program. After the viewing, each student was asked to rate their craving for a Sub-U-Like sandwich on a scale of 0 to 10.\n(a) What kind of study is this? [ANS] A. An experiment because the study investigator controlled the amount of exposure to advertisements the participating students received. B. An experiment because the study investigator compared the degree of craving between different amounts of exposure to advertisements in the study. C. An observational study because the study investigator observed the students\u2019 ratings on their craving for the Sub-U-Like sandwich. D. An observational study because there was no control group.\n(b) What are the subjects in the \"Sub-U-Like\" study? [ANS] A. The effect of the length and repetition of the TV commercials B. One, three, or five commercials during the 50-minute television program C. The 50-minute television program D. The 60 students E. The craving for a Sub-U-Like sandwich on a scale of 0 to 10\n(c) What is/are the response variables in the \"Sub-U-Like\" study? [ANS] A. One, three, or five commercials during the 50-minute television program B. The 50-minute television program C. The length and repetition of the TV advertisements D. Craving for Sub-U-Like on a scale of 0 to 10 E. The 60 students\n(d) How many treatments are there in the \"Sub-U-Like\" study? Enter your answer as a number (e.g. e.g. 1), not as text. Answer: [ANS]", "answer_v2": [ "A", "D", "D", "6" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "NV" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [] ], "problem_v3": "A study investigated the effect of the length and the repetition of TV advertisements on students' desire to eat at a Sub-U-Like sandwich franchise. Sixty students watched a 50-minute television program that showed at least one commercial for Sub-U-Like during advertisement breaks. Some students saw a 30-second commerical, others a 90-second commerical. The same commerical was shown one, three, or five times during the program. After the viewing, each student was asked to rate their craving for a Sub-U-Like sandwich on a scale of 0 to 10.\n(a) What kind of study is this? [ANS] A. An experiment because the study investigator controlled the amount of exposure to advertisements the participating students received. B. An observational study because the study investigator observed the students\u2019 ratings on their craving for the Sub-U-Like sandwich. C. An experiment because the study investigator compared the degree of craving between different amounts of exposure to advertisements in the study. D. An observational study because there was no control group.\n(b) What are the subjects in the \"Sub-U-Like\" study? [ANS] A. The craving for a Sub-U-Like sandwich on a scale of 0 to 10 B. The effect of the length and repetition of the TV commercials C. The 50-minute television program D. One, three, or five commercials during the 50-minute television program E. The 60 students\n(c) What is/are the response variables in the \"Sub-U-Like\" study? [ANS] A. One, three, or five commercials during the 50-minute television program B. Craving for Sub-U-Like on a scale of 0 to 10 C. The 60 students D. The length and repetition of the TV advertisements E. The 50-minute television program\n(d) How many treatments are there in the \"Sub-U-Like\" study? Enter your answer as a number (e.g. e.g. 1), not as text. Answer: [ANS]", "answer_v3": [ "A", "E", "B", "6" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "NV" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [] ] }, { "id": "Statistics_0011", "subject": "Statistics", "topic": "Experimental design", "subtopic": "Concepts", "level": "3", "keywords": [], "problem_v1": "A small-scale experiment was carried out to determine operating conditions for leaching uranium from ore. Three processing factors---acid concentration ($x_1$), grind size ($x_2$), and processing time ($x_3$)---were varied. The response of interest was the percentage uranium recovery ($y$). Eight different combinations of $x_1$, $x_2$, and $x_3$ values were applied to eight samples of ore, giving the following data. \\begin{array}{rrrc} \\hline Conc & Grind & Time & \\% U recovery \\\\ (x_1) & (x_2) & (x_3) & (y) \\\\ \\hline-1 &-1 &-1 & \\\\-1 &-1 & 1 & \\\\-1 & 1 &-1 & \\\\-1 & 1 & 1 & \\\\ 1 &-1 &-1 & \\\\ 1 &-1 & 1 & \\\\ 1 & 1 &-1 & \\\\ 1 & 1 & 1 & \\\\ \\hline \\end{array} \\begin{array}{rrrc} \\hline Conc & Grind & Time & \\% U recovery \\\\ (x_1) & (x_2) & (x_3) & (y) \\\\ \\hline-1 &-1 &-1 & \\\\-1 &-1 & 1 & \\\\-1 & 1 &-1 & \\\\-1 & 1 & 1 & \\\\ 1 &-1 &-1 & \\\\ 1 &-1 & 1 & \\\\ 1 & 1 &-1 & \\\\ 1 & 1 & 1 & \\\\ \\hline \\end{array} In this table,-1 and 1 code ''low'' and ''high'' levels for each explanatory variable.\n[A] What is an experimental unit in this experiment? [ANS] A. an operating condition B. percentage uranium recovery C. a small-scale experiment D. a processing factor like $x_1$ E. a sample of ore\n[E] What is the response variable in this experiment? [ANS] A. percentage uranium recovery B. a sample of ore C. a level of $x_1$ D. the levels of $x_1$, $x_2$, and $x_3$ defining an operating condition E. a small-scale experiment", "answer_v1": [ "E", "A" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "A small-scale experiment was carried out to determine operating conditions for leaching uranium from ore. Three processing factors---acid concentration ($x_1$), grind size ($x_2$), and processing time ($x_3$)---were varied. The response of interest was the percentage uranium recovery ($y$). Eight different combinations of $x_1$, $x_2$, and $x_3$ values were applied to eight samples of ore, giving the following data. \\begin{array}{rrrc} \\hline Conc & Grind & Time & \\% U recovery \\\\ (x_1) & (x_2) & (x_3) & (y) \\\\ \\hline-1 &-1 &-1 & \\\\-1 &-1 & 1 & \\\\-1 & 1 &-1 & \\\\-1 & 1 & 1 & \\\\ 1 &-1 &-1 & \\\\ 1 &-1 & 1 & \\\\ 1 & 1 &-1 & \\\\ 1 & 1 & 1 & \\\\ \\hline \\end{array} \\begin{array}{rrrc} \\hline Conc & Grind & Time & \\% U recovery \\\\ (x_1) & (x_2) & (x_3) & (y) \\\\ \\hline-1 &-1 &-1 & \\\\-1 &-1 & 1 & \\\\-1 & 1 &-1 & \\\\-1 & 1 & 1 & \\\\ 1 &-1 &-1 & \\\\ 1 &-1 & 1 & \\\\ 1 & 1 &-1 & \\\\ 1 & 1 & 1 & \\\\ \\hline \\end{array} In this table,-1 and 1 code ''low'' and ''high'' levels for each explanatory variable.\n[A] What is an experimental unit in this experiment? [ANS] A. a processing factor like $x_1$ B. a small-scale experiment C. a sample of ore D. an operating condition E. percentage uranium recovery\n[E] What is the response variable in this experiment? [ANS] A. a sample of ore B. a level of $x_1$ C. the levels of $x_1$, $x_2$, and $x_3$ defining an operating condition D. percentage uranium recovery E. a small-scale experiment", "answer_v2": [ "C", "D" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "A small-scale experiment was carried out to determine operating conditions for leaching uranium from ore. Three processing factors---acid concentration ($x_1$), grind size ($x_2$), and processing time ($x_3$)---were varied. The response of interest was the percentage uranium recovery ($y$). Eight different combinations of $x_1$, $x_2$, and $x_3$ values were applied to eight samples of ore, giving the following data. \\begin{array}{rrrc} \\hline Conc & Grind & Time & \\% U recovery \\\\ (x_1) & (x_2) & (x_3) & (y) \\\\ \\hline-1 &-1 &-1 & \\\\-1 &-1 & 1 & \\\\-1 & 1 &-1 & \\\\-1 & 1 & 1 & \\\\ 1 &-1 &-1 & \\\\ 1 &-1 & 1 & \\\\ 1 & 1 &-1 & \\\\ 1 & 1 & 1 & \\\\ \\hline \\end{array} \\begin{array}{rrrc} \\hline Conc & Grind & Time & \\% U recovery \\\\ (x_1) & (x_2) & (x_3) & (y) \\\\ \\hline-1 &-1 &-1 & \\\\-1 &-1 & 1 & \\\\-1 & 1 &-1 & \\\\-1 & 1 & 1 & \\\\ 1 &-1 &-1 & \\\\ 1 &-1 & 1 & \\\\ 1 & 1 &-1 & \\\\ 1 & 1 & 1 & \\\\ \\hline \\end{array} In this table,-1 and 1 code ''low'' and ''high'' levels for each explanatory variable.\n[A] What is an experimental unit in this experiment? [ANS] A. percentage uranium recovery B. an operating condition C. a sample of ore D. a small-scale experiment E. a processing factor like $x_1$\n[E] What is the response variable in this experiment? [ANS] A. a small-scale experiment B. a sample of ore C. the levels of $x_1$, $x_2$, and $x_3$ defining an operating condition D. a level of $x_1$ E. percentage uranium recovery", "answer_v3": [ "C", "E" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Statistics_0012", "subject": "Statistics", "topic": "Experimental design", "subtopic": "Concepts", "level": "4", "keywords": [ "statistics", "fractional factorial" ], "problem_v1": "The $2^{5-2}$ design with 5 factors A, B, C, D, E is based on 2 generators that can be equivalently written as I=ABC=ADE. Deduce the full defining relation before procedure to the questions below. Part a) Suppose we use the 3 columns A,B,D to generate the 8 treatment combinations. (a1) How is the column C expressed in terms of A,B,D? [ANS]\n(a2) How is the column E expressed in terms of A,B,D? [ANS]\nPart b) Suppose we use the 3 columns C,D,E to generate the 8 treatment combinations. (b1) How is the column A expressed in terms of C,D,E? [ANS]\n(b2) How is the column B expressed in terms of C,D,E? [ANS]\nPart c) What are the treatment combinations in your quarter fraction? (The notation abc, for example, indicates that factors A, B, and C are at their+1 levels. There is no d or e in abc, indicating that factors D and E are at their-1 levels.) [ANS] A. e ac bce ab d acde bcd abde B. e cde bd bc a ac abde abcde C. c ae b abce cde ad bde abcd D. ce a be abc cd ade bd abcde E. None of the above\nPart d) Which of the following 2-way interactions are not aliased to a main effect? Choose all that apply [ANS] A. CD B. DE C. BD D. BC E. AD F. None of the above\nPart e) There are 7 estimable quantities: 5 main effects and 10 2-way interactions. Which main effect(s) is/are aliased to two 2-way interactions? Choose all that apply [ANS] A. E B. A C. C D. A,B,C E. A,D,E F. D G. B H. None of the above", "answer_v1": [ "AB", "AD", "DE", "CDE", "D", "AC", "B" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "MCS", "MCM", "MCS" ], "options_v1": [ [], [], [], [], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ] ], "problem_v2": "The $2^{5-2}$ design with 5 factors A, B, C, D, E is based on 2 generators that can be equivalently written as I=-ABC=-ADE. Deduce the full defining relation before procedure to the questions below. Part a) Suppose we use the 3 columns A,B,D to generate the 8 treatment combinations. (a1) How is the column C expressed in terms of A,B,D? [ANS]\n(a2) How is the column E expressed in terms of A,B,D? [ANS]\nPart b) Suppose we use the 3 columns C,D,E to generate the 8 treatment combinations. (b1) How is the column A expressed in terms of C,D,E? [ANS]\n(b2) How is the column B expressed in terms of C,D,E? [ANS]\nPart c) What are the treatment combinations in your quarter fraction? (The notation abc, for example, indicates that factors A, B, and C are at their+1 levels. There is no d or e in abc, indicating that factors D and E are at their-1 levels.) [ANS] A. e ac bce ab d acde bcd abde B. e cde bd bc a ac abde abcde C. c ae b abce cde ad bde abcd D. (1) ace bc abe de acd bcde abd E. ce a be abc cd ade bd abcde F. None of the above\nPart d) Which of the following 2-way interactions are not aliased to a main effect? Choose all that apply [ANS] A. CD B. BC C. AD D. BD E. DE F. None of the above\nPart e) There are 7 estimable quantities: 5 main effects and 10 2-way interactions. Which main effect(s) is/are aliased to two 2-way interactions? Choose all that apply [ANS] A. A,D,E B. D C. C D. E E. A F. A,B,C G. B H. None of the above", "answer_v2": [ "-AB", "-AD", "-DE", "CDE", "D", "AD", "E" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "MCS", "MCM", "MCS" ], "options_v2": [ [], [], [], [], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ] ], "problem_v3": "The $2^{5-2}$ design with 5 factors A, B, C, D, E is based on 2 generators that can be equivalently written as I=-ABC=ADE. Deduce the full defining relation before procedure to the questions below. Part a) Suppose we use the 3 columns A,B,D to generate the 8 treatment combinations. (a1) How is the column C expressed in terms of A,B,D? [ANS]\n(a2) How is the column E expressed in terms of A,B,D? [ANS]\nPart b) Suppose we use the 3 columns C,D,E to generate the 8 treatment combinations. (b1) How is the column A expressed in terms of C,D,E? [ANS]\n(b2) How is the column B expressed in terms of C,D,E? [ANS]\nPart c) What are the treatment combinations in your quarter fraction? (The notation abc, for example, indicates that factors A, B, and C are at their+1 levels. There is no d or e in abc, indicating that factors D and E are at their-1 levels.) [ANS] A. ce a be abc cd ade bd abcde B. c ae b abce cde ad bde abcd C. e cde bd bc a ac abde abcde D. e ac bce ab d acde bcd abde E. None of the above\nPart d) Which of the following 2-way interactions are not aliased to a main effect? Choose all that apply [ANS] A. CD B. DE C. BD D. AD E. BC F. None of the above\nPart e) There are 7 estimable quantities: 5 main effects and 10 2-way interactions. Which main effect(s) is/are aliased to two 2-way interactions? Choose all that apply [ANS] A. C B. E C. A,B,C D. A E. A,D,E F. D G. B H. None of the above", "answer_v3": [ "-AB", "AD", "DE", "-CDE", "D", "AC", "D" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "MCS", "MCM", "MCS" ], "options_v3": [ [], [], [], [], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ] ] }, { "id": "Statistics_0013", "subject": "Statistics", "topic": "Sample survey methods", "subtopic": "Concepts", "level": "3", "keywords": [ "statistic", "producing data", "sample" ], "problem_v1": "A television station is interested in predicting whether or not voters are in favor of an increase in the state sales tax. It asks its viewers to phone in and indicate whether they support or are opposed to an increase in the state sales tax in order to generate additional revenue for education. Of the 2633 viewers who phone in, 1474 (55.98\\%) are opposed to the increase.\n(a) In this case, the sample obtained is [ANS] A. a stratified random sample. B. a probability sample in which each person in the population has the same chance of being in the sample. C. a simple random sample. D. probably biased.\n(b) The sample is [ANS] A. all people who will vote on the sales tax increase on the data of the vote. B. the 1474 viewers were opposed to the increase. C. all regular viewers of the television station who own a phone and have participated in similar phone surveys in the past. D. the 2633 viewers who phoned in.\n(c) The population of interest is [ANS] A. all people who will vote on the sales tax increase on the date of the vote. B. the 1474 viewers who were opposed. C. all regular viewers of the television station who own a phone and have participated in similar phone surveys in the past. D. the 2633 viewers who phoned in.", "answer_v1": [ "D", "D", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "A television station is interested in predicting whether or not voters are in favor of an increase in the state sales tax. It asks its viewers to phone in and indicate whether they support or are opposed to an increase in the state sales tax in order to generate additional revenue for education. Of the 2633 viewers who phone in, 1474 (55.98\\%) are opposed to the increase.\n(a) The population of interest is [ANS] A. all regular viewers of the television station who own a phone and have participated in similar phone surveys in the past. B. the 1474 viewers who were opposed. C. the 2633 viewers who phoned in. D. all people who will vote on the sales tax increase on the date of the vote.\n(b) In this case, the sample obtained is [ANS] A. a stratified random sample. B. probably biased. C. a simple random sample. D. a probability sample in which each person in the population has the same chance of being in the sample.\n(c) The sample is [ANS] A. the 2633 viewers who phoned in. B. all regular viewers of the television station who own a phone and have participated in similar phone surveys in the past. C. all people who will vote on the sales tax increase on the data of the vote. D. the 1474 viewers were opposed to the increase.", "answer_v2": [ "D", "B", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "A television station is interested in predicting whether or not voters are in favor of an increase in the state sales tax. It asks its viewers to phone in and indicate whether they support or are opposed to an increase in the state sales tax in order to generate additional revenue for education. Of the 2633 viewers who phone in, 1474 (55.98\\%) are opposed to the increase.\n(a) The population of interest is [ANS] A. all people who will vote on the sales tax increase on the date of the vote. B. the 2633 viewers who phoned in. C. all regular viewers of the television station who own a phone and have participated in similar phone surveys in the past. D. the 1474 viewers who were opposed.\n(b) In this case, the sample obtained is [ANS] A. a probability sample in which each person in the population has the same chance of being in the sample. B. probably biased. C. a simple random sample. D. a stratified random sample.\n(c) The sample is [ANS] A. the 2633 viewers who phoned in. B. all regular viewers of the television station who own a phone and have participated in similar phone surveys in the past. C. the 1474 viewers were opposed to the increase. D. all people who will vote on the sales tax increase on the data of the vote.", "answer_v3": [ "A", "B", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0014", "subject": "Statistics", "topic": "Sample survey methods", "subtopic": "Concepts", "level": "3", "keywords": [ "statistic", "producing data", "sample" ], "problem_v1": "A Senator wants to know what the voters of his state think of proposed legislation on gun control. He mails a questionnaire on the subject to an SRS of 2500 voters in his state. His staff reports that 448 questionnaires have been returned, 343 of which support the legislation.\n(a) This is an example of [ANS] A. a survey with little bias because it was the voters who elected the senator. B. a survey containing nonresponse. C. a survey with little bias because a large SRS was used. D. all of the above.\n(b) The sample is [ANS] A. the 2500 voters receiving the questionnaire B. the voters in his state. C. the 343 letters supporting the legislation. D. the 448 letters received.\n(c) The population is [ANS] A. the voters in his state. B. the 2500 voters receiving the questionnaire. C. the 343 letters supporting the legislation. D. the 448 letters received.", "answer_v1": [ "B", "D", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "A Senator wants to know what the voters of his state think of proposed legislation on gun control. He mails a questionnaire on the subject to an SRS of 2500 voters in his state. His staff reports that 448 questionnaires have been returned, 343 of which support the legislation.\n(a) The population is [ANS] A. the 343 letters supporting the legislation. B. the 2500 voters receiving the questionnaire. C. the 448 letters received. D. the voters in his state.\n(b) This is an example of [ANS] A. a survey with little bias because it was the voters who elected the senator. B. a survey with little bias because a large SRS was used. C. a survey containing nonresponse. D. all of the above.\n(c) The sample is [ANS] A. the 448 letters received. B. the 343 letters supporting the legislation. C. the 2500 voters receiving the questionnaire D. the voters in his state.", "answer_v2": [ "D", "C", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "A Senator wants to know what the voters of his state think of proposed legislation on gun control. He mails a questionnaire on the subject to an SRS of 2500 voters in his state. His staff reports that 448 questionnaires have been returned, 343 of which support the legislation.\n(a) The population is [ANS] A. the voters in his state. B. the 448 letters received. C. the 343 letters supporting the legislation. D. the 2500 voters receiving the questionnaire.\n(b) This is an example of [ANS] A. a survey with little bias because a large SRS was used. B. a survey with little bias because it was the voters who elected the senator. C. a survey containing nonresponse. D. all of the above.\n(c) The sample is [ANS] A. the 448 letters received. B. the 343 letters supporting the legislation. C. the voters in his state. D. the 2500 voters receiving the questionnaire", "answer_v3": [ "A", "C", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0015", "subject": "Statistics", "topic": "Sample survey methods", "subtopic": "Concepts", "level": "3", "keywords": [ "statistic", "producing data", "sample" ], "problem_v1": "In order to assess the opinion of students at the Ohio State University on campus safety, a reporter for the student newspaper interviews 15 students he meets walking on the campus late at night who are willing to give their opinion.\n(a) The sample obtained is [ANS] A. a probability sample of students with night classes. B. a stratified random sample of students feeling safe. C. a simple random sample of students feeling safe. D. probably biased.\n(b) The method of sampling used is [ANS] A. simple random sampling. B. a census. C. the Gallup Poll. D. voluntary response.\n(c) The sample is [ANS] A. the 15 students interviewed. B. all those students walking on campus late at night. C. all students at universities with safety issues. D. all students approached by the reporter.", "answer_v1": [ "D", "D", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "In order to assess the opinion of students at the Ohio State University on campus safety, a reporter for the student newspaper interviews 15 students he meets walking on the campus late at night who are willing to give their opinion.\n(a) The sample is [ANS] A. all students at universities with safety issues. B. all those students walking on campus late at night. C. all students approached by the reporter. D. the 15 students interviewed.\n(b) The sample obtained is [ANS] A. a probability sample of students with night classes. B. probably biased. C. a simple random sample of students feeling safe. D. a stratified random sample of students feeling safe.\n(c) The method of sampling used is [ANS] A. voluntary response. B. the Gallup Poll. C. simple random sampling. D. a census.", "answer_v2": [ "D", "B", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "In order to assess the opinion of students at the Ohio State University on campus safety, a reporter for the student newspaper interviews 15 students he meets walking on the campus late at night who are willing to give their opinion.\n(a) The sample is [ANS] A. the 15 students interviewed. B. all students approached by the reporter. C. all students at universities with safety issues. D. all those students walking on campus late at night.\n(b) The sample obtained is [ANS] A. a stratified random sample of students feeling safe. B. probably biased. C. a simple random sample of students feeling safe. D. a probability sample of students with night classes.\n(c) The method of sampling used is [ANS] A. voluntary response. B. the Gallup Poll. C. a census. D. simple random sampling.", "answer_v3": [ "A", "B", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0016", "subject": "Statistics", "topic": "Sample survey methods", "subtopic": "Concepts", "level": "3", "keywords": [ "statistic", "producing data", "sample" ], "problem_v1": "A marketing research firm wishes to determine if the residents of Caldwell, Idaho, would be interested in a new downtown restaurant. The firm selects a simple random sample of 175 phone numbers from the Caldwell phone book and calls each household. Only 36 of those called are willing to participate in the survey, and 23 participants would support a new downtown restaurant.\n(a) The population of interest is [ANS] A. the 175 phone numbers chosen. B. all households in the Caldwell phone book. C. all residents of Caldwell. D. the 36 households that participated in the study. E. None of the above.\n(b) The chance that all 175 phone numbers chosen are located in one particular neighborhood in Caldwell is [ANS] A. reasonably large due to the ''cluster'' effect. B. exactly 0. Simple random sampling will spread out the locations of the phone numbers selected. C. the same as for any other set of 175 phone numbers. D. 175 divided by the size of the population of Caldwell. E. None of the above.\n(c) The sample in this survey is [ANS] A. the 175 phone numbers chosen. B. the 36 households that participated in the study. C. all residents of Caldwell. D. all households in the Caldwell phone book. E. None of the above.", "answer_v1": [ "C", "C", "B" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "A marketing research firm wishes to determine if the residents of Caldwell, Idaho, would be interested in a new downtown restaurant. The firm selects a simple random sample of 135 phone numbers from the Caldwell phone book and calls each household. Only 72 of those called are willing to participate in the survey, and 47 participants would support a new downtown restaurant.\n(a) The sample in this survey is [ANS] A. all residents of Caldwell. B. the 72 households that participated in the study. C. the 135 phone numbers chosen. D. all households in the Caldwell phone book. E. None of the above.\n(b) The population of interest is [ANS] A. the 72 households that participated in the study. B. all households in the Caldwell phone book. C. the 135 phone numbers chosen. D. all residents of Caldwell. E. None of the above.\n(c) The chance that all 135 phone numbers chosen are located in one particular neighborhood in Caldwell is [ANS] A. the same as for any other set of 135 phone numbers. B. reasonably large due to the ''cluster'' effect. C. 135 divided by the size of the population of Caldwell. D. exactly 0. Simple random sampling will spread out the locations of the phone numbers selected. E. None of the above.", "answer_v2": [ "B", "D", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "A marketing research firm wishes to determine if the residents of Caldwell, Idaho, would be interested in a new downtown restaurant. The firm selects a simple random sample of 155 phone numbers from the Caldwell phone book and calls each household. Only 30 of those called are willing to participate in the survey, and 20 participants would support a new downtown restaurant.\n(a) The sample in this survey is [ANS] A. all residents of Caldwell. B. the 155 phone numbers chosen. C. the 30 households that participated in the study. D. all households in the Caldwell phone book. E. None of the above.\n(b) The population of interest is [ANS] A. the 30 households that participated in the study. B. all households in the Caldwell phone book. C. all residents of Caldwell. D. the 155 phone numbers chosen. E. None of the above.\n(c) The chance that all 155 phone numbers chosen are located in one particular neighborhood in Caldwell is [ANS] A. the same as for any other set of 155 phone numbers. B. exactly 0. Simple random sampling will spread out the locations of the phone numbers selected. C. 155 divided by the size of the population of Caldwell. D. reasonably large due to the ''cluster'' effect. E. None of the above.", "answer_v3": [ "C", "C", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Statistics_0017", "subject": "Statistics", "topic": "Sample survey methods", "subtopic": "Concepts", "level": "3", "keywords": [ "statistic", "producing data", "sample" ], "problem_v1": "For each problem, select the best response.\n(a) A news release for a diet product company reports: There's good news for the 65 million Americans currently on a diet. Its study showed that people who lose weight can keep it off. The sample was 20 graduates of the company's program who endorse it in commercials. The results of the sample are probably [ANS] A. biased, overstating the effectiveness of the diet. B. biased, understating the effectiveness of the diet. C. unbiased, but they could be more accurate. A larger sample size should be used. D. None of the above.\n(b) Simple random sampling [ANS] A. reduces bias resulting from the behavior of the interviewer. B. offsets bias resulting from undercoverage and nonresponse. C. guarantees valid results. D. reduces bias resulting from poorly worded questions. E. None of the above.", "answer_v1": [ "A", "E" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "For each problem, select the best response.\n(a) Simple random sampling [ANS] A. offsets bias resulting from undercoverage and nonresponse. B. reduces bias resulting from poorly worded questions. C. guarantees valid results. D. reduces bias resulting from the behavior of the interviewer. E. None of the above.\n(b) A news release for a diet product company reports: There's good news for the 65 million Americans currently on a diet. Its study showed that people who lose weight can keep it off. The sample was 20 graduates of the company's program who endorse it in commercials. The results of the sample are probably [ANS] A. unbiased, but they could be more accurate. A larger sample size should be used. B. biased, understating the effectiveness of the diet. C. biased, overstating the effectiveness of the diet. D. None of the above.", "answer_v2": [ "E", "C" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D" ] ], "problem_v3": "For each problem, select the best response.\n(a) Simple random sampling [ANS] A. offsets bias resulting from undercoverage and nonresponse. B. reduces bias resulting from the behavior of the interviewer. C. guarantees valid results. D. reduces bias resulting from poorly worded questions. E. None of the above.\n(b) A news release for a diet product company reports: There's good news for the 65 million Americans currently on a diet. Its study showed that people who lose weight can keep it off. The sample was 20 graduates of the company's program who endorse it in commercials. The results of the sample are probably [ANS] A. biased, overstating the effectiveness of the diet. B. biased, understating the effectiveness of the diet. C. unbiased, but they could be more accurate. A larger sample size should be used. D. None of the above.", "answer_v3": [ "E", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0018", "subject": "Statistics", "topic": "Sample survey methods", "subtopic": "Concepts", "level": "3", "keywords": [ "statistic", "producing data", "sample" ], "problem_v1": "(a) A sample of households in a community is selected at random from the telephone directory. In this community, 3\\% of households have no telephone, 14\\% have only cell phones, and another 26\\% have unlisted telephone numbers. The sample will certainly suffer from [ANS] A. nonresponse. B. undercoverage. C. false responses. D. None of the above.\n(b) A committee on community relations in a college town plans to survey local businesses about the importance of students as customers. From telephone book listings, the committee chooses 235 businesses at random. Of these, 79 return the questionnaire mailed by the committee. The population for this study is [ANS] A. the 79 businesses that returned the questionnaire. B. all businesses in the college town. C. the 235 businesses chosen. D. None of the above.\n(c) Archaeologists plan to examine a sample of 3-meter-square plots near an ancient Greek city for artifacts visible in the ground. They choose separate samples of plots from floodplain, coast, foothills, and high hills. What kind of sample is this? [ANS] A. A stratified random sample. B. A simple random sample. C. A voluntary response sample. D. None of the above.", "answer_v1": [ "B", "B", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "(a) An opinion poll contacts 1334 adults and asks them, \" Which political party do you think has better ideas for leading the country in the 21st century?\" In all, 690 of the 1334 say, \" The Democrats.\" The sample in this setting is [ANS] A. the 690 people who chose the Democrats. B. all 235 million adults in the United States. C. the 1334 people interviewed. D. None of the above.\n(b) A sample of households in a community is selected at random from the telephone directory. In this community, 4\\% of households have no telephone, 17\\% have only cell phones, and another 24\\% have unlisted telephone numbers. The sample will certainly suffer from [ANS] A. false responses. B. nonresponse. C. undercoverage. D. None of the above.\n(c) A committee on community relations in a college town plans to survey local businesses about the importance of students as customers. From telephone book listings, the committee chooses 177 businesses at random. Of these, 66 return the questionnaire mailed by the committee. The population for this study is [ANS] A. all businesses in the college town. B. the 66 businesses that returned the questionnaire. C. the 177 businesses chosen. D. None of the above.", "answer_v2": [ "C", "C", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "(a) A committee on community relations in a college town plans to survey local businesses about the importance of students as customers. From telephone book listings, the committee chooses 272 businesses at random. Of these, 96 return the questionnaire mailed by the committee. The population for this study is [ANS] A. the 96 businesses that returned the questionnaire. B. the 272 businesses chosen. C. all businesses in the college town. D. None of the above.\n(b) Archaeologists plan to examine a sample of 2-meter-square plots near an ancient Greek city for artifacts visible in the ground. They choose separate samples of plots from floodplain, coast, foothills, and high hills. What kind of sample is this? [ANS] A. A voluntary response sample. B. A stratified random sample. C. A simple random sample. D. None of the above.\n(c) An opinion poll contacts 1549 adults and asks them, \" Which political party do you think has better ideas for leading the country in the 21st century?\" In all, 541 of the 1549 say, \" The Democrats.\" The sample in this setting is [ANS] A. all 235 million adults in the United States. B. the 541 people who chose the Democrats. C. the 1549 people interviewed. D. None of the above.", "answer_v3": [ "C", "B", "C" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0019", "subject": "Statistics", "topic": "Sample survey methods", "subtopic": "Concepts", "level": "3", "keywords": [ "statistic", "producing data", "sample" ], "problem_v1": "For each problem, select the best response.\n(a) In order to take a sample of 1200 people from a population, I first divide the population into men and women, and then take a simple random sample of 500 men and a separate simple random sample of 700 women. This is an example of a [ANS] A. stratified random sample. B. a multistage sample. C. convenience sampling. D. randomized comparative experiment. E. a simple random sample.\n(b) A small college has 500 male and 600 female undergraduates. A simple random sample of 50 of the male undergraduates is selected, and, separately, a simple random sample of 60 of the female undergraduates is selected. The two samples are combined to give an overall sample of 110 students. The overall sample is [ANS] A. a stratified random sample. B. a simple random sample. C. convenience sampling. D. a systematic sample. E. a multistage sample.\n(c) A simple random sample of size n is defined to be [ANS] A. a sample of size n chosen in such a way that every set of n units in the population has an equal chance to be the sample actually selected. B. a sample of size n chosen in such a way that every unit in the population has the same chance of being selected. C. a sample of size n chosen in such a way that every unit in the population has a known nonzero chance of being selected. D. All of the above. They are essentially identical definitions.", "answer_v1": [ "A", "A", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D" ] ], "problem_v2": "For each problem, select the best response.\n(a) A simple random sample of size n is defined to be [ANS] A. a sample of size n chosen in such a way that every unit in the population has a known nonzero chance of being selected. B. a sample of size n chosen in such a way that every unit in the population has the same chance of being selected. C. a sample of size n chosen in such a way that every set of n units in the population has an equal chance to be the sample actually selected. D. All of the above. They are essentially identical definitions.\n(b) In order to take a sample of 1200 people from a population, I first divide the population into men and women, and then take a simple random sample of 500 men and a separate simple random sample of 700 women. This is an example of a [ANS] A. stratified random sample. B. a simple random sample. C. randomized comparative experiment. D. a multistage sample. E. convenience sampling.\n(c) A small college has 500 male and 600 female undergraduates. A simple random sample of 50 of the male undergraduates is selected, and, separately, a simple random sample of 60 of the female undergraduates is selected. The two samples are combined to give an overall sample of 110 students. The overall sample is [ANS] A. a simple random sample. B. convenience sampling. C. a stratified random sample. D. a multistage sample. E. a systematic sample.", "answer_v2": [ "C", "A", "C" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "For each problem, select the best response.\n(a) A simple random sample of size n is defined to be [ANS] A. a sample of size n chosen in such a way that every set of n units in the population has an equal chance to be the sample actually selected. B. a sample of size n chosen in such a way that every unit in the population has the same chance of being selected. C. a sample of size n chosen in such a way that every unit in the population has a known nonzero chance of being selected. D. All of the above. They are essentially identical definitions.\n(b) In order to take a sample of 1200 people from a population, I first divide the population into men and women, and then take a simple random sample of 500 men and a separate simple random sample of 700 women. This is an example of a [ANS] A. stratified random sample. B. convenience sampling. C. a simple random sample. D. randomized comparative experiment. E. a multistage sample.\n(c) A small college has 500 male and 600 female undergraduates. A simple random sample of 50 of the male undergraduates is selected, and, separately, a simple random sample of 60 of the female undergraduates is selected. The two samples are combined to give an overall sample of 110 students. The overall sample is [ANS] A. a systematic sample. B. a simple random sample. C. convenience sampling. D. a stratified random sample. E. a multistage sample.", "answer_v3": [ "A", "A", "D" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Statistics_0020", "subject": "Statistics", "topic": "Sample survey methods", "subtopic": "Concepts", "level": "1", "keywords": [ "statistic", "parameter", "sampling distribution", "sample" ], "problem_v1": "For the following problems, select the best response.\n(a) The number of hours a light bulb burns before failing varies from bulb to bulb. The distribution of burnout times is strongly skewed to the right. The central limit theorem says that [ANS] A. the average burnout time of a large number of bulbs has a distribution of the same shape (strongly skewed) as the distribution for individual bulbs. B. as we look at more and more bulbs, their average burnout time gets ever closer to the mean for all bulbs of this type. C. the average burnout time of a large number of bulbs has a distribution that is close to Normal. D. None of the above.\n(b) A study of voting chose 663 registered voters at random shortly after an election. Of these, 72\\% said they had voted in the election. Election records show that only 56\\% of registered voters voted in the election. The boldface number is a [ANS] A. parameter. B. statistic. C. sampling distribution. D. None of the above.\n(c) Annual returns on the more than 5000 common stocks available to investors vary a lot. In a recent year, the mean return was 8.3\\% and the standard deviation of return was 28.5\\%. The law of large numbers says that [ANS] A. if you invest in a large number of stocks chosen at random, your average return will have approximately a Normal distribution. B. as you invest in more and more stocks chosen at random that year, your average return on these stocks gets ever closer to 8.3\\%. C. you can get an average return higher than the mean 8.3\\% by investing in a large number of stocks. D. None of the above.", "answer_v1": [ "C", "A", "B" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "For the following problems, select the best response.\n(a) The Bureau of Labor Statistics announces that last month it interviewed all members of the labor force in a sample of 60000 households; 4.9\\% of the people interviewed were unemployed. The boldface number is a [ANS] A. sampling distribution. B. parameter. C. statistic. D. None of the above.\n(b) The number of hours a light bulb burns before failing varies from bulb to bulb. The distribution of burnout times is strongly skewed to the right. The central limit theorem says that [ANS] A. as we look at more and more bulbs, their average burnout time gets ever closer to the mean for all bulbs of this type. B. the average burnout time of a large number of bulbs has a distribution of the same shape (strongly skewed) as the distribution for individual bulbs. C. the average burnout time of a large number of bulbs has a distribution that is close to Normal. D. None of the above.\n(c) A study of voting chose 663 registered voters at random shortly after an election. Of these, 72\\% said they had voted in the election. Election records show that only 56\\% of registered voters voted in the election. The boldface number is a [ANS] A. sampling distribution. B. statistic. C. parameter. D. None of the above.", "answer_v2": [ "C", "C", "C" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "For the following problems, select the best response.\n(a) A study of voting chose 663 registered voters at random shortly after an election. Of these, 72\\% said they had voted in the election. Election records show that only 56\\% of registered voters voted in the election. The boldface number is a [ANS] A. sampling distribution. B. parameter. C. statistic. D. None of the above.\n(b) Annual returns on the more than 5000 common stocks available to investors vary a lot. In a recent year, the mean return was 8.3\\% and the standard deviation of return was 28.5\\%. The law of large numbers says that [ANS] A. you can get an average return higher than the mean 8.3\\% by investing in a large number of stocks. B. if you invest in a large number of stocks chosen at random, your average return will have approximately a Normal distribution. C. as you invest in more and more stocks chosen at random that year, your average return on these stocks gets ever closer to 8.3\\%. D. None of the above.\n(c) The Bureau of Labor Statistics announces that last month it interviewed all members of the labor force in a sample of 60000 households; 4.9\\% of the people interviewed were unemployed. The boldface number is a [ANS] A. statistic. B. parameter. C. sampling distribution. D. None of the above.", "answer_v3": [ "B", "C", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0021", "subject": "Statistics", "topic": "Sample survey methods", "subtopic": "Concepts", "level": "2", "keywords": [ "Statistics", "Sampling" ], "problem_v1": "A public opinion poll in Ohio wants to determine whether registered voters in the state approve of a measure to ban smoking in all public areas. They randomly select 50 voters from each county in the state and ask whether they approve or disapprove of the measure. This is an example of [ANS] A. SIMPLE RANDOM SAMPLE B. STRATIFIED RANDOM SAMPLE At a party there are 30 students over age 21 and 20 students under age 21. You randomly select 3 of those over 21 and 2 of those under 21 to interview about attitudes toward alcohol. This is an example of [ANS] A. STRATIFIED RANDOM SAMPLE B. SIMPLE RANDOM SAMPLE The Ministry of Health in the Canadian Province of Ontario conducted the Ontario Health Survey by conducting interviews with 30,000 randomly selected men and 35,000 randomly selected women who reside in Ontario. This is an example of [ANS] A. SIMPLE RANDOM SAMPLE B. STRATIFIED RANDOM SAMPLE", "answer_v1": [ "B", "A", "B" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B" ], [ "A", "B" ], [ "A", "B" ] ], "problem_v2": "A public opinion poll in Ohio wants to determine whether registered voters in the state approve of a measure to ban smoking in all public areas. They randomly select 50 voters in the state and ask whether they approve or disapprove of the measure. This is an example of [ANS] A. SIMPLE RANDOM SAMPLE B. STRATIFIED RANDOM SAMPLE At a party there are 30 students over age 21 and 20 students under age 21. You randomly select 5 students to interview about attitudes toward alcohol. This is an example of [ANS] A. STRATIFIED RANDOM SAMPLE B. SIMPLE RANDOM SAMPLE The Ministry of Health in the Canadian Province of Ontario conducted the Ontario Health Survey by conducting interviews with 65,000 randomly selected residents of Ontario. This is an example of [ANS] A. SIMPLE RANDOM SAMPLE B. STRATIFIED RANDOM SAMPLE", "answer_v2": [ "A", "B", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B" ], [ "A", "B" ], [ "A", "B" ] ], "problem_v3": "A public opinion poll in Ohio wants to determine whether registered voters in the state approve of a measure to ban smoking in all public areas. They randomly select 50 voters in the state and ask whether they approve or disapprove of the measure. This is an example of [ANS] A. SIMPLE RANDOM SAMPLE B. STRATIFIED RANDOM SAMPLE At a party there are 30 students over age 21 and 20 students under age 21. You randomly select 5 students to interview about attitudes toward alcohol. This is an example of [ANS] A. STRATIFIED RANDOM SAMPLE B. SIMPLE RANDOM SAMPLE The Ministry of Health in the Canadian Province of Ontario conducted the Ontario Health Survey by conducting interviews with 30,000 randomly selected men and 35,000 randomly selected women who reside in Ontario. This is an example of [ANS] A. SIMPLE RANDOM SAMPLE B. STRATIFIED RANDOM SAMPLE", "answer_v3": [ "A", "B", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B" ], [ "A", "B" ], [ "A", "B" ] ] }, { "id": "Statistics_0022", "subject": "Statistics", "topic": "Sample survey methods", "subtopic": "Concepts", "level": "2", "keywords": [ "statistics", "introduction" ], "problem_v1": "Identify each quantity as a parameter or statistic. Write \"parameter\" or \"statistic\" (without quotations).\n(a) $\\mu$ answer: [ANS]\n(b) $\\sigma$ answer: [ANS]\n(c) $\\overline{x}$ answer: [ANS]\n(d) $s$ answer: [ANS]", "answer_v1": [ "PARAMETER", "PARAMETER", "STATISTIC", "statistic" ], "answer_type_v1": [ "EX", "EX", "EX", "EX" ], "options_v1": [ [ "PARAMETER", "STATISTIC" ], [ "PARAMETER", "STATISTIC" ], [ "PARAMETER", "STATISTIC" ], [ "PARAMETER", "STATISTIC" ] ], "problem_v2": "Identify each quantity as a parameter or statistic. Write \"parameter\" or \"statistic\" (without quotations).\n(a) $s$ answer: [ANS]\n(b) $\\mu$ answer: [ANS]\n(c) $\\sigma$ answer: [ANS]\n(d) $\\overline{x}$ answer: [ANS]", "answer_v2": [ "STATISTIC", "PARAMETER", "PARAMETER", "statistic" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "PARAMETER", "STATISTIC" ], [ "PARAMETER", "STATISTIC" ], [ "PARAMETER", "STATISTIC" ], [ "PARAMETER", "STATISTIC" ] ], "problem_v3": "Identify each quantity as a parameter or statistic. Write \"parameter\" or \"statistic\" (without quotations).\n(a) $\\sigma$ answer: [ANS]\n(b) $\\overline{x}$ answer: [ANS]\n(c) $s$ answer: [ANS]\n(d) $\\mu$ answer: [ANS]", "answer_v3": [ "PARAMETER", "STATISTIC", "STATISTIC", "parameter" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "PARAMETER", "STATISTIC" ], [ "PARAMETER", "STATISTIC" ], [ "PARAMETER", "STATISTIC" ], [ "PARAMETER", "STATISTIC" ] ] }, { "id": "Statistics_0023", "subject": "Statistics", "topic": "Sample survey methods", "subtopic": "Concepts", "level": "3", "keywords": [ "percent" ], "problem_v1": "Since 1966, 45\\% of the No.1 draftpicks in the NBA have been centers.\nidentify the population [ANS] A. all No.1 draftpick centers in the NBA since 1966 B. all NBA players since 1966 C. all No.1 draftpicks in the NBA since 1966 D. None of the above\nidentify the specified attribute [ANS] A. being a center B. being a No.1 draftpicks in the NBA C. being an NBA player D. None of the above\nis the proportion 0.45 (45\\%) a population proportion or a sample proportion? [ANS] A. population proportion B. sample proportion C. None of the above", "answer_v1": [ "C", "A", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C" ] ], "problem_v2": "Since 1966, 45\\% of the No.1 draftpicks in the NBA have been centers.\nidentify the population [ANS] A. all No.1 draftpicks in the NBA since 1966 B. all NBA players since 1966 C. all No.1 draftpick centers in the NBA since 1966 D. None of the above\nidentify the specified attribute [ANS] A. being a center B. being an NBA player C. being a No.1 draftpicks in the NBA D. None of the above\nis the proportion 0.45 (45\\%) a population proportion or a sample proportion? [ANS] A. sample proportion B. population proportion C. None of the above", "answer_v2": [ "A", "A", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C" ] ], "problem_v3": "Since 1966, 45\\% of the No.1 draftpicks in the NBA have been centers.\nidentify the population [ANS] A. all No.1 draftpick centers in the NBA since 1966 B. all No.1 draftpicks in the NBA since 1966 C. all NBA players since 1966 D. None of the above\nidentify the specified attribute [ANS] A. being a No.1 draftpicks in the NBA B. being a center C. being an NBA player D. None of the above\nis the proportion 0.45 (45\\%) a population proportion or a sample proportion? [ANS] A. sample proportion B. population proportion C. None of the above", "answer_v3": [ "B", "B", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C" ] ] }, { "id": "Statistics_0024", "subject": "Statistics", "topic": "Sample survey methods", "subtopic": "Concepts", "level": "3", "keywords": [ "Statistics", "Sampling" ], "problem_v1": "A sample of 30 dentists from Seattle is taken to estimate the median income of all Seattle residents. Is this study [ANS] A. REPRESENTATIVE? B. NON-REPRESENTATIVE? A simple random sample of men over age 18 is taken to estimate the mean weight of all adult males. Is this study [ANS] A. REPRESENTATIVE? B. NON-REPRESENTATIVE? A telephone survey is conducted during the day in order to determine the chances of a certain candidate winning an election. Is this study [ANS] A. REPRESENTATIVE? B. NON-REPRESENTATIVE? A simple random sample of 40 U.S. citizens is taken in order to estimate the mean cholesterol level of U.S. citizens. Is this study [ANS] A. REPRESENTATIVE? B. NON-REPRESENTATIVE?", "answer_v1": [ "B", "A", "B", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B" ], [ "A", "B" ], [ "A", "B" ], [ "A", "B" ] ], "problem_v2": "A simple random sample of 30 residents from Seattle is taken to estimate the median income of all Seattle residents. Is this study [ANS] A. REPRESENTATIVE? B. NON-REPRESENTATIVE? A sample of professional football players is taken to estimate the mean weight of all adult males. Is this study [ANS] A. REPRESENTATIVE? B. NON-REPRESENTATIVE? A simple random sample of voters is taken in order to determine the chances of a certain candidate winning an election. Is this study [ANS] A. REPRESENTATIVE? B. NON-REPRESENTATIVE? Using a sample of 40 patients from a local hospital, researchers measured cholesterol level in an attempt to estimate the mean cholesterol level of U.S. citizens. Is this study [ANS] A. REPRESENTATIVE? B. NON-REPRESENTATIVE?", "answer_v2": [ "A", "B", "A", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B" ], [ "A", "B" ], [ "A", "B" ], [ "A", "B" ] ], "problem_v3": "A simple random sample of 30 residents from Seattle is taken to estimate the median income of all Seattle residents. Is this study [ANS] A. REPRESENTATIVE? B. NON-REPRESENTATIVE? A sample of professional football players is taken to estimate the mean weight of all adult males. Is this study [ANS] A. REPRESENTATIVE? B. NON-REPRESENTATIVE? A telephone survey is conducted during the day in order to determine the chances of a certain candidate winning an election. Is this study [ANS] A. REPRESENTATIVE? B. NON-REPRESENTATIVE? A simple random sample of 40 U.S. citizens is taken in order to estimate the mean cholesterol level of U.S. citizens. Is this study [ANS] A. REPRESENTATIVE? B. NON-REPRESENTATIVE?", "answer_v3": [ "A", "B", "B", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B" ], [ "A", "B" ], [ "A", "B" ], [ "A", "B" ] ] }, { "id": "Statistics_0025", "subject": "Statistics", "topic": "Sample survey methods", "subtopic": "Concepts", "level": "1", "keywords": [ "statistic", "sample", "definition" ], "problem_v1": "A choral conductor has 1301 singers in her choir, some of them are professional singers. The conductor wants to estimate what percentage of singers are professional, but can't ask all 1301, so she instead asks 632 singers in front and finds 24 who are professionals.\n(a) Identify the sample. [ANS] A. singers who were asked B. 24/1301 C. 632 D. None of the above.\n(b) Identify the statistic. [ANS] A. 1301 B. entire choir members C. 24/632 D. None of the above.\n(c) Identify the parameter [ANS] A. 24 B. percentage of entire singers who are professional C. range of values in which 95\\% values fall D. None of the above.\n(d) What specific type of bias does this scenario demonstrate? [ANS] A. Voluntary response sampling B. Convenience Sampling C. It is representative of the population. D. None of the above.", "answer_v1": [ "A", "C", "B", "B" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "A choral conductor has 1947 singers in her choir, some of them are professional singers. The conductor wants to estimate what percentage of singers are professional, but can't ask all 1947, so she instead asks 632 singers in front and finds 17 who are professionals.\n(a) Identify the population [ANS] A. entire singers B. percentage of entire singers who are professional C. 17/1947 D. None of the above.\n(b) What specific type of bias does this scenario demonstrate? [ANS] A. It is representative of the population. B. Voluntary response sampling C. Convenience Sampling D. None of the above.\n(c) Identify the parameter [ANS] A. percentage of entire singers who are professional B. 17 C. range of values in which 95\\% values fall D. None of the above.\n(d) Identify the statistic. [ANS] A. entire choir members B. 1947 C. 17/632 D. None of the above.", "answer_v2": [ "A", "C", "A", "C" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "A choral conductor has 1206 singers in her choir, some of them are professional singers. The conductor wants to estimate what percentage of singers are professional, but can't ask all 1206, so she instead asks 634 singers in front and finds 27 who are professionals.\n(a) Identify the parameter [ANS] A. 27 B. range of values in which 95\\% values fall C. percentage of entire singers who are professional D. None of the above.\n(b) Identify the sample. [ANS] A. 634 B. singers who were asked C. 27/1206 D. None of the above.\n(c) Identify the population [ANS] A. 27/1206 B. entire singers C. percentage of entire singers who are professional D. None of the above.\n(d) What specific type of bias does this scenario demonstrate? [ANS] A. Voluntary response sampling B. Convenience Sampling C. It is representative of the population. D. None of the above.", "answer_v3": [ "C", "B", "B", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0026", "subject": "Statistics", "topic": "Sample survey methods", "subtopic": "Concepts", "level": "3", "keywords": [ "percent" ], "problem_v1": "A survey of 4000 Americans found that 80 \\% agreed with the president on a certain issue.\nIdentify the population [ANS] A. The 4000 Americans interviewed B. All American Voters C. All Americans D. The 3200 Americans who agreed\nIdentify the attribute of interest. [ANS] A. Agreeing with the President B. Being an American Voter C. Being one of the 4000 who agreed D. Being one of the 3200 that agreed\nIs the proportion 0.80 (80\\%) a population proportion or a sample proportion? [ANS] A. Sample proportion B. Population proportion C. None of the above", "answer_v1": [ "C", "A", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C" ] ], "problem_v2": "A survey of 4000 Americans found that 80 \\% agreed with the president on a certain issue.\nIdentify the population [ANS] A. All Americans B. All American Voters C. The 4000 Americans interviewed D. The 3200 Americans who agreed\nIdentify the attribute of interest. [ANS] A. Agreeing with the President B. Being one of the 4000 who agreed C. Being an American Voter D. Being one of the 3200 that agreed\nIs the proportion 0.80 (80\\%) a population proportion or a sample proportion? [ANS] A. Population proportion B. Sample proportion C. None of the above", "answer_v2": [ "A", "A", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C" ] ], "problem_v3": "A survey of 4000 Americans found that 80 \\% agreed with the president on a certain issue.\nIdentify the population [ANS] A. The 4000 Americans interviewed B. All Americans C. All American Voters D. The 3200 Americans who agreed\nIdentify the attribute of interest. [ANS] A. Being an American Voter B. Agreeing with the President C. Being one of the 4000 who agreed D. Being one of the 3200 that agreed\nIs the proportion 0.80 (80\\%) a population proportion or a sample proportion? [ANS] A. Population proportion B. Sample proportion C. None of the above", "answer_v3": [ "B", "B", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C" ] ] }, { "id": "Statistics_0027", "subject": "Statistics", "topic": "Sample survey methods", "subtopic": "Concepts", "level": "4", "keywords": [ "sample surveys", "data display", "causality" ], "problem_v1": "Is there an association between the use of computerized systems and hospital mortality? In a large-scale hospital study, the study investigator randomly sampled 35 hospitals from all hospitals in Texas. Within each of the sampled hospitals, 500 patients who were hospitalized during the past year were randomly chosen. Their medical records were retrieved and whether they died while hospitalized was noted. It was found that on average, hospitals (26 of them) in which health care information is gathered and stored on computers had a lower patient death rate than hospitals (9 of them) that still rely on the 'paper' system (using paper forms and handwritten notes).\nPart I What kind of study is this? [ANS] A. An observational study. B. An experiment. C. None of the above.\nPart II What sampling method was employed in selecting the 17500 patients? [ANS] A. Stratified random sampling. B. Simple random sampling. C. Multistage sampling. D. Systematic sampling.\nPart III Which of the following is the population of interest to the study investigator? Choose the most appropriate answer. [ANS] A. All hospitals in Texas. B. The 17500 patients whose medical records were retrieved in the study. C. The 35 Texas hospitals that were selected in the study. D. All patients who were admitted to the 35 selected hospitals in Texas.\nPart IV Which of the following statements is correct about the average patient death rate of the 26 hospitals in which health care information is gathered and stored on computers? [ANS] A. It is a statistic. B. It is a parameter. C. It is a variable of interest.\nPart V What graphical display is appropriate for comparing the distribution of patient death rates between hospitals that use computer systems and those that rely on the 'paper' system? [ANS] A. A bar chart. B. Side-by-side boxplots. C. A stem-and-leaf display. D. A scatterplot.\nPart VI Mr. Prudence read about the study results and concluded from there that hospitals should adopt computerized automation of notes and records because it increases patients chance of survival. Which of the following is a correct statement about the conclusion of Mr. Prudence? [ANS] A. It is a valid conclusion because the study results showed that hospitals which use computerized systems have a lower patient death rate. B. It is a valid conclusion because the study was based on a very large number of patients. C. It is an invalid conclusion because this study does not necessarily prove a causal relationship between the use of computerized system and hospital mortality. D. Both A) and B).", "answer_v1": [ "A", "C", "A", "A", "B", "C" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Is there an association between the use of computerized systems and hospital mortality? In a large-scale hospital study, the study investigator randomly sampled 35 hospitals from all hospitals in Texas. Within each of the sampled hospitals, 500 patients who were hospitalized during the past year were randomly chosen. Their medical records were retrieved and whether they died while hospitalized was noted. It was found that on average, hospitals (26 of them) in which health care information is gathered and stored on computers had a lower patient death rate than hospitals (9 of them) that still rely on the 'paper' system (using paper forms and handwritten notes).\nPart I What kind of study is this? [ANS] A. An observational study. B. An experiment. C. None of the above.\nPart II What sampling method was employed in selecting the 17500 patients? [ANS] A. Stratified random sampling. B. Simple random sampling. C. Systematic sampling. D. Multistage sampling.\nPart III Which of the following is the population of interest to the study investigator? Choose the most appropriate answer. [ANS] A. The 35 Texas hospitals that were selected in the study. B. The 17500 patients whose medical records were retrieved in the study. C. All patients who were admitted to the 35 selected hospitals in Texas. D. All hospitals in Texas.\nPart IV Which of the following statements is correct about the average patient death rate of the 26 hospitals in which health care information is gathered and stored on computers? [ANS] A. It is a variable of interest. B. It is a parameter. C. It is a statistic.\nPart V What graphical display is appropriate for comparing the distribution of patient death rates between hospitals that use computer systems and those that rely on the 'paper' system? [ANS] A. A stem-and-leaf display. B. Side-by-side boxplots. C. A bar chart. D. A scatterplot.\nPart VI Mr. Prudence read about the study results and concluded from there that hospitals should adopt computerized automation of notes and records because it increases patients chance of survival. Which of the following is a correct statement about the conclusion of Mr. Prudence? [ANS] A. It is a valid conclusion because the study results showed that hospitals which use computerized systems have a lower patient death rate. B. It is a valid conclusion because the study was based on a very large number of patients. C. It is an invalid conclusion because this study does not necessarily prove a causal relationship between the use of computerized system and hospital mortality. D. Both A) and B).", "answer_v2": [ "A", "D", "D", "C", "B", "C" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Is there an association between the use of computerized systems and hospital mortality? In a large-scale hospital study, the study investigator randomly sampled 35 hospitals from all hospitals in Texas. Within each of the sampled hospitals, 500 patients who were hospitalized during the past year were randomly chosen. Their medical records were retrieved and whether they died while hospitalized was noted. It was found that on average, hospitals (26 of them) in which health care information is gathered and stored on computers had a lower patient death rate than hospitals (9 of them) that still rely on the 'paper' system (using paper forms and handwritten notes).\nPart I What kind of study is this? [ANS] A. An observational study. B. An experiment. C. None of the above.\nPart II What sampling method was employed in selecting the 17500 patients? [ANS] A. Stratified random sampling. B. Multistage sampling. C. Systematic sampling. D. Simple random sampling.\nPart III Which of the following is the population of interest to the study investigator? Choose the most appropriate answer. [ANS] A. All hospitals in Texas. B. All patients who were admitted to the 35 selected hospitals in Texas. C. The 35 Texas hospitals that were selected in the study. D. The 17500 patients whose medical records were retrieved in the study.\nPart IV Which of the following statements is correct about the average patient death rate of the 26 hospitals in which health care information is gathered and stored on computers? [ANS] A. It is a variable of interest. B. It is a statistic. C. It is a parameter.\nPart V What graphical display is appropriate for comparing the distribution of patient death rates between hospitals that use computer systems and those that rely on the 'paper' system? [ANS] A. A stem-and-leaf display. B. A bar chart. C. Side-by-side boxplots. D. A scatterplot.\nPart VI Mr. Prudence read about the study results and concluded from there that hospitals should adopt computerized automation of notes and records because it increases patients chance of survival. Which of the following is a correct statement about the conclusion of Mr. Prudence? [ANS] A. It is a valid conclusion because the study was based on a very large number of patients. B. It is a valid conclusion because the study results showed that hospitals which use computerized systems have a lower patient death rate. C. It is an invalid conclusion because this study does not necessarily prove a causal relationship between the use of computerized system and hospital mortality. D. Both A) and B).", "answer_v3": [ "A", "B", "A", "B", "C", "C" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0028", "subject": "Statistics", "topic": "Sample survey methods", "subtopic": "Concepts", "level": "2", "keywords": [ "statistics", "descriptive statistics", "experiment" ], "problem_v1": "Which of the following must be avoided in designing a questionnaire? [ANS] A. Leading questions B. Demographic questions C. Open-ended questions D. Dichotomous questions\nWhen the population is divided into mutually exclusive sets, and then a simple random sample is drawn from each set, this is called: [ANS] A. Slection Bias B. Simple random sampling C. Stratified random sampling D. Cluster sampling", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Which of the following must be avoided in designing a questionnaire? [ANS] A. Open-ended questions B. Demographic questions C. Dichotomous questions D. Leading questions\nWhich of the following is not an example of primary data? [ANS] A. Financial data tapes that contain data compiled from the New York Stock Exchange B. Data published by the New York Stock Exchange C. Data published by the U.S. Census Bureau D. Data published by Statistics Canada", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Which of the following must be avoided in designing a questionnaire? [ANS] A. Open-ended questions B. Leading questions C. Demographic questions D. Dichotomous questions\nWhich of the following statements is correct in questionnaire design? [ANS] A. A mixture of dichotomous, multiple-choice, and open-ended questions may be used B. The questionnaire should be kept as short as possible, and the questions themselves should also be kept short C. Leading questions must be avoided D. All of the above are correct statements", "answer_v3": [ "B", "D" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0029", "subject": "Statistics", "topic": "Sample survey methods", "subtopic": "Concepts", "level": "2", "keywords": [ "statistics", "descriptive statistics", "experiment" ], "problem_v1": "Select True or False, depending on whether the corresponding statement is true or false.\n[ANS] 1. Self-administered questionnaires usually have a high response rate and may have a relatively high number of correct responses. [ANS] 2. 'Wouldn't you agree that foreign cars are better than American cars?' is an example of leading questions. [ANS] 3. A 95\\% confidence interval for average wage rates in a random sample of 40 workers is developed. This illustrates the characteristic of sampling error. [ANS] 4. A person receives a mail questionnaire and places it in the wastebasket. This illustrates the characteristic of nonresponse error.", "answer_v1": [ "F", "T", "T", "T" ], "answer_type_v1": [ "TF", "TF", "TF", "TF" ], "options_v1": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v2": "Select True or False, depending on whether the corresponding statement is true or false.\n[ANS] 1. 'Wouldn't you agree that foreign cars are better than American cars?' is an example of leading questions. [ANS] 2. A person receives a mail questionnaire and places it in the wastebasket. This illustrates the characteristic of nonresponse error. [ANS] 3. Self-administered questionnaires usually have a high response rate and may have a relatively high number of correct responses. [ANS] 4. The simplest method of collecting data is by direct observation.", "answer_v2": [ "T", "T", "F", "T" ], "answer_type_v2": [ "TF", "TF", "TF", "TF" ], "options_v2": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v3": "Select True or False, depending on whether the corresponding statement is true or false.\n[ANS] 1. The response rate of a survey is the proportion of all people who were selected but did not complete the survey. [ANS] 2. The simplest method of collecting data is by direct observation. [ANS] 3. Self-administered questionnaires usually have a high response rate and may have a relatively high number of correct responses. [ANS] 4. In designing a questionnaire, demographic and open-ended questions must be avoided.", "answer_v3": [ "F", "T", "F", "F" ], "answer_type_v3": [ "TF", "TF", "TF", "TF" ], "options_v3": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ] }, { "id": "Statistics_0030", "subject": "Statistics", "topic": "Sample survey methods", "subtopic": "Sampling bias", "level": "3", "keywords": [], "problem_v1": "For the purposes of this question a household corresponds to a single residential address, while household size is the number of individuals living at that address. Say you conduct a simple random sample of individuals living in Vancouver. Those sampled are asked to report their household sizes. The sample average of household size will be: [ANS] A. systematically biased downwards as an estimator of the average size of Vancouver households. B. sensible as an estimator of the average size of Vancouver households. C. systematically biased upwards as an estimator of the average size of Vancouver households.", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C" ] ], "problem_v2": "For the purposes of this question a household corresponds to a single residential address, while household size is the number of individuals living at that address. Say you conduct a simple random sample of individuals living in Vancouver. Those sampled are asked to report their household sizes. The sample average of household size will be: [ANS] A. systematically biased upwards as an estimator of the average size of Vancouver households. B. sensible as an estimator of the average size of Vancouver households. C. systematically biased downwards as an estimator of the average size of Vancouver households.", "answer_v2": [ "A" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C" ] ], "problem_v3": "For the purposes of this question a household corresponds to a single residential address, while household size is the number of individuals living at that address. Say you conduct a simple random sample of individuals living in Vancouver. Those sampled are asked to report their household sizes. The sample average of household size will be: [ANS] A. systematically biased downwards as an estimator of the average size of Vancouver households. B. systematically biased upwards as an estimator of the average size of Vancouver households. C. sensible as an estimator of the average size of Vancouver households.", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C" ] ] }, { "id": "Statistics_0031", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Classifying data", "level": "3", "keywords": [ "Continuous", "Discrete", "statistics", "introduction", "concepts" ], "problem_v1": "Determine whether the following examples are discrete or continuous data sets. Write \"DISCRETE\" for discrete and \"CONTINUOUS\" for continuous. (without quotations)\n(a) The temperature in any given location. answer: [ANS]\n(b) The length of time it takes to fill up your gas tank. answer: [ANS]\n(c) The number of customers waiting in line at the grocery store. answer: [ANS]\n(d) The length of time needed for a student to complete a homework assignment. answer: [ANS]", "answer_v1": [ "CONTINUOUS", "CONTINUOUS", "DISCRETE", "CONTINUOUS" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "DISCRETE", "CONTINUOUS" ], [ "DISCRETE", "CONTINUOUS" ], [ "DISCRETE", "CONTINUOUS" ], [ "DISCRETE", "CONTINUOUS" ] ], "problem_v2": "Determine whether the following examples are discrete or continuous data sets. Write \"DISCRETE\" for discrete and \"CONTINUOUS\" for continuous. (without quotations)\n(a) The number of voters who vote Democratic. answer: [ANS]\n(b) The distance traveled by a city bus each day. answer: [ANS]\n(c) The number of students applying to graduate schools. answer: [ANS]\n(d) The number of customers waiting in line at the grocery store. answer: [ANS]", "answer_v2": [ "DISCRETE", "CONTINUOUS", "DISCRETE", "DISCRETE" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "DISCRETE", "CONTINUOUS" ], [ "DISCRETE", "CONTINUOUS" ], [ "DISCRETE", "CONTINUOUS" ], [ "DISCRETE", "CONTINUOUS" ] ], "problem_v3": "Determine whether the following examples are discrete or continuous data sets. Write \"DISCRETE\" for discrete and \"CONTINUOUS\" for continuous. (without quotations)\n(a) The number of errors found on a student's research paper. answer: [ANS]\n(b) The length of time needed for a student to complete a homework assignment. answer: [ANS]\n(c) The number of students applying to graduate schools. answer: [ANS]\n(d) The length of time it takes to fill up your gas tank. answer: [ANS]", "answer_v3": [ "DISCRETE", "CONTINUOUS", "DISCRETE", "CONTINUOUS" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "DISCRETE", "CONTINUOUS" ], [ "DISCRETE", "CONTINUOUS" ], [ "DISCRETE", "CONTINUOUS" ], [ "DISCRETE", "CONTINUOUS" ] ] }, { "id": "Statistics_0032", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Classifying data", "level": "1", "keywords": [ "statistics", "quantitative data", "categorical data" ], "problem_v1": "Determine whether the following examples of data are quantitative or categorical. Write \"QUANTITATIVE\" for quantitative and \"CATEGORICAL\" for categorical (without quotations).\n(a) How long it takes you to run a mile. Answer: [ANS]\n(b) The occupation of your neighbors. Answer: [ANS]\n(c) The time it takes for your car to get an oil change. Answer: [ANS]\n(d) The condition of a used car you're thinking about purchasing. Answer: [ANS]", "answer_v1": [ "QUANTITATIVE", "CATEGORICAL", "QUANTITATIVE", "CATEGORICAL" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "QUANTITATIVE", "CATEGORICAL" ], [ "QUANTITATIVE", "CATEGORICAL" ], [ "QUANTITATIVE", "CATEGORICAL" ], [ "QUANTITATIVE", "CATEGORICAL" ] ], "problem_v2": "Determine whether the following examples of data are quantitative or categorical. Write \"QUANTITATIVE\" for quantitative and \"CATEGORICAL\" for categorical (without quotations).\n(a) The amount of bacteria on a piece of moldy bread. Answer: [ANS]\n(b) The condition of a used car you're thinking about purchasing. Answer: [ANS]\n(c) The marital status of your coworkers. Answer: [ANS]\n(d) The time it takes for your car to get an oil change. Answer: [ANS]", "answer_v2": [ "QUANTITATIVE", "CATEGORICAL", "CATEGORICAL", "QUANTITATIVE" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "QUANTITATIVE", "CATEGORICAL" ], [ "QUANTITATIVE", "CATEGORICAL" ], [ "QUANTITATIVE", "CATEGORICAL" ], [ "QUANTITATIVE", "CATEGORICAL" ] ], "problem_v3": "Determine whether the following examples of data are quantitative or categorical. Write \"QUANTITATIVE\" for quantitative and \"CATEGORICAL\" for categorical (without quotations).\n(a) The marital status of your coworkers. Answer: [ANS]\n(b) How long it takes you to run a mile. Answer: [ANS]\n(c) The occupation of your neighbors. Answer: [ANS]\n(d) The time it takes for your car to get an oil change. Answer: [ANS]", "answer_v3": [ "CATEGORICAL", "QUANTITATIVE", "CATEGORICAL", "QUANTITATIVE" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "QUANTITATIVE", "CATEGORICAL" ], [ "QUANTITATIVE", "CATEGORICAL" ], [ "QUANTITATIVE", "CATEGORICAL" ], [ "QUANTITATIVE", "CATEGORICAL" ] ] }, { "id": "Statistics_0033", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Classifying data", "level": "1", "keywords": [ "statistics", "quantitative data", "categorical data" ], "problem_v1": "Select the best response.\n(a) Here are the amounts of money (cents) in coins carried by 10 students in a statistics class: \\begin{array}{ccccccccccc} \\mbox{50} & \\mbox{35} & \\mbox{0} & \\mbox{97} & \\mbox{76} & \\mbox{0} & \\mbox{0} & \\mbox{87} & \\mbox{23} & \\mbox{65} \\end{array} To make a stemplot of these data, you would use stems [ANS] A. 0, 2, 3, 5, 6, 7, 8, 9. B. 00, 10, 20, 30, 40, 50, 60, 70, 80, 90. C. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. D. None of the above.\n(b) A study of recent college graduates records the sex and total college debt in dollars for 10,000 people a year after they graduate from college. Which of the following is a true statement? [ANS] A. Sex is a quantitative variable and college debt is a categorical variable. B. Sex and college debt are both categorical variables. C. Sex and college debt are both quantitative variables. D. Sex is a categorical variable and college debt is a quantitative variable. E. All of the above.\n(c) A political party's data bank includes the zip codes of past donors, such as\n\\begin{array}{cccccccc} \\mbox{47906} & \\mbox{34236} & \\mbox{53075} & \\mbox{10010} & \\mbox{90210} & \\mbox{75204} & \\mbox{30304} & \\mbox{99709} \\end{array} Zip code is a [ANS] A. Unit of Measurement. B. Categorical variable. C. Quantitative variable. D. None of the above.", "answer_v1": [ "C", "D", "B" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Select the best response.\n(a) To display the distribution of grades (A, B, C, D, F) in a course, it would be correct to use [ANS] A. a bar graph but not a pie chart. B. a pie chart but not a bar graph. C. either a pie chart or a bar graph. D. None of the above.\n(b) Here are the amounts of money (cents) in coins carried by 10 students in a statistics class: \\begin{array}{ccccccccccc} \\mbox{50} & \\mbox{35} & \\mbox{0} & \\mbox{97} & \\mbox{76} & \\mbox{0} & \\mbox{0} & \\mbox{87} & \\mbox{23} & \\mbox{65} \\end{array} To make a stemplot of these data, you would use stems [ANS] A. 00, 10, 20, 30, 40, 50, 60, 70, 80, 90. B. 0, 2, 3, 5, 6, 7, 8, 9. C. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. D. None of the above.\n(c) A study of recent college graduates records the sex and total college debt in dollars for 10,000 people a year after they graduate from college. Which of the following is a true statement? [ANS] A. Sex is a categorical variable and college debt is a quantitative variable. B. Sex and college debt are both quantitative variables. C. Sex is a quantitative variable and college debt is a categorical variable. D. Sex and college debt are both categorical variables. E. All of the above.", "answer_v2": [ "C", "C", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Select the best response.\n(a) A study of recent college graduates records the sex and total college debt in dollars for 10,000 people a year after they graduate from college. Which of the following is a true statement? [ANS] A. Sex is a categorical variable and college debt is a quantitative variable. B. Sex and college debt are both quantitative variables. C. Sex and college debt are both categorical variables. D. Sex is a quantitative variable and college debt is a categorical variable. E. All of the above.\n(b) A political party's data bank includes the zip codes of past donors, such as\n\\begin{array}{cccccccc} \\mbox{47906} & \\mbox{34236} & \\mbox{53075} & \\mbox{10010} & \\mbox{90210} & \\mbox{75204} & \\mbox{30304} & \\mbox{99709} \\end{array} Zip code is a [ANS] A. Quantitative variable. B. Unit of Measurement. C. Categorical variable. D. None of the above.\n(c) To display the distribution of grades (A, B, C, D, F) in a course, it would be correct to use [ANS] A. either a pie chart or a bar graph. B. a pie chart but not a bar graph. C. a bar graph but not a pie chart. D. None of the above.", "answer_v3": [ "A", "C", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0034", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Classifying data", "level": "1", "keywords": [ "statistics", "quantitative data", "categorical data" ], "problem_v1": "For each problem, select the best response.\n(a) A description of different houses on the market includes the following three variables. Which of the variables is quantitative? [ANS] A. The school district. B. The monthly electric bill. C. The street number. D. The exterior paint colors. E. None of the above.\n(b) In order to rate TV shows, phone surveys are sometimes used. Such a survey might record several variables, some of which are listed below. Which of these variables are categorical? [ANS] A. The number of times the show has been watched in the last month. B. The number of persons watching the show. C. The ages of all persons watching the show. D. The name of the show (if any) being watched. E. All of the above.\n(c) A description of different houses on the market includes the following three variables. Which of the variables is quantitative? [ANS] A. The school district. B. The monthly electric bill. C. The street number. D. The exterior paint colors. E. None of the above.", "answer_v1": [ "B", "D", "B" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "For each problem, select the best response.\n(a) A survey records many variables of interest to the researchers conducting the survey. Below are some of the variables from a survey conducted by the U.S. Postal Service. Which of the variables is categorical? [ANS] A. Total household income, before taxes, in 1993. B. Age of respondent. C. Number of people, both adults and children, living in the household. D. County of residence. E. None of the above.\n(b) A description of different houses on the market includes the following three variables. Which of the variables is quantitative? [ANS] A. The street number. B. The monthly electric bill. C. The school district. D. The exterior paint colors. E. None of the above.\n(c) A description of different houses on the market includes the following three variables. Which of the variables is quantitative? [ANS] A. The street number. B. The monthly electric bill. C. The school district. D. The exterior paint colors. E. None of the above.", "answer_v2": [ "D", "B", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "For each problem, select the best response.\n(a) In order to rate TV shows, phone surveys are sometimes used. Such a survey might record several variables, some of which are listed below. Which of these variables are categorical? [ANS] A. The name of the show (if any) being watched. B. The ages of all persons watching the show. C. The number of persons watching the show. D. The number of times the show has been watched in the last month. E. All of the above.\n(b) A professor records the values of several variables for each student in her class. These include the variables listed below. Which of these variables is categorical? [ANS] A. The total number of points earned in the class (i.e., the total of the points on all exams and quizzes in the course. The maximum number of points possible is 500). B. Final grade for the course (A, B, C, D, or F). C. The number of lectures the student missed. D. Score on the final exam (out of 200 points). E. None of the above.\n(c) A description of different houses on the market includes the following three variables. Which of the variables is quantitative? [ANS] A. The street number. B. The school district. C. The monthly electric bill. D. The exterior paint colors. E. None of the above.", "answer_v3": [ "A", "B", "C" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Statistics_0035", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Classifying data", "level": "2", "keywords": [ "Statistics" ], "problem_v1": "Of the variables you have studied so far, which type yields nonnumerical data? [ANS] A. Quantitative continuous B. Quantitative discrete C. Qualitative D. None of the above", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "Of the variables you have studied so far, which type yields nonnumerical data? [ANS] A. Qualitative B. Quantitative discrete C. Quantitative continuous D. None of the above", "answer_v2": [ "A" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "Of the variables you have studied so far, which type yields nonnumerical data? [ANS] A. Quantitative continuous B. Qualitative C. Quantitative discrete D. None of the above", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0036", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Classifying data", "level": "2", "keywords": [], "problem_v1": "Which of the following selected variables, associated with clinical trials of a drug, are quantitative variables? CHECK ALL THAT APPLY. [ANS] A. Smoking history (cigarettes per year) B. Gender-Male, Female C. Age (in years)-$<$ 20, 20-30, 30-40, 40-50, 50-60, 60 or above D. Dosage form-1=tablet, 2=capsule, 3=liquid solution, 4=other E. Body mass index ($kg/m^2$)", "answer_v1": [ "AE" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Which of the following selected variables, associated with clinical trials of a drug, are quantitative variables? CHECK ALL THAT APPLY. [ANS] A. Body mass index ($kg/m^2$) B. Gender-Male, Female C. Age (in years)-$<$ 20, 20-30, 30-40, 40-50, 50-60, 60 or above D. Smoking history (cigarettes per year) E. Dosage form-1=tablet, 2=capsule, 3=liquid solution, 4=other", "answer_v2": [ "AD" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Which of the following selected variables, associated with clinical trials of a drug, are quantitative variables? CHECK ALL THAT APPLY. [ANS] A. Dosage form-1=tablet, 2=capsule, 3=liquid solution, 4=other B. Body mass index ($kg/m^2$) C. Gender-Male, Female D. Age (in years)-$<$ 20, 20-30, 30-40, 40-50, 50-60, 60 or above E. Smoking history (cigarettes per year)", "answer_v3": [ "BE" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E" ] ] }, { "id": "Statistics_0037", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Classifying data", "level": "3", "keywords": [ "statistics", "descriptive statistics", "interval", "nominal", "ordinal" ], "problem_v1": "Determine whether the following possible responses should be classified as interval, nominal or ordinal data.\n[ANS] 1. Amount of time you spend per week on your homework [ANS] 2. Lily's travel time from her dorm to the student union at the University of Virginia [ANS] 3. The status of your Statistics professor; Assistant Professor, Associate Professor, or Full Professor [ANS] 4. The availability of parking on campus; Insufficient, Moderate, or Abundant", "answer_v1": [ "INTERVAL", "INTERVAL", "ORDINAL", "Ordinal" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "Ordinal", "Nominal", "Interval" ], [ "Ordinal", "Nominal", "Interval" ], [ "Ordinal", "Nominal", "Interval" ], [ "Ordinal", "Nominal", "Interval" ] ], "problem_v2": "Determine whether the following possible responses should be classified as interval, nominal or ordinal data.\n[ANS] 1. Lily's travel time from her dorm to the student union at the University of Virginia [ANS] 2. The availability of parking on campus; Insufficient, Moderate, or Abundant [ANS] 3. Amount of time you spend per week on your homework [ANS] 4. Whether you are a US citizen", "answer_v2": [ "INTERVAL", "ORDINAL", "INTERVAL", "Nominal" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "Ordinal", "Nominal", "Interval" ], [ "Ordinal", "Nominal", "Interval" ], [ "Ordinal", "Nominal", "Interval" ], [ "Ordinal", "Nominal", "Interval" ] ], "problem_v3": "Determine whether the following possible responses should be classified as interval, nominal or ordinal data.\n[ANS] 1. Your marital status [ANS] 2. Whether you are a US citizen [ANS] 3. Amount of time you spend per week on your homework [ANS] 4. Heidi's favorite brand of tennis balls", "answer_v3": [ "NOMINAL", "NOMINAL", "INTERVAL", "Nominal" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "Ordinal", "Nominal", "Interval" ], [ "Ordinal", "Nominal", "Interval" ], [ "Ordinal", "Nominal", "Interval" ], [ "Ordinal", "Nominal", "Interval" ] ] }, { "id": "Statistics_0038", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Classifying data", "level": "3", "keywords": [ "statistics", "descriptive statistics" ], "problem_v1": "Select True or False, depending on whether the corresponding statement is true or false.\n[ANS] 1. Time series data are often graphically depicted on a line chart, which is a plot of the variable of interest over time. [ANS] 2. Professor Hogg graduated from the University of Iowa with a code value=2 while Professor Maas graduated from Michigan State University with a code value=1. The scale of measurement likely represented by this information is ratio. [ANS] 3. Quantitative variables usually represent membership in groups or categories. [ANS] 4. An automobile insurance agent believes that company A is more reliable than company B. The scale of measurement that this information represents is the ordinal scale.", "answer_v1": [ "T", "F", "F", "T" ], "answer_type_v1": [ "TF", "TF", "TF", "TF" ], "options_v1": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v2": "Select True or False, depending on whether the corresponding statement is true or false.\n[ANS] 1. Professor Hogg graduated from the University of Iowa with a code value=2 while Professor Maas graduated from Michigan State University with a code value=1. The scale of measurement likely represented by this information is ratio. [ANS] 2. An automobile insurance agent believes that company A is more reliable than company B. The scale of measurement that this information represents is the ordinal scale. [ANS] 3. Time series data are often graphically depicted on a line chart, which is a plot of the variable of interest over time. [ANS] 4. Bar and pie charts are graphical techniques for nominal data. The former focus the attention on the frequency of the occurrences of the categories, and the later emphasize the proportion of occurrences in each category.", "answer_v2": [ "F", "T", "T", "T" ], "answer_type_v2": [ "TF", "TF", "TF", "TF" ], "options_v2": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v3": "Select True or False, depending on whether the corresponding statement is true or false.\n[ANS] 1. The graphical technique used to describe the relationship between two interval variables is the scatter diagram. [ANS] 2. Bar and pie charts are graphical techniques for nominal data. The former focus the attention on the frequency of the occurrences of the categories, and the later emphasize the proportion of occurrences in each category. [ANS] 3. Time series data are often graphically depicted on a line chart, which is a plot of the variable of interest over time. [ANS] 4. ATP singles rankings for tennis players is an example of an interval scale.", "answer_v3": [ "T", "T", "T", "F" ], "answer_type_v3": [ "TF", "TF", "TF", "TF" ], "options_v3": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ] }, { "id": "Statistics_0039", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Classifying data", "level": "3", "keywords": [ "statistics", "descriptive statistics", "variable", "quantitative" ], "problem_v1": "In a questionnaire, respondents are asked to mark their gender as male or female. Gender is an example of a: [ANS] A. qualitative variable B. qualitative or quantitative variable, depending on how the respondents answered the question C. quantitative variable D. None of the above answers is correct\nQualitative data: [ANS] A. must be nonnumeric B. indicate either how much or how many C. are labels used to identify attributes of elements D. cannot be numeric", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "In a questionnaire, respondents are asked to mark their gender as male or female. Gender is an example of a: [ANS] A. qualitative variable B. quantitative variable C. qualitative or quantitative variable, depending on how the respondents answered the question D. None of the above answers is correct\nQuantitative Data: [ANS] A. are always numeric B. are always nonnumeric C. may be either numeric or nonnumeric D. None of the above answers is correct", "answer_v2": [ "A", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "In a questionnaire, respondents are asked to mark their gender as male or female. Gender is an example of a: [ANS] A. qualitative or quantitative variable, depending on how the respondents answered the question B. qualitative variable C. quantitative variable D. None of the above answers is correct\nOrdinary arithmetic operations are meaningful: [ANS] A. either with quantitative or qualitative data B. only with quantitative data C. only with qualitative data D. None of the above answers is correct", "answer_v3": [ "B", "B" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0040", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Classifying data", "level": "3", "keywords": [ "statistics", "descriptive statistics", "interval", "nominal", "ordinal" ], "problem_v1": "Before leaving a particular restaurant, patrons are asked to respond to a questionaire containing the questions given below. For each question, indicate whether the possible responses are Interval, Nominal, or Ordinal.\n[ANS] 1. Which of the following attributes of this restaurant do you find most attractive: service, prices, quality of food, menu options? [ANS] 2. Would your overall rating of this restaurant be excellent, good, fair, or poor? [ANS] 3. If you've eaten here before, how long has it been since your last visit? [ANS] 4. Would you recommend this restaurant to a friend?", "answer_v1": [ "N", "O", "I", "N" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "I", "N", "O" ], [ "I", "N", "O" ], [ "I", "N", "O" ], [ "I", "N", "O" ] ], "problem_v2": "Before leaving a particular restaurant, patrons are asked to respond to a questionaire containing the questions given below. For each question, indicate whether the possible responses are Interval, Nominal, or Ordinal.\n[ANS] 1. Would your overall rating of this restaurant be excellent, good, fair, or poor? [ANS] 2. Would you recommend this restaurant to a friend? [ANS] 3. Which of the following attributes of this restaurant do you find most attractive: service, prices, quality of food, menu options? [ANS] 4. What is the approximate distance of the restaurant from your home?", "answer_v2": [ "O", "N", "N", "I" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "I", "N", "O" ], [ "I", "N", "O" ], [ "I", "N", "O" ], [ "I", "N", "O" ] ], "problem_v3": "Before leaving a particular restaurant, patrons are asked to respond to a questionaire containing the questions given below. For each question, indicate whether the possible responses are Interval, Nominal, or Ordinal.\n[ANS] 1. Have you eaten at this restaurant previously? [ANS] 2. What is the approximate distance of the restaurant from your home? [ANS] 3. Which of the following attributes of this restaurant do you find most attractive: service, prices, quality of food, menu options? [ANS] 4. Do you consider our prices to be high, average, or low?", "answer_v3": [ "N", "I", "N", "O" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "I", "N", "O" ], [ "I", "N", "O" ], [ "I", "N", "O" ], [ "I", "N", "O" ] ] }, { "id": "Statistics_0041", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Classifying data", "level": "3", "keywords": [ "statistics", "descriptive statistics" ], "problem_v1": "Which of the following statements is false? [ANS] A. The intervals in a frequency distribution may overlap to ensure that each observation is assigned to an interval B. The number of class intervals we select in a frequency distribution depends entirely on the number of observations in the data set C. Although the frequency distribution provides information about how the numbers in the data set are distributed, the information is more easily understood and imparted by drawing a histogram D. A frequency distribution counts the number of observations that fall into each of a series of intervals, called classes that cover the complete range of observations\nIn general, incomes of employees in large firms tend to be: [ANS] A. positively skewed B. symmetric C. negatively skewed D. none of the above", "answer_v1": [ "A", "A" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Which of the following statements is false? [ANS] A. Although the frequency distribution provides information about how the numbers in the data set are distributed, the information is more easily understood and imparted by drawing a histogram B. The number of class intervals we select in a frequency distribution depends entirely on the number of observations in the data set C. A frequency distribution counts the number of observations that fall into each of a series of intervals, called classes that cover the complete range of observations D. The intervals in a frequency distribution may overlap to ensure that each observation is assigned to an interval\nWhich of the following statements is false? [ANS] A. All calculations are permitted on nominal data B. The only permissible calculations on ordinal data are ones involving a ranking process C. All calculations are permitted on interval data D. The most important aspect of ordinal data is the order of the data values", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Which of the following statements is false? [ANS] A. Although the frequency distribution provides information about how the numbers in the data set are distributed, the information is more easily understood and imparted by drawing a histogram B. The intervals in a frequency distribution may overlap to ensure that each observation is assigned to an interval C. The number of class intervals we select in a frequency distribution depends entirely on the number of observations in the data set D. A frequency distribution counts the number of observations that fall into each of a series of intervals, called classes that cover the complete range of observations\nA modal class is the class that includes: [ANS] A. the largest number of observations B. the largest observation in the data set C. the smallest number of observations D. the smallest observation in the data set", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0043", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "mean", "median", "application" ], "problem_v1": "Here is a list of some cities\u2019 population:\n${72704000,\\;\\;72120000,\\;\\;75396000,\\;\\;1200,\\;\\;75305000,\\;\\;71261000,\\;\\;71130000,\\;\\;74547000,\\;\\;79392000}$ The mean of these cities\u2019 population is [ANS]. The median of these cities\u2019 population is [ANS]. Which number, mean or median, is a better way to represent these cities\u2019 population? [ANS]", "answer_v1": [ " 65761800", " 72704000", "median" ], "answer_type_v1": [ "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "median", "mean" ] ], "problem_v2": "Here is a list of some cities\u2019 population:\n${2000,\\;\\;73341000,\\;\\;79468000,\\;\\;73166000,\\;\\;71830000}$ The mean of these cities\u2019 population is [ANS]. The median of these cities\u2019 population is [ANS]. Which number, mean or median, is a better way to represent these cities\u2019 population? [ANS]", "answer_v2": [ " 59561400", " 73166000", "median" ], "answer_type_v2": [ "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "median", "mean" ] ], "problem_v3": "Here is a list of some cities\u2019 population:\n${75486000,\\;\\;1600,\\;\\;72061000,\\;\\;73461000,\\;\\;78086000}$ The mean of these cities\u2019 population is [ANS]. The median of these cities\u2019 population is [ANS]. Which number, mean or median, is a better way to represent these cities\u2019 population? [ANS]", "answer_v3": [ " 59819120", " 73461000", "median" ], "answer_type_v3": [ "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "median", "mean" ] ] }, { "id": "Statistics_0044", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "3", "keywords": [ "Frequency", "Relative", "statistics", "introduction", "concepts" ], "problem_v1": "Complete the table below. $\\begin{array}{ccc}\\hline Books read within the past year & Frequency & Relative Frequency \\\\ \\hline none & 8 & [ANS] \\\\ \\hline 0-4 & 12 & [ANS] \\\\ \\hline 5-9 & 13 & [ANS] \\\\ \\hline 10-14 & [ANS] & 0.225806451612903 \\\\ \\hline 15-19 & 11 & [ANS] \\\\ \\hline 20-25 & 4 & [ANS] \\\\ \\hline total & 62 & 1 \\\\ \\hline \\end{array}$", "answer_v1": [ "0.129032258064516", "0.193548387096774", "0.209677419354839", "14", "0.17741935483871", "0.0645161290322581" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Complete the table below. $\\begin{array}{ccc}\\hline Books read within the past year & Frequency & Relative Frequency \\\\ \\hline none & 1 & [ANS] \\\\ \\hline 0-4 & 19 & [ANS] \\\\ \\hline 5-9 & 3 & [ANS] \\\\ \\hline 10-14 & [ANS] & 0.222222222222222 \\\\ \\hline 15-19 & 15 & [ANS] \\\\ \\hline 20-25 & 4 & [ANS] \\\\ \\hline total & 54 & 1 \\\\ \\hline \\end{array}$", "answer_v2": [ "0.0185185185185185", "0.351851851851852", "0.0555555555555556", "12", "0.277777777777778", "0.0740740740740741" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Complete the table below. $\\begin{array}{ccc}\\hline Books read within the past year & Frequency & Relative Frequency \\\\ \\hline none & 4 & [ANS] \\\\ \\hline 0-4 & 13 & [ANS] \\\\ \\hline 5-9 & 6 & [ANS] \\\\ \\hline 10-14 & [ANS] & 0.254901960784314 \\\\ \\hline 15-19 & 11 & [ANS] \\\\ \\hline 20-25 & 4 & [ANS] \\\\ \\hline total & 51 & 1 \\\\ \\hline \\end{array}$", "answer_v3": [ "0.0784313725490196", "0.254901960784314", "0.117647058823529", "13", "0.215686274509804", "0.0784313725490196" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Statistics_0045", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "3", "keywords": [ "Measure", "Percentile", "statistics", "measures", "percentile" ], "problem_v1": "Here is a list of 25 scores on a Math midterm exam: $38.5, \\ \\ 41.5, \\ \\ 52, \\ \\ 52.5, \\ \\ 61, \\ \\ 63, \\ \\ 63.5, \\ \\ 68, \\ \\ 69, \\ \\ 69,$ $78.5, \\ \\ 79, \\ \\ 80, \\ \\ 83, \\ \\ 87, \\ \\ 88.5, \\ \\ 88.5, \\ \\ 91, \\ \\ 91.5, \\ \\ 92,$ $92.5, \\ \\ 94, \\ \\ 94, \\ \\ 97, \\ \\ 97$ Find $P_{74}$: $\\ $ [ANS]", "answer_v1": [ "91.5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Here is a list of 25 scores on a Math midterm exam: $38.5, \\ \\ 41.5, \\ \\ 52, \\ \\ 52.5, \\ \\ 61, \\ \\ 63, \\ \\ 63.5, \\ \\ 68, \\ \\ 69, \\ \\ 69,$ $78.5, \\ \\ 79, \\ \\ 80, \\ \\ 83, \\ \\ 87, \\ \\ 88.5, \\ \\ 88.5, \\ \\ 91, \\ \\ 91.5, \\ \\ 92,$ $92.5, \\ \\ 94, \\ \\ 94, \\ \\ 97, \\ \\ 97$ Find $P_{24}$: $\\ $ [ANS]", "answer_v2": [ "63.25" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Here is a list of 25 scores on a Math midterm exam: $38.5, \\ \\ 41.5, \\ \\ 52, \\ \\ 52.5, \\ \\ 61, \\ \\ 63, \\ \\ 63.5, \\ \\ 68, \\ \\ 69, \\ \\ 69,$ $78.5, \\ \\ 79, \\ \\ 80, \\ \\ 83, \\ \\ 87, \\ \\ 88.5, \\ \\ 88.5, \\ \\ 91, \\ \\ 91.5, \\ \\ 92,$ $92.5, \\ \\ 94, \\ \\ 94, \\ \\ 97, \\ \\ 97$ Find $P_{42}$: $\\ $ [ANS]", "answer_v3": [ "78.5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0046", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "Measure", "Mean", "z Score", "statistics", "measures", "mean", "standard deviation", "z score" ], "problem_v1": "IQ scores have a mean of 100 and a standard deviation of 15. John has an IQ of 124. What is the difference between John's IQ and the mean? [ANS]\nConvert John's IQ score to a z score: [ANS]", "answer_v1": [ "24", "1.6" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "IQ scores have a mean of 100 and a standard deviation of 15. Mary has an IQ of 103. What is the difference between Mary's IQ and the mean? [ANS]\nConvert Mary's IQ score to a z score: [ANS]", "answer_v2": [ "3", "0.2" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "IQ scores have a mean of 100 and a standard deviation of 15. Nick has an IQ of 112. What is the difference between Nick's IQ and the mean? [ANS]\nConvert Nick's IQ score to a z score: [ANS]", "answer_v3": [ "12", "0.8" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0047", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "3", "keywords": [ "Measure", "Mean", "Standard Deviation", "z Score", "statistics", "measures", "mean", "standard deviation", "z score" ], "problem_v1": "John took 4 courses last semester: History, Physics, Calculus, and Biology. The means and standard deviations for the final exams, and John's scores are given in the table below. Convert John's score into z scores.\n$\\begin{array}{ccccc}\\hline Subject & Mean & Stand. dev. & John's score & John's z score \\\\ \\hline History & 53 & 16 & 61 & [ANS] \\\\ \\hline Physics & 60 & 14 & 60 & [ANS] \\\\ \\hline Calculus & 70 & 12 & 73 & [ANS] \\\\ \\hline Biology & 77 & 10 & 89.5 & [ANS] \\\\ \\hline \\end{array}$ On what exam did John have the highest relative score? (Enter the subject.) [ANS]", "answer_v1": [ "0.5", "0", "0.25", "1.25", "Biology" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [], [ "History", "Physics", "Calculus", "Biology" ] ], "problem_v2": "Nick took 4 courses last semester: Calculus, Spanish, History, and Biology. The means and standard deviations for the final exams, and Nick's scores are given in the table below. Convert Nick's score into z scores.\n$\\begin{array}{ccccc}\\hline Subject & Mean & Stand. dev. & Nick's score & Nick's z score \\\\ \\hline Calculus & 70 & 12 & 70 & [ANS] \\\\ \\hline Spanish & 44 & 12 & 56 & [ANS] \\\\ \\hline History & 53 & 16 & 37 & [ANS] \\\\ \\hline Biology & 77 & 10 & 89.5 & [ANS] \\\\ \\hline \\end{array}$ On what exam did Nick have the highest relative score? (Enter the subject.) [ANS]", "answer_v2": [ "0", "1", "-1", "1.25", "Biology" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [], [ "History", "Physics", "Calculus", "Biology" ] ], "problem_v3": "Mary took 4 courses last semester: Biology, Spanish, Calculus, and Physics. The means and standard deviations for the final exams, and Mary's scores are given in the table below. Convert Mary's score into z scores.\n$\\begin{array}{ccccc}\\hline Subject & Mean & Stand. dev. & Mary's score & Mary's z score \\\\ \\hline Biology & 77 & 10 & 94.5 & [ANS] \\\\ \\hline Spanish & 44 & 12 & 59 & [ANS] \\\\ \\hline Calculus & 70 & 12 & 64 & [ANS] \\\\ \\hline Physics & 60 & 14 & 56.5 & [ANS] \\\\ \\hline \\end{array}$ On what exam did Mary have the highest relative score? (Enter the subject.) [ANS]", "answer_v3": [ "1.75", "1.25", "-0.5", "-0.25", "Biology" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [], [ "History", "Physics", "Calculus", "Biology" ] ] }, { "id": "Statistics_0048", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "Measure", "Variance", "Standard Deviation", "Range", "statistics", "introduction", "concepts" ], "problem_v1": "Given the data set below, calculate the range, sample variance, and sample standard deviation.\n21, \\ 26, \\ 36, \\ 16, \\ 30, \\ 16, \\ 44, \\ 9, \\ 52 range $=$ [ANS] variance $=$ [ANS] standard deviation $=$ [ANS]", "answer_v1": [ "43", "200.194444444444", "14.1490086028826" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Given the data set below, calculate the range, sample variance, and sample standard deviation.\n33, \\ 5, \\ 39, \\ 23, \\ 51, \\ 20, \\ 20, \\ 27, \\ 45 range $=$ [ANS] variance $=$ [ANS] standard deviation $=$ [ANS]", "answer_v2": [ "46", "204.194444444444", "14.2896621529148" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Given the data set below, calculate the range, sample variance, and sample standard deviation.\n45, \\ 17, \\ 21, \\ 17, \\ 6, \\ 36, \\ 55, \\ 29, \\ 26 range $=$ [ANS] variance $=$ [ANS] standard deviation $=$ [ANS]", "answer_v3": [ "49", "232.75", "15.2561463023924" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Statistics_0049", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "Measure", "Decile", "statistics", "measures" ], "problem_v1": "Find the indicated decile of the following data set 32, \\ 26, \\ 25, \\ 27, \\ 41, \\ 13, \\ 22, \\ 37, \\ 54, \\ 60, \\ 22, \\ 47 $D_{7}=$ [ANS]", "answer_v1": [ "41" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the indicated decile of the following data set 40, \\ 17, \\ 17, \\ 22, \\ 30, \\ 19, \\ 23, \\ 35, \\ 15, \\ 26, \\ 47, \\ 53 $D_{1}=$ [ANS]", "answer_v2": [ "17" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the indicated decile of the following data set 13, \\ 35, \\ 58, \\ 46, \\ 23, \\ 18, \\ 20, \\ 41, \\ 29, \\ 21, \\ 18, \\ 52 $D_{3}=$ [ANS]", "answer_v3": [ "20" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0050", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "statistics", "introduction", "concepts", "introduction" ], "problem_v1": "For each of the given the data sets below, calculate the mean, variance, and standard deviation.\n(a) $\\ 76, \\ 59, \\ 63, \\ 73, \\ 31, \\ 34, \\ 58, \\ 58, \\ 38$ mean $=$ [ANS] variance $=$ [ANS] standard deviation $=$ [ANS]\n(b) $\\ 50, \\ 53, \\ 45, \\ 50, \\ 48, \\ 48$ mean $=$ [ANS] variance $=$ [ANS] standard deviation $=$ [ANS]\n(c) $\\ 3.3, \\ 3.5, \\ 2.4, \\ 2.4, \\ 3.2$ mean $=$ [ANS] variance $=$ [ANS] standard deviation $=$ [ANS]", "answer_v1": [ "54.4444444444444", "270.777777777778", "16.4553267295966", "49", "7.2", "2.68328157299975", "2.96", "0.273", "0.522494019104525" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "For each of the given the data sets below, calculate the mean, variance, and standard deviation.\n(a) $\\ 10, \\ 94, \\ 16, \\ 35, \\ 95, \\ 33, \\ 20, \\ 34, \\ 57$ mean $=$ [ANS] variance $=$ [ANS] standard deviation $=$ [ANS]\n(b) $\\ 41, \\ 53, \\ 49, \\ 57, \\ 44, \\ 43$ mean $=$ [ANS] variance $=$ [ANS] standard deviation $=$ [ANS]\n(c) $\\ 2.8, \\ 3.7, \\ 2.5, \\ 3, \\ 3.7$ mean $=$ [ANS] variance $=$ [ANS] standard deviation $=$ [ANS]", "answer_v2": [ "43.7777777777778", "1010.94444444444", "31.7953525604678", "47.8333333333333", "39.3666666666667", "6.27428614797466", "3.14", "0.293", "0.541294744108974" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "For each of the given the data sets below, calculate the mean, variance, and standard deviation.\n(a) $\\ 33, \\ 61, \\ 29, \\ 56, \\ 22, \\ 36, \\ 82, \\ 92, \\ 88$ mean $=$ [ANS] variance $=$ [ANS] standard deviation $=$ [ANS]\n(b) $\\ 44, \\ 46, \\ 45, \\ 40, \\ 52, \\ 60$ mean $=$ [ANS] variance $=$ [ANS] standard deviation $=$ [ANS]\n(c) $\\ 4.4, \\ 3.9, \\ 2.3, \\ 2.9, \\ 3.9$ mean $=$ [ANS] variance $=$ [ANS] standard deviation $=$ [ANS]", "answer_v3": [ "55.4444444444444", "729.027777777778", "27.0005143983921", "47.8333333333333", "50.5666666666667", "7.11102430502573", "3.48", "0.732000000000001", "0.855569985448298" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Statistics_0051", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "4", "keywords": [ "statistics", "two sample", "inference", "t score" ], "problem_v1": "Scrapie is a degenerative disease of the nervous system. A study of the substance IDX as a treatment for scrapie used as subjects 28 infected hamsters. Fourteen, chosen at random, were injected with IDX. The other 14 were untreated. The researchers recorded how long each hamster lived. They reported: Thus, although all infected control hamsters had died by 94 days after infection (mean $\\pm$ SEM=87.8 $\\pm$ 1.58), IDX-treated hamsters lived up to 116 days (mean $\\pm$ SEM=107.3 $\\pm$ 2.27). Readers are supposed to know that SEM stands for the standard error of the mean.\n(a) For the group injected with IDX, find:\nThe mean, $\\bar x$=[ANS]\nThe standard deviation, $s$=[ANS]\n(b) For the untreated group, find:\nThe mean, $\\bar x$=[ANS]\nThe standard deviation, $s$=[ANS]", "answer_v1": [ "107.3", "8.5", "87.8", "5.9" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Scrapie is a degenerative disease of the nervous system. A study of the substance IDX as a treatment for scrapie used as subjects 16 infected hamsters. Eight, chosen at random, were injected with IDX. The other 8 were untreated. The researchers recorded how long each hamster lived. They reported: Thus, although all infected control hamsters had died by 95 days after infection (mean $\\pm$ SEM=88.8 $\\pm$ 1.84), IDX-treated hamsters lived up to 110 days (mean $\\pm$ SEM=100.5 $\\pm$ 3.22). Readers are supposed to know that SEM stands for the standard error of the mean.\n(a) For the group injected with IDX, find:\nThe mean, $\\bar x$=[ANS]\nThe standard deviation, $s$=[ANS]\n(b) For the untreated group, find:\nThe mean, $\\bar x$=[ANS]\nThe standard deviation, $s$=[ANS]", "answer_v2": [ "100.5", "9.1", "88.8", "5.2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Scrapie is a degenerative disease of the nervous system. A study of the substance IDX as a treatment for scrapie used as subjects 20 infected hamsters. Ten, chosen at random, were injected with IDX. The other 10 were untreated. The researchers recorded how long each hamster lived. They reported: Thus, although all infected control hamsters had died by 94 days after infection (mean $\\pm$ SEM=87.8 $\\pm$ 1.71), IDX-treated hamsters lived up to 112 days (mean $\\pm$ SEM=103.8 $\\pm$ 2.47). Readers are supposed to know that SEM stands for the standard error of the mean.\n(a) For the group injected with IDX, find:\nThe mean, $\\bar x$=[ANS]\nThe standard deviation, $s$=[ANS]\n(b) For the untreated group, find:\nThe mean, $\\bar x$=[ANS]\nThe standard deviation, $s$=[ANS]", "answer_v3": [ "103.8", "7.8", "87.8", "5.4" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Statistics_0052", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "3", "keywords": [ "statistics", "Inference", "one proportion" ], "problem_v1": "An article in the Washington Post on March 16, 1993 stated that nearly 45 percent of all Americans have brown eyes. A random sample of $n=80$ C of I students found 32 with brown eyes. Give the numerical value of the statistic $\\hat{p}$. $\\hat{p}=$ [ANS]", "answer_v1": [ "32/80" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "An article in the Washington Post on March 16, 1993 stated that nearly 45 percent of all Americans have brown eyes. A random sample of $n=50$ C of I students found 30 with brown eyes. Give the numerical value of the statistic $\\hat{p}$. $\\hat{p}=$ [ANS]", "answer_v2": [ "30/50" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "An article in the Washington Post on March 16, 1993 stated that nearly 45 percent of all Americans have brown eyes. A random sample of $n=60$ C of I students found 30 with brown eyes. Give the numerical value of the statistic $\\hat{p}$. $\\hat{p}=$ [ANS]", "answer_v3": [ "30/60" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0053", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "statistics", "skewed", "five number summary", "median", "boxplot" ], "problem_v1": "For each problem, select the best response.\n(a) Which of the following is least affected if an extreme high outlier is added to your data? [ANS] A. The mean. B. The standard deviation. C. The median. D. None of the above.\n(b) What percent of the observations in a distribution lie between the first quartile and the third quartile? [ANS] A. 50\\%. B. 25\\%. C. 75\\%. D. None of the above.\n(c) To make a boxplot of a distribution, you must know [ANS] A. the mean and the standard deviation. B. the five-number summary. C. all of the individual observations. D. None of the above.", "answer_v1": [ "C", "A", "B" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "For each problem, select the best response.\n(a) If a distribution is skewed to the right, [ANS] A. the mean and median are equal. B. the mean is less than the median. C. the mean is greater than the median. D. None of the above.\n(b) Which of the following is least affected if an extreme high outlier is added to your data? [ANS] A. The standard deviation. B. The mean. C. The median. D. None of the above.\n(c) What percent of the observations in a distribution lie between the first quartile and the third quartile? [ANS] A. 75\\%. B. 25\\%. C. 50\\%. D. None of the above.", "answer_v2": [ "C", "C", "C" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "For each problem, select the best response.\n(a) What percent of the observations in a distribution lie between the first quartile and the third quartile? [ANS] A. 75\\%. B. 50\\%. C. 25\\%. D. None of the above.\n(b) To make a boxplot of a distribution, you must know [ANS] A. all of the individual observations. B. the mean and the standard deviation. C. the five-number summary. D. None of the above.\n(c) If a distribution is skewed to the right, [ANS] A. the mean is greater than the median. B. the mean is less than the median. C. the mean and median are equal. D. None of the above.", "answer_v3": [ "B", "C", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0054", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "statistics", "mean", "median" ], "problem_v1": "Calculate the mean and median of the following data: 31, \\ 47, \\ 48, \\ 24, \\ 54, \\ 43, \\ 18, \\ 57\nMean=[ANS]\nMedian=[ANS]", "answer_v1": [ "40.25", "45" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Calculate the mean and median of the following data: 41, \\ 10, \\ 42, \\ 31, \\ 45, \\ 20, \\ 23, \\ 44\nMean=[ANS]\nMedian=[ANS]", "answer_v2": [ "32", "36" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Calculate the mean and median of the following data: 48, \\ 19, \\ 23, \\ 34, \\ 13, \\ 39, \\ 53, \\ 37\nMean=[ANS]\nMedian=[ANS]", "answer_v3": [ "33.25", "35.5" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0055", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "statistics", "skewed", "mean", "median" ], "problem_v1": "Calculate the mean and median of the following grades on a math test: 96, \\ 88, \\ 87, \\ 86, \\ 84, \\ 82, \\ 80, \\ 76, \\ 69, \\ 68, \\ 61\nMean=[ANS]\nMedian=[ANS]\nIs this data set skewed to the right, symmetric, or skewed to the left? [ANS]\n(Enter \"SKEWED RIGHT\", \"SYMMETRIC\", or \"SKEWED LEFT\");", "answer_v1": [ "79.7273", "82", "SKEWED LEFT" ], "answer_type_v1": [ "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "SKEWED RIGHT", "SYMMETRIC", "SKEWED LEFT" ] ], "problem_v2": "Calculate the mean and median of the following grades on a math test: 91, \\ 74, \\ 75, \\ 73, \\ 72, \\ 71, \\ 68, \\ 67, \\ 64, \\ 61, \\ 54\nMean=[ANS]\nMedian=[ANS]\nIs this data set skewed to the right, symmetric, or skewed to the left? [ANS]\n(Enter \"SKEWED RIGHT\", \"SYMMETRIC\", or \"SKEWED LEFT\");", "answer_v2": [ "70", "71", "SKEWED LEFT" ], "answer_type_v2": [ "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "SKEWED RIGHT", "SYMMETRIC", "SKEWED LEFT" ] ], "problem_v3": "Calculate the mean and median of the following grades on a math test: 88, \\ 83, \\ 85, \\ 82, \\ 78, \\ 75, \\ 73, \\ 71, \\ 65, \\ 64, \\ 57\nMean=[ANS]\nMedian=[ANS]\nIs this data set skewed to the right, symmetric, or skewed to the left? [ANS]\n(Enter \"SKEWED RIGHT\", \"SYMMETRIC\", or \"SKEWED LEFT\");", "answer_v3": [ "74.6364", "75", "SKEWED LEFT" ], "answer_type_v3": [ "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "SKEWED RIGHT", "SYMMETRIC", "SKEWED LEFT" ] ] }, { "id": "Statistics_0056", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "statistics", "quartile", "Q1", "Q2", "five number summary", "median" ], "problem_v1": "Find the five-number summary for the following 10 values: 31,\\ 28,\\ 32,\\ 26,\\ 42,\\ 39,\\ 36,\\ 24,\\ 41,\\ 34 NOTE: Different books define the word \"quartile\" in different ways which can lead to somewhat different results. In this exercise we define the first quartile to be the median of the bottom half of the data set, and the third quartile to be the median of the top half. Find the minimum: [ANS]\nFind $Q_{1}$: [ANS]\nFind the median: [ANS]\nFind $Q_{3}$: [ANS]\nFind the maximum: [ANS]", "answer_v1": [ "24", "28", "33", "39", "42" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Find the five-number summary for the following 10 values: 23,\\ 22,\\ 26,\\ 22,\\ 32,\\ 30,\\ 28,\\ 19,\\ 32,\\ 28 NOTE: Different books define the word \"quartile\" in different ways which can lead to somewhat different results. In this exercise we define the first quartile to be the median of the bottom half of the data set, and the third quartile to be the median of the top half. Find the minimum: [ANS]\nFind $Q_{1}$: [ANS]\nFind the median: [ANS]\nFind $Q_{3}$: [ANS]\nFind the maximum: [ANS]", "answer_v2": [ "19", "22", "27", "30", "32" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Find the five-number summary for the following 10 values: 40,\\ 40,\\ 29,\\ 27,\\ 26,\\ 24,\\ 23,\\ 32,\\ 21,\\ 37 NOTE: Different books define the word \"quartile\" in different ways which can lead to somewhat different results. In this exercise we define the first quartile to be the median of the bottom half of the data set, and the third quartile to be the median of the top half. Find the minimum: [ANS]\nFind $Q_{1}$: [ANS]\nFind the median: [ANS]\nFind $Q_{3}$: [ANS]\nFind the maximum: [ANS]", "answer_v3": [ "21", "24", "28", "37", "40" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Statistics_0057", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "1", "keywords": [ "statistics", "mean" ], "problem_v1": "Animals and people that take in more energy than they expend will get fatter. Here are data on 12 rhesus monkeys: 6 lean monkeys (4\\% to 9\\% body fat) and 6 obese monkeys (13\\% to 14\\% body fat). The data report the energy expended in 24 hours (kilojoules per minute) and the lean body mass (kilograms, leaving out fat) for each monkey.\n$\\begin{array}{cc}\\hline Lean & Obese \\\\ \\hline \\end{array}$\n$\\begin{array}{cccc}\\hline Body Mass & Energy & Body Mass & Energy \\\\ \\hline 6.8 & 1.17 & 8.1 & 0.91 \\\\ \\hline 7.6 & 1.04 & 9.6 & 1.35 \\\\ \\hline 9.1 & 1.44 & 10.4 & 1.15 \\\\ \\hline 10.2 & 1.66 & 12.5 & 1.48 \\\\ \\hline 9.3 & 1.05 & 12.2 & 1.25 \\\\ \\hline 9.2 & 1.14 & 10.4 & 1.32 \\\\ \\hline \\end{array}$\n(a) What is the mean lean body mass of the lean monkeys? ANSWER $\\bar{x}=$ [ANS] kilograms. (b) What is the mean lean body mass of the obese monkeys? ANSWER $\\bar{x}=$ [ANS] kilograms.", "answer_v1": [ "8.7", "10.5333" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Animals and people that take in more energy than they expend will get fatter. Here are data on 12 rhesus monkeys: 6 lean monkeys (4\\% to 9\\% body fat) and 6 obese monkeys (13\\% to 14\\% body fat). The data report the energy expended in 24 hours (kilojoules per minute) and the lean body mass (kilograms, leaving out fat) for each monkey.\n$\\begin{array}{cc}\\hline Lean & Obese \\\\ \\hline \\end{array}$\n$\\begin{array}{cccc}\\hline Body Mass & Energy & Body Mass & Energy \\\\ \\hline 6 & 1.22 & 7.7 & 0.93 \\\\ \\hline 8 & 1.02 & 9.3 & 1.38 \\\\ \\hline 8.6 & 1.42 & 10.6 & 1.11 \\\\ \\hline 9.8 & 1.64 & 12 & 1.41 \\\\ \\hline 10 & 1.06 & 12.2 & 1.21 \\\\ \\hline 9.2 & 1.15 & 10.7 & 1.32 \\\\ \\hline \\end{array}$\n(a) What is the mean lean body mass of the lean monkeys? ANSWER $\\bar{x}=$ [ANS] kilograms. (b) What is the mean lean body mass of the obese monkeys? ANSWER $\\bar{x}=$ [ANS] kilograms.", "answer_v2": [ "8.6", "10.4167" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Animals and people that take in more energy than they expend will get fatter. Here are data on 12 rhesus monkeys: 6 lean monkeys (4\\% to 9\\% body fat) and 6 obese monkeys (13\\% to 14\\% body fat). The data report the energy expended in 24 hours (kilojoules per minute) and the lean body mass (kilograms, leaving out fat) for each monkey.\n$\\begin{array}{cc}\\hline Lean & Obese \\\\ \\hline \\end{array}$\n$\\begin{array}{cccc}\\hline Body Mass & Energy & Body Mass & Energy \\\\ \\hline 6.3 & 1.1 & 8.3 & 0.92 \\\\ \\hline 7.6 & 1.06 & 10 & 1.39 \\\\ \\hline 8.8 & 1.5 & 10.9 & 1.17 \\\\ \\hline 10.1 & 1.71 & 12.2 & 1.45 \\\\ \\hline 9.2 & 1.06 & 11.8 & 1.24 \\\\ \\hline 9.2 & 1.14 & 10.4 & 1.37 \\\\ \\hline \\end{array}$\n(a) What is the mean lean body mass of the lean monkeys? ANSWER $\\bar{x}=$ [ANS] kilograms. (b) What is the mean lean body mass of the obese monkeys? ANSWER $\\bar{x}=$ [ANS] kilograms.", "answer_v3": [ "8.53333", "10.6" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0058", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [], "problem_v1": "Find the average (mean) of each group of numbers. a) 6, 9, 9 has mean [ANS]\nb)-5, 5,-6 has mean [ANS]\nc) 8, 12, 14, 17, 29 has mean [ANS]\nd)-17, 20,-19, 26, 25,-53 has mean [ANS]", "answer_v1": [ "8", "-2", "16", "-3" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Find the average (mean) of each group of numbers. a) 3, 12, 6 has mean [ANS]\nb)-7, 9,-8 has mean [ANS]\nc) 6, 10, 16, 13, 35 has mean [ANS]\nd)-15, 23,-21, 25, 23,-47 has mean [ANS]", "answer_v2": [ "7", "-2", "16", "-2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Find the average (mean) of each group of numbers. a) 4, 9, 8 has mean [ANS]\nb)-6, 5,-5 has mean [ANS]\nc) 9, 14, 18, 15, 14 has mean [ANS]\nd)-17, 15,-18, 29, 27,-48 has mean [ANS]", "answer_v3": [ "7", "-2", "14", "-2" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Statistics_0059", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [], "problem_v1": "Marietta and Patricia were planning a special dinner for their parents. They wanted to serve a grain or pasta. At the supermarket they examined the package labels and found these calories per serving size: 200 for brown rice, 230 for orzo, 200 for taboule, 180 for kasha, 210 for macaroni, 210 for linguini, 180 for white rice, and 180 for vermicelli. The mean of the data was [ANS]\nThe median of the data was [ANS]\nThe mode of the data was [ANS]", "answer_v1": [ "198.75", "200", "180" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Marietta and Patricia were planning a special dinner for their parents. They wanted to serve a grain or pasta. At the supermarket they examined the package labels and found these calories per serving size: 170 for brown rice, 210 for orzo, 170 for taboule, 150 for kasha, 180 for macaroni, 180 for linguini, 150 for white rice, and 150 for vermicelli. The mean of the data was [ANS]\nThe median of the data was [ANS]\nThe mode of the data was [ANS]", "answer_v2": [ "170", "170", "150" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Marietta and Patricia were planning a special dinner for their parents. They wanted to serve a grain or pasta. At the supermarket they examined the package labels and found these calories per serving size: 180 for brown rice, 210 for orzo, 180 for taboule, 160 for kasha, 190 for macaroni, 190 for linguini, 160 for white rice, and 160 for vermicelli. The mean of the data was [ANS]\nThe median of the data was [ANS]\nThe mode of the data was [ANS]", "answer_v3": [ "178.75", "180", "160" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Statistics_0060", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "Statistics", "Mean", "measures", "mode", "median" ], "problem_v1": "For the given data, find $\\Sigma x$, $n$, and $\\overline{x}$:\nx_1=20, \\ x_2=17, \\ x_3=18, \\ x_4=17, \\ x_5=22, \\ x_6=18, \\ x_7=12, \\ x_8=18\n$\\Sigma x=$ [ANS]\n$n=$ [ANS]\n$\\overline{x}=$ [ANS]", "answer_v1": [ "142", "8", "17.75" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "For the given data, find $\\Sigma x$, $n$, and $\\overline{x}$:\nx_1=10, \\ x_2=24, \\ x_3=8, \\ x_4=10, \\ x_5=16, \\ x_6=11, \\ x_7=4, \\ x_8=10\n$\\Sigma x=$ [ANS]\n$n=$ [ANS]\n$\\overline{x}=$ [ANS]", "answer_v2": [ "93", "8", "11.625" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "For the given data, find $\\Sigma x$, $n$, and $\\overline{x}$:\nx_1=17, \\ x_2=8, \\ x_3=11, \\ x_4=13, \\ x_5=13, \\ x_6=13, \\ x_7=17, \\ x_8=14\n$\\Sigma x=$ [ANS]\n$n=$ [ANS]\n$\\overline{x}=$ [ANS]", "answer_v3": [ "106", "8", "13.25" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Statistics_0061", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "4", "keywords": [], "problem_v1": "Calculate the mode, mean, and median of the following data:\n18, \\ 20, \\ 17, \\ 18, \\ 17, \\ 999, \\ 18, \\ 12 Mode=[ANS]\nMean=[ANS]\nMedian=[ANS]\nWhich measure of center does not work well here? [ANS] A. Median B. Mean C. Mode D. All of the above", "answer_v1": [ "18", "139.875", "18", "B" ], "answer_type_v1": [ "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v2": "Calculate the mode, mean, and median of the following data:\n10, \\ 10, \\ 24, \\ 8, \\ 10, \\ 999, \\ 11, \\ 6 Mode=[ANS]\nMean=[ANS]\nMedian=[ANS]\nWhich measure of center does not work well here? [ANS] A. Mode B. Mean C. Median D. All of the above", "answer_v2": [ "10", "134.75", "10", "B" ], "answer_type_v2": [ "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v3": "Calculate the mode, mean, and median of the following data:\n13, \\ 999, \\ 7, \\ 11, \\ 13, \\ 13, \\ 17, \\ 14 Mode=[ANS]\nMean=[ANS]\nMedian=[ANS]\nWhich measure of center does not work well here? [ANS] A. Mode B. Median C. Mean D. All of the above", "answer_v3": [ "13", "135.875", "13", "C" ], "answer_type_v3": [ "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0062", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "3", "keywords": [ "statistics", "introduction", "normal distribution" ], "problem_v1": "A standardized variable always has [ANS] A. mean 0 and changing standard deviation B. changing mean and standard deviation 1 C. mean 0 and standard deviation 1 D. changing mean and changing standard deviation\nThe z-score corresponding to an observed value of a variable tells you the number of standard deviations that the observation is from the mean [ANS] A. True B. False\nA positive z-score indicates that the observation is [ANS] A. above the mean B. below the mean", "answer_v1": [ "C", "A", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B" ], [ "A", "B" ] ], "problem_v2": "A standardized variable always has [ANS] A. mean 0 and standard deviation 1 B. changing mean and standard deviation 1 C. mean 0 and changing standard deviation D. changing mean and changing standard deviation\nThe z-score corresponding to an observed value of a variable tells you the number of standard deviations that the observation is from the mean [ANS] A. True B. False\nA positive z-score indicates that the observation is [ANS] A. above the mean B. below the mean", "answer_v2": [ "A", "A", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B" ], [ "A", "B" ] ], "problem_v3": "A standardized variable always has [ANS] A. mean 0 and changing standard deviation B. mean 0 and standard deviation 1 C. changing mean and standard deviation 1 D. changing mean and changing standard deviation\nThe z-score corresponding to an observed value of a variable tells you the number of standard deviations that the observation is from the mean [ANS] A. True B. False\nA positive z-score indicates that the observation is [ANS] A. above the mean B. below the mean", "answer_v3": [ "B", "A", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B" ], [ "A", "B" ] ] }, { "id": "Statistics_0063", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "3", "keywords": [ "Statistics", "Quartiles", "measures" ], "problem_v1": "Calculate the 5 number summary and the interquartile range of the following data: 32, \\ 65, \\ 40, \\ 58, \\ 52, \\ 36, \\ 24, \\ 21,\\ 41,\\ 48,\\ 75,\\ 16,\\-29,\\ 51,\\ 44, \\ 82\nQ1=[ANS]\nQ2=[ANS]\nQ3=[ANS]\nMin=[ANS]\nMax=[ANS]\nIQR=[ANS]\nThere is a potential outlier in this data set [ANS] A. True B. False", "answer_v1": [ "28", "42.5", "55", "-29", "82", "27", "A" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [], [], [], [ "A", "B" ] ], "problem_v2": "Calculate the 5 number summary and the interquartile range of the following data: 20, \\ 44, \\ 11, \\-5, \\ 19, \\ 29, \\ 15, \\ 1,\\ 27,\\ 37,\\ 51,\\ 23,\\ 31,\\ 3,\\ 61, \\-50\nQ1=[ANS]\nQ2=[ANS]\nQ3=[ANS]\nMin=[ANS]\nMax=[ANS]\nIQR=[ANS]\nThere is a potential outlier in this data set [ANS] A. True B. False", "answer_v2": [ "7", "21.5", "34", "-50", "61", "27", "A" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [], [], [], [ "A", "B" ] ], "problem_v3": "Calculate the 5 number summary and the interquartile range of the following data: 26, \\ 22, \\ 37, \\ 30, \\ 51, \\-43, \\ 64, \\ 10,\\ 27,\\ 18,\\ 7,\\ 2,\\ 44,\\ 38,\\ 68, \\ 34\nQ1=[ANS]\nQ2=[ANS]\nQ3=[ANS]\nMin=[ANS]\nMax=[ANS]\nIQR=[ANS]\nThere is a potential outlier in this data set [ANS] A. True B. False", "answer_v3": [ "14", "28.5", "41", "-43", "68", "27", "A" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [], [], [], [ "A", "B" ] ] }, { "id": "Statistics_0064", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "3", "keywords": [ "statistics", "measures" ], "problem_v1": "The length (in pages) of math research projects is given below. Using this information, calculate the range, variance, and standard deviation.\n27, \\ 21, \\ 26, \\ 31, \\ 21, \\ 36, \\ 23, \\ 14, \\ 299 range $=$ [ANS]\nvariance $=$ [ANS]\nstandard deviation $=$ [ANS]\nThe lack of what property of the standard deviation accounts for its sensitivity to the one data value that is significantly larger than the other ones? [ANS] A. Resistance B. Decreasing property C. All of the above", "answer_v1": [ "285", "8389.25", "91.5928490658523", "A" ], "answer_type_v1": [ "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B", "C" ] ], "problem_v2": "The length (in pages) of math research projects is given below. Using this information, calculate the range, variance, and standard deviation.\n25, \\ 31, \\ 10, \\ 35, \\ 28, \\ 299, \\ 25, \\ 26, \\ 38 range $=$ [ANS]\nvariance $=$ [ANS]\nstandard deviation $=$ [ANS]\nThe lack of what property of the standard deviation accounts for its sensitivity to the one data value that is significantly larger than the other ones? [ANS] A. Decreasing property B. Resistance C. All of the above", "answer_v2": [ "289", "8267.77777777778", "90.927321404393", "B" ], "answer_type_v2": [ "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B", "C" ] ], "problem_v3": "The length (in pages) of math research projects is given below. Using this information, calculate the range, variance, and standard deviation.\n299, \\ 30, \\ 22, \\ 22, \\ 23, \\ 11, \\ 26, \\ 36, \\ 25 range $=$ [ANS]\nvariance $=$ [ANS]\nstandard deviation $=$ [ANS]\nThe lack of what property of the standard deviation accounts for its sensitivity to the one data value that is significantly larger than the other ones? [ANS] A. Resistance B. Decreasing property C. All of the above", "answer_v3": [ "288", "8425.11111111111", "91.7884040122232", "A" ], "answer_type_v3": [ "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B", "C" ] ] }, { "id": "Statistics_0065", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "average" ], "problem_v1": "Consider the report cards for Sam and Samantha given below.\n$\\begin{array}{cccc}\\hline Sam & Samantha \\\\ \\hline Credits & Letter Grade & Credits & Letter Grade \\\\ \\hline 5 & D & 4 & C \\\\ \\hline 4 & C & 2 & C \\\\ \\hline 4 & C & 4 & F \\\\ \\hline 5 & D & 3 & F \\\\ \\hline 2 & C & 3 & D \\\\ \\hline \\end{array}$ D\n(a) Calculate Sam's GPA. [ANS]\n(b) Calculate Samantha's GPA. [ANS]", "answer_v1": [ "30/20", "15/16" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Consider the report cards for Sam and Samantha given below.\n$\\begin{array}{cccc}\\hline Sam & Samantha \\\\ \\hline Credits & Letter Grade & Credits & Letter Grade \\\\ \\hline 1 & D & 4 & D \\\\ \\hline 6 & F & 3 & C \\\\ \\hline 1 & D & 5 & F \\\\ \\hline 3 & C & 2 & D \\\\ \\hline 6 & F & 2 & C \\\\ \\hline \\end{array}$ C\n(a) Calculate Sam's GPA. [ANS]\n(b) Calculate Samantha's GPA. [ANS]", "answer_v2": [ "8/17", "16/16" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Consider the report cards for Sam and Samantha given below.\n$\\begin{array}{cccc}\\hline Sam & Samantha \\\\ \\hline Credits & Letter Grade & Credits & Letter Grade \\\\ \\hline 2 & D & 2 & B \\\\ \\hline 4 & A & 2 & B \\\\ \\hline 2 & A & 1 & F \\\\ \\hline 4 & A & 4 & D \\\\ \\hline 2 & D & 6 & B \\\\ \\hline \\end{array}$ B\n(a) Calculate Sam's GPA. [ANS]\n(b) Calculate Samantha's GPA. [ANS]", "answer_v3": [ "44/14", "34/15" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0066", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "average" ], "problem_v1": "Suppose there are three sections of a Statistics course taught by the same instructor. The class averages for each section on Test #1 are displayed in the table below.\n$\\begin{array}{ccc}\\hline & Class Size & Class Average \\\\ \\hline Section 01 & 25 & 85 \\\\ \\hline Section 02 & 20 & 76 \\\\ \\hline Section 03 & 21 & 76 \\\\ \\hline \\end{array}$ 76 What is the average test score for all sections combined? [ANS]", "answer_v1": [ "5241/66" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose there are three sections of a Statistics course taught by the same instructor. The class averages for each section on Test #1 are displayed in the table below.\n$\\begin{array}{ccc}\\hline & Class Size & Class Average \\\\ \\hline Section 01 & 8 & 77 \\\\ \\hline Section 02 & 29 & 89 \\\\ \\hline Section 03 & 10 & 76 \\\\ \\hline \\end{array}$ 76 What is the average test score for all sections combined? [ANS]", "answer_v2": [ "3957/47" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose there are three sections of a Statistics course taught by the same instructor. The class averages for each section on Test #1 are displayed in the table below.\n$\\begin{array}{ccc}\\hline & Class Size & Class Average \\\\ \\hline Section 01 & 14 & 81 \\\\ \\hline Section 02 & 21 & 74 \\\\ \\hline Section 03 & 13 & 77 \\\\ \\hline \\end{array}$ 77 What is the average test score for all sections combined? [ANS]", "answer_v3": [ "3689/48" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0068", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "average" ], "problem_v1": "Most credit cards charge interest based on the average daily balance per billing cycle. In this case each balance within the billing cycle is weighted by the number of days it exists. Suppose your credit card has a 30 day billing cycle and the balances over these 30 days are given in the table below.\n$\\begin{array}{cccc}\\hline Days & Transaction & balance & # days \\\\ \\hline 1-8 & remaining balance & \\$1500 & 8 \\\\ \\hline 9-14 & \\$400 purchase & \\$1900 & 6 \\\\ \\hline 15-20 & \\$400 purchase & \\$2300 & 6 \\\\ \\hline 21-30 & \\$1400 payment & \\$900 & 10 \\\\ \\hline \\end{array}$ 10\n(a) Calculate the average daily balance for these 30 days. [ANS]\n(b) What would have been your average daily balance if you paid \\$1400 on day 17 instead of \\$1400 on day 21? [ANS]", "answer_v1": [ "46200/30", "40600/30" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Most credit cards charge interest based on the average daily balance per billing cycle. In this case each balance within the billing cycle is weighted by the number of days it exists. Suppose your credit card has a 30 day billing cycle and the balances over these 30 days are given in the table below.\n$\\begin{array}{cccc}\\hline Days & Transaction & balance & # days \\\\ \\hline 1-2 & remaining balance & \\$700 & 2 \\\\ \\hline 3-11 & \\$1000 purchase & \\$1700 & 9 \\\\ \\hline 12-14 & \\$400 purchase & \\$2100 & 3 \\\\ \\hline 15-30 & \\$400 payment & \\$1700 & 16 \\\\ \\hline \\end{array}$ 16\n(a) Calculate the average daily balance for these 30 days. [ANS]\n(b) What would have been your average daily balance if you paid \\$700 on day 14 instead of \\$400 on day 15? [ANS]", "answer_v2": [ "50200/30", "44700/30" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Most credit cards charge interest based on the average daily balance per billing cycle. In this case each balance within the billing cycle is weighted by the number of days it exists. Suppose your credit card has a 30 day billing cycle and the balances over these 30 days are given in the table below.\n$\\begin{array}{cccc}\\hline Days & Transaction & balance & # days \\\\ \\hline 1-4 & remaining balance & \\$1100 & 4 \\\\ \\hline 5-10 & \\$300 purchase & \\$1400 & 6 \\\\ \\hline 11-14 & \\$400 purchase & \\$1800 & 4 \\\\ \\hline 15-30 & \\$1500 payment & \\$300 & 16 \\\\ \\hline \\end{array}$ 16\n(a) Calculate the average daily balance for these 30 days. [ANS]\n(b) What would have been your average daily balance if you paid \\$1700 on day 14 instead of \\$1500 on day 15? [ANS]", "answer_v3": [ "24800/30", "19900/30" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0069", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "five number summary" ], "problem_v1": "39, 31, 33, 37, 18, 19, 30, 30, 21 Consider the following data set. Give the five number summary listing values in numerical order: [ANS], [ANS], [ANS], [ANS], [ANS].", "answer_v1": [ "18", "20", "30", "35", "39" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "7, 47, 11, 19, 48, 18, 12, 19, 30 Consider the following data set. Give the five number summary listing values in numerical order: [ANS], [ANS], [ANS], [ANS], [ANS].", "answer_v2": [ "7", "11.5", "19", "38.5", "48" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "18, 32, 17, 29, 13, 20, 42, 46, 45 Consider the following data set. Give the five number summary listing values in numerical order: [ANS], [ANS], [ANS], [ANS], [ANS].", "answer_v3": [ "13", "17.5", "29", "43.5", "46" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Statistics_0070", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "4", "keywords": [], "problem_v1": "A statistician had a data set containing 13 data points written in his research notebook. He spilled coffee on his notebook and now he cannot read two of the data values. He remembers that the sample mean of the original data set was 28.538 and the sample median was 26. Use your power of deduction and the 11 still readable data values given below, to determine the range of possible values for the two lost data points.\n27, \\ 26, \\ 23, \\ 31, \\ 36, \\ 14, \\ 21, \\ 41, \\ 47, \\ 53, \\ 21 Smallest possible lost data value: $=$ [ANS]\nLargest possible lost data value: $=$ [ANS]", "answer_v1": [ "13", "18" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A statistician had a data set containing 13 data points written in his research notebook. He spilled coffee on his notebook and now he cannot read two of the data values. He remembers that the sample mean of the original data set was 30.462 and the sample median was 28. Use your power of deduction and the 11 still readable data values given below, to determine the range of possible values for the two lost data points.\n42, \\ 25, \\ 25, \\ 28, \\ 38, \\ 26, \\ 31, \\ 49, \\ 10, \\ 35, \\ 54 Smallest possible lost data value: $=$ [ANS]\nLargest possible lost data value: $=$ [ANS]", "answer_v2": [ "13", "20" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A statistician had a data set containing 13 data points written in his research notebook. He spilled coffee on his notebook and now he cannot read two of the data values. He remembers that the sample mean of the original data set was 28.538 and the sample median was 25. Use your power of deduction and the 11 still readable data values given below, to determine the range of possible values for the two lost data points.\n11, \\ 30, \\ 55, \\ 42, \\ 26, \\ 22, \\ 23, \\ 36, \\ 50, \\ 25, \\ 22 Smallest possible lost data value: $=$ [ANS]\nLargest possible lost data value: $=$ [ANS]", "answer_v3": [ "11", "18" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0071", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "center measures", "mean", "median", "mode" ], "problem_v1": "Consider the following data set. Find the mean, median and mode(s). Record answers that are not integers to two decimal places.\nNOTES: (1) If there is more than one mode, list them all, separated by commas. (2) If there is no mode, enter \"None\".\nData set: 5, 11, 11, 3, 3, 10, 19, 17, 7, 12, 18, 18, 16, 10, 19, 19, 5, 6, 9\nMean: [ANS]\nMedian: [ANS]\nMode: [ANS]", "answer_v1": [ "11.4736842105263", "11", "19" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Consider the following data set. Find the mean, median and mode(s). Record answers that are not integers to two decimal places.\nNOTES: (1) If there is more than one mode, list them all, separated by commas. (2) If there is no mode, enter \"None\".\nData set: 3, 3, 2, 7, 10, 2, 18, 7, 8, 14, 19, 10, 11, 18, 2, 9, 9, 15, 19, 9, 2\nMean: [ANS]\nMedian: [ANS]\nMode: [ANS]", "answer_v2": [ "9.38095238095238", "9", "2" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Consider the following data set. Find the mean, median and mode(s). Record answers that are not integers to two decimal places.\nNOTES: (1) If there is more than one mode, list them all, separated by commas. (2) If there is no mode, enter \"None\".\nData set: 13, 6, 11, 5, 7, 17, 19, 18, 5, 6, 6, 1, 12, 20, 16, 13, 3, 6, 13\nMean: [ANS]\nMedian: [ANS]\nMode: [ANS]", "answer_v3": [ "10.3684210526316", "11", "6" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Statistics_0072", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "4", "keywords": [], "problem_v1": "Consider the data set given below:\n27, \\ 26, \\ 23, \\ 31, \\ 36, \\ 14, \\ 21, \\ 41, \\ 47, \\ 53, \\ 21 a)Find the minimum value of the data set: [ANS]\nb)Find the maximum value of the data set: [ANS]\nc)Find the arithmetic mean of the data set: [ANS]\nd)Find the median of the data set: [ANS]\nLet's include one more data point in the data set. Suppose this new data value lies between the values 14 and 53, inclusively. e)Find the smallest possible value of the mean of the new data set: [ANS]\nf)Find the largest possible value of the mean of the new data set: [ANS]\ng)Find the smallest possible value of the median of the new data set: [ANS]\nh)Find the largest possible value of the median of the new data set: [ANS]", "answer_v1": [ "14", "53", "30.909", "27", "29.5", "32.75", "26.5", "29" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Consider the data set given below:\n42, \\ 25, \\ 25, \\ 28, \\ 38, \\ 26, \\ 31, \\ 49, \\ 10, \\ 35, \\ 54 a)Find the minimum value of the data set: [ANS]\nb)Find the maximum value of the data set: [ANS]\nc)Find the arithmetic mean of the data set: [ANS]\nd)Find the median of the data set: [ANS]\nLet's include one more data point in the data set. Suppose this new data value lies between the values 10 and 54, inclusively. e)Find the smallest possible value of the mean of the new data set: [ANS]\nf)Find the largest possible value of the mean of the new data set: [ANS]\ng)Find the smallest possible value of the median of the new data set: [ANS]\nh)Find the largest possible value of the median of the new data set: [ANS]", "answer_v2": [ "10", "54", "33", "31", "31.083", "34.75", "29.5", "33" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Consider the data set given below:\n11, \\ 30, \\ 55, \\ 42, \\ 26, \\ 22, \\ 23, \\ 36, \\ 50, \\ 25, \\ 22 a)Find the minimum value of the data set: [ANS]\nb)Find the maximum value of the data set: [ANS]\nc)Find the arithmetic mean of the data set: [ANS]\nd)Find the median of the data set: [ANS]\nLet's include one more data point in the data set. Suppose this new data value lies between the values 11 and 55, inclusively. e)Find the smallest possible value of the mean of the new data set: [ANS]\nf)Find the largest possible value of the mean of the new data set: [ANS]\ng)Find the smallest possible value of the median of the new data set: [ANS]\nh)Find the largest possible value of the median of the new data set: [ANS]", "answer_v3": [ "11", "55", "31.091", "26", "29.417", "33.083", "25.5", "28" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Statistics_0073", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "statistics", "descriptive statistics", "spread" ], "problem_v1": "Use the table below to find the standard deviation of the following data set: Use at least three decimal places of accuracy at each step. Use at least three decimal places of accuracy at each step.\n22, 23, 33, 35, 39, 40\n$\\begin{array}{ccc}\\hline X & X-\\bar{x} & \\left(X-\\bar{x}\\right)^2 \\\\ \\hline 22 & [ANS] & [ANS] \\\\ \\hline 23 & [ANS] & [ANS] \\\\ \\hline 33 & [ANS] & [ANS] \\\\ \\hline 35 & [ANS] & [ANS] \\\\ \\hline 39 & [ANS] & [ANS] \\\\ \\hline 40 & [ANS] & [ANS] \\\\ \\hline & & \\\\ \\hline & SUM & [ANS] \\\\ \\hline & \\frac{SUM}{N-1} & [ANS] \\\\ \\hline & \\sqrt{\\frac{SUM}{N-1}} & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "-10", "100", "-9", "81", "1", "1", "3", "9", "7", "49", "8", "64", "304", "60.8", "7.79743547584717" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Use the table below to find the standard deviation of the following data set: Use at least three decimal places of accuracy at each step. Use at least three decimal places of accuracy at each step.\n13, 16, 22, 23, 48, 48\n$\\begin{array}{ccc}\\hline X & X-\\bar{x} & \\left(X-\\bar{x}\\right)^2 \\\\ \\hline 13 & [ANS] & [ANS] \\\\ \\hline 16 & [ANS] & [ANS] \\\\ \\hline 22 & [ANS] & [ANS] \\\\ \\hline 23 & [ANS] & [ANS] \\\\ \\hline 48 & [ANS] & [ANS] \\\\ \\hline 48 & [ANS] & [ANS] \\\\ \\hline & & \\\\ \\hline & SUM & [ANS] \\\\ \\hline & \\frac{SUM}{N-1} & [ANS] \\\\ \\hline & \\sqrt{\\frac{SUM}{N-1}} & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "-15.3333333333333", "235.111111111111", "-12.3333333333333", "152.111111111111", "-6.33333333333333", "40.1111111111111", "-5.33333333333333", "28.4444444444444", "19.6666666666667", "386.777777777778", "19.6666666666667", "386.777777777778", "1229.33333333333", "245.866666666667", "15.6801360538315" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Use the table below to find the standard deviation of the following data set: Use at least three decimal places of accuracy at each step. Use at least three decimal places of accuracy at each step.\n18, 21, 22, 24, 32, 34\n$\\begin{array}{ccc}\\hline X & X-\\bar{x} & \\left(X-\\bar{x}\\right)^2 \\\\ \\hline 18 & [ANS] & [ANS] \\\\ \\hline 21 & [ANS] & [ANS] \\\\ \\hline 22 & [ANS] & [ANS] \\\\ \\hline 24 & [ANS] & [ANS] \\\\ \\hline 32 & [ANS] & [ANS] \\\\ \\hline 34 & [ANS] & [ANS] \\\\ \\hline & & \\\\ \\hline & SUM & [ANS] \\\\ \\hline & \\frac{SUM}{N-1} & [ANS] \\\\ \\hline & \\sqrt{\\frac{SUM}{N-1}} & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "-7.16666666666667", "51.3611111111111", "-4.16666666666667", "17.3611111111111", "-3.16666666666667", "10.0277777777778", "-1.16666666666667", "1.36111111111111", "6.83333333333333", "46.6944444444444", "8.83333333333333", "78.0277777777778", "204.833333333333", "40.9666666666667", "6.40052081214229" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Statistics_0074", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "mean", "standard deviation" ], "problem_v1": "Consider the following data set. Find the mean and standard deviation. Data set: 69, 72, 79, 47, 50, 68, 68\nMean: [ANS]\nStandard deviation: [ANS]", "answer_v1": [ "64.7142857142857", "11.7432858223226" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Consider the following data set. Find the mean and standard deviation. Data set: 95, 36, 50, 96, 49\nMean: [ANS]\nStandard deviation: [ANS]", "answer_v2": [ "65.2", "28.208154849263" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Consider the following data set. Find the mean and standard deviation. Data set: 71, 46, 66, 40, 51\nMean: [ANS]\nStandard deviation: [ANS]", "answer_v3": [ "54.8", "13.2174127574197" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0075", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "3", "keywords": [ "statistics", "frequency" ], "problem_v1": "The U.S. Bureau of the Census conducts nationwide surveys on characteristics of U.S. households. Following are data on the number of people per household for a sample of 50 households. Construct a grouped data table for these household sizes.\n$\\begin{array}{cccccccccc}\\hline 5 & 5 & 4 & 6 & 3 & 1 & 4 & 4 & 4 & 5 \\\\ \\hline 4 & 3 & 4 & 3 & 3 & 4 & 4 & 1 & 2 & 3 \\\\ \\hline 4 & 4 & 4 & 2 & 4 & 5 & 5 & 3 & 1 & 2 \\\\ \\hline 3 & 1 & 3 & 6 & 6 & 4 & 2 & 3 & 6 & 2 \\\\ \\hline 2 & 2 & 2 & 3 & 2 & 4 & 4 & 2 & 1 & 4 \\\\ \\hline \\end{array}$\n$\\begin{array}{ccc}\\hline Household size & Frequency & Relative Frequency \\\\ \\hline 1 & [ANS] & [ANS] \\\\ \\hline 2 & [ANS] & [ANS] \\\\ \\hline 3 & [ANS] & [ANS] \\\\ \\hline 4 & [ANS] & [ANS] \\\\ \\hline 5 & [ANS] & [ANS] \\\\ \\hline 6 & [ANS] & [ANS] \\\\ \\hline 7 & [ANS] & [ANS] \\\\ \\hline Total & 50 & 1 \\\\ \\hline \\end{array}$", "answer_v1": [ "5", "0.1", "10", "0.2", "10", "0.2", "16", "0.32", "5", "0.1", "4", "0.08", "0", "0" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "The U.S. Bureau of the Census conducts nationwide surveys on characteristics of U.S. households. Following are data on the number of people per household for a sample of 50 households. Construct a grouped data table for these household sizes.\n$\\begin{array}{cccccccccc}\\hline 1 & 7 & 1 & 3 & 7 & 1 & 2 & 3 & 5 & 3 \\\\ \\hline 4 & 4 & 5 & 2 & 2 & 2 & 5 & 2 & 3 & 4 \\\\ \\hline 3 & 1 & 1 & 2 & 4 & 3 & 5 & 3 & 3 & 5 \\\\ \\hline 5 & 3 & 4 & 7 & 1 & 4 & 3 & 5 & 7 & 3 \\\\ \\hline 1 & 2 & 3 & 3 & 2 & 2 & 3 & 6 & 6 & 4 \\\\ \\hline \\end{array}$\n$\\begin{array}{ccc}\\hline Household size & Frequency & Relative Frequency \\\\ \\hline 1 & [ANS] & [ANS] \\\\ \\hline 2 & [ANS] & [ANS] \\\\ \\hline 3 & [ANS] & [ANS] \\\\ \\hline 4 & [ANS] & [ANS] \\\\ \\hline 5 & [ANS] & [ANS] \\\\ \\hline 6 & [ANS] & [ANS] \\\\ \\hline 7 & [ANS] & [ANS] \\\\ \\hline Total & 50 & 1 \\\\ \\hline \\end{array}$", "answer_v2": [ "7", "0.14", "9", "0.18", "14", "0.28", "7", "0.14", "7", "0.14", "2", "0.04", "4", "0.08" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "The U.S. Bureau of the Census conducts nationwide surveys on characteristics of U.S. households. Following are data on the number of people per household for a sample of 50 households. Construct a grouped data table for these household sizes.\n$\\begin{array}{cccccccccc}\\hline 2 & 5 & 2 & 4 & 2 & 1 & 5 & 6 & 6 & 4 \\\\ \\hline 2 & 3 & 3 & 5 & 7 & 6 & 5 & 1 & 3 & 5 \\\\ \\hline 5 & 3 & 3 & 4 & 6 & 5 & 3 & 4 & 7 & 6 \\\\ \\hline 1 & 3 & 7 & 6 & 2 & 5 & 4 & 2 & 4 & 4 \\\\ \\hline 2 & 2 & 3 & 3 & 1 & 6 & 3 & 3 & 1 & 3 \\\\ \\hline \\end{array}$\n$\\begin{array}{ccc}\\hline Household size & Frequency & Relative Frequency \\\\ \\hline 1 & [ANS] & [ANS] \\\\ \\hline 2 & [ANS] & [ANS] \\\\ \\hline 3 & [ANS] & [ANS] \\\\ \\hline 4 & [ANS] & [ANS] \\\\ \\hline 5 & [ANS] & [ANS] \\\\ \\hline 6 & [ANS] & [ANS] \\\\ \\hline 7 & [ANS] & [ANS] \\\\ \\hline Total & 50 & 1 \\\\ \\hline \\end{array}$", "answer_v3": [ "5", "0.1", "8", "0.16", "12", "0.24", "7", "0.14", "8", "0.16", "7", "0.14", "3", "0.06" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Statistics_0076", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "3", "keywords": [ "Frequency", "Relative", "Cumulative" ], "problem_v1": "$\\begin{array}{cc}\\hline Grade on Statistics Exam & Frequency \\\\ \\hline Below 50 & 8 \\\\ \\hline 50-59 & 6 \\\\ \\hline 60-69 & 11 \\\\ \\hline 70-79 & 17 \\\\ \\hline 80-89 & 13 \\\\ \\hline 90-100 & 8 \\\\ \\hline \\end{array}$\nGiven the frequency table above, construct the following:\n(a) The relative frequency table that corresponds with the above table.\n$\\begin{array}{cc}\\hline Grade on Statistics Exam & Relative Frequency \\\\ \\hline Below 50 & [ANS] \\\\ \\hline 50-59 & [ANS] \\\\ \\hline 60-69 & [ANS] \\\\ \\hline 70-79 & [ANS] \\\\ \\hline 80-89 & [ANS] \\\\ \\hline 90-100 & [ANS] \\\\ \\hline \\end{array}$\n(b) The cumulative frequency table that corresponds with the above table.\n$\\begin{array}{cc}\\hline Grade on Statistics Exam & Cumulative Frequency \\\\ \\hline Below 50 & [ANS] \\\\ \\hline 50-59 & [ANS] \\\\ \\hline 60-69 & [ANS] \\\\ \\hline 70-79 & [ANS] \\\\ \\hline 80-89 & [ANS] \\\\ \\hline 90-100 & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "0.126984126984127", "0.0952380952380952", "0.174603174603175", "0.26984126984127", "0.206349206349206", "0.126984126984127", "8", "14", "25", "42", "55", "63" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "$\\begin{array}{cc}\\hline Grade on Statistics Exam & Frequency \\\\ \\hline Below 50 & 1 \\\\ \\hline 50-59 & 10 \\\\ \\hline 60-69 & 6 \\\\ \\hline 70-79 & 13 \\\\ \\hline 80-89 & 20 \\\\ \\hline 90-100 & 8 \\\\ \\hline \\end{array}$\nGiven the frequency table above, construct the following:\n(a) The relative frequency table that corresponds with the above table.\n$\\begin{array}{cc}\\hline Grade on Statistics Exam & Relative Frequency \\\\ \\hline Below 50 & [ANS] \\\\ \\hline 50-59 & [ANS] \\\\ \\hline 60-69 & [ANS] \\\\ \\hline 70-79 & [ANS] \\\\ \\hline 80-89 & [ANS] \\\\ \\hline 90-100 & [ANS] \\\\ \\hline \\end{array}$\n(b) The cumulative frequency table that corresponds with the above table.\n$\\begin{array}{cc}\\hline Grade on Statistics Exam & Cumulative Frequency \\\\ \\hline Below 50 & [ANS] \\\\ \\hline 50-59 & [ANS] \\\\ \\hline 60-69 & [ANS] \\\\ \\hline 70-79 & [ANS] \\\\ \\hline 80-89 & [ANS] \\\\ \\hline 90-100 & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "0.0172413793103448", "0.172413793103448", "0.103448275862069", "0.224137931034483", "0.344827586206897", "0.137931034482759", "1", "11", "17", "30", "50", "58" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "$\\begin{array}{cc}\\hline Grade on Statistics Exam & Frequency \\\\ \\hline Below 50 & 4 \\\\ \\hline 50-59 & 7 \\\\ \\hline 60-69 & 8 \\\\ \\hline 70-79 & 16 \\\\ \\hline 80-89 & 12 \\\\ \\hline 90-100 & 8 \\\\ \\hline \\end{array}$\nGiven the frequency table above, construct the following:\n(a) The relative frequency table that corresponds with the above table.\n$\\begin{array}{cc}\\hline Grade on Statistics Exam & Relative Frequency \\\\ \\hline Below 50 & [ANS] \\\\ \\hline 50-59 & [ANS] \\\\ \\hline 60-69 & [ANS] \\\\ \\hline 70-79 & [ANS] \\\\ \\hline 80-89 & [ANS] \\\\ \\hline 90-100 & [ANS] \\\\ \\hline \\end{array}$\n(b) The cumulative frequency table that corresponds with the above table.\n$\\begin{array}{cc}\\hline Grade on Statistics Exam & Cumulative Frequency \\\\ \\hline Below 50 & [ANS] \\\\ \\hline 50-59 & [ANS] \\\\ \\hline 60-69 & [ANS] \\\\ \\hline 70-79 & [ANS] \\\\ \\hline 80-89 & [ANS] \\\\ \\hline 90-100 & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "0.0727272727272727", "0.127272727272727", "0.145454545454545", "0.290909090909091", "0.218181818181818", "0.145454545454545", "4", "11", "19", "35", "47", "55" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Statistics_0077", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "statistics", "mean", "median" ], "problem_v1": "Given the following table, compute the median of the grouped data.\n$\\begin{array}{ccc}\\hline Class & Frequency & Cumulative Frequency \\\\ \\hline [7,13) & 1 & [ANS] \\\\ \\hline [13,19) & 4 & [ANS] \\\\ \\hline [19,25) & 7 & [ANS] \\\\ \\hline [25,31) & 8 & [ANS] \\\\ \\hline [31,37) & 6 & [ANS] \\\\ \\hline [37,43) & 4 & [ANS] \\\\ \\hline [43,49) & 1 & [ANS] \\\\ \\hline \\end{array}$\nWhat is the median of the grouped data? [ANS]", "answer_v1": [ "1", "5", "12", "20", "26", "30", "31", "27.625" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Given the following table, compute the median of the grouped data.\n$\\begin{array}{ccc}\\hline Class & Frequency & Cumulative Frequency \\\\ \\hline [14,16) & 0 & [ANS] \\\\ \\hline [16,18) & 3 & [ANS] \\\\ \\hline [18,20) & 11 & [ANS] \\\\ \\hline [20,22) & 8 & [ANS] \\\\ \\hline [22,24) & 2 & [ANS] \\\\ \\hline [24,26) & 3 & [ANS] \\\\ \\hline [26,28) & 2 & [ANS] \\\\ \\hline \\end{array}$\nWhat is the median of the grouped data? [ANS]", "answer_v2": [ "0", "3", "14", "22", "24", "27", "29", "20.125" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Given the following table, compute the median of the grouped data.\n$\\begin{array}{ccc}\\hline Class & Frequency & Cumulative Frequency \\\\ \\hline [7,10) & 0 & [ANS] \\\\ \\hline [10,13) & 4 & [ANS] \\\\ \\hline [13,16) & 7 & [ANS] \\\\ \\hline [16,19) & 8 & [ANS] \\\\ \\hline [19,22) & 9 & [ANS] \\\\ \\hline [22,25) & 5 & [ANS] \\\\ \\hline [25,28) & 0 & [ANS] \\\\ \\hline \\end{array}$\nWhat is the median of the grouped data? [ANS]", "answer_v3": [ "0", "4", "11", "19", "28", "33", "33", "18.0625" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Statistics_0078", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "statistics", "numerical", "descriptive statistics", "variance" ], "problem_v1": "A population of grades for a statistics class of six students is given below:\n83 \\quad 74 \\quad 71 \\quad 84 \\quad 89 \\quad 65 Find the variance for this population. Population Variance=[ANS]", "answer_v1": [ "69.2222222222222" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A population of grades for a statistics class of six students is given below:\n70 \\quad 70 \\quad 90 \\quad 63 \\quad 91 \\quad 85 Find the variance for this population. Population Variance=[ANS]", "answer_v2": [ "119.138888888889" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A population of grades for a statistics class of six students is given below:\n84 \\quad 90 \\quad 83 \\quad 63 \\quad 70 \\quad 71 Find the variance for this population. Population Variance=[ANS]", "answer_v3": [ "89.1388888888889" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0079", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "4", "keywords": [ "statistics", "numerical", "descriptive statistics", "mean" ], "problem_v1": "Suppose that the mean score of a class of 34 students was 77. The 18 male students in the class had a mean score of 70. What was the mean score for the 16 female students? Mean Score for Female Students=[ANS]", "answer_v1": [ "84.875" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that the mean score of a class of 28 students was 74. The 20 male students in the class had a mean score of 64. What was the mean score for the 8 female students? Mean Score for Female Students=[ANS]", "answer_v2": [ "99" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that the mean score of a class of 30 students was 75. The 18 male students in the class had a mean score of 66. What was the mean score for the 12 female students? Mean Score for Female Students=[ANS]", "answer_v3": [ "88.5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0080", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "statistics", "descriptive statistics", "standard deviation", "variance" ], "problem_v1": "Select True or False, depending on whether the corresponding statement is true or false.\n[ANS] 1. The standard deviation will always exceed that of the variance. [ANS] 2. The standard deviation is the positive square root of the variance. [ANS] 3. The standard deviation is expressed in terms of the original units of measurement but the variance is not. [ANS] 4. The value of the standard deviation may be either positive or negative, while the value of the variance will always be positive.", "answer_v1": [ "F", "T", "T", "F" ], "answer_type_v1": [ "TF", "TF", "TF", "TF" ], "options_v1": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v2": "Select True or False, depending on whether the corresponding statement is true or false.\n[ANS] 1. The standard deviation is the positive square root of the variance. [ANS] 2. The value of the standard deviation may be either positive or negative, while the value of the variance will always be positive. [ANS] 3. The standard deviation will always exceed that of the variance. [ANS] 4. The difference between the largest and smallest values in an ordered data set is called the range.", "answer_v2": [ "T", "F", "F", "T" ], "answer_type_v2": [ "TF", "TF", "TF", "TF" ], "options_v2": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v3": "Select True or False, depending on whether the corresponding statement is true or false.\n[ANS] 1. Quartiles divide the values in a data set into four parts of equal size. [ANS] 2. The difference between the largest and smallest values in an ordered data set is called the range. [ANS] 3. The standard deviation will always exceed that of the variance. [ANS] 4. The interquartile range is found by taking the difference between the 1st and 3rd quartiles and dividing that value by 2.", "answer_v3": [ "T", "T", "F", "F" ], "answer_type_v3": [ "TF", "TF", "TF", "TF" ], "options_v3": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ] }, { "id": "Statistics_0081", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "statistics", "numerical", "descriptive statistics", "range", "variance" ], "problem_v1": "A sample of prices for six cordless drills available at a local home improvement store is given below:\n242 \\quad 216 \\quad 202 \\quad 250 \\quad 286 \\quad 188 Find the range and variance for this sample. Sample Range=[ANS]\nSample Variance=[ANS]", "answer_v1": [ "98", "1284.26666666667" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A sample of prices for six cordless drills available at a local home improvement store is given below:\n196 \\quad 200 \\quad 270 \\quad 176 \\quad 294 \\quad 250 Find the range and variance for this sample. Sample Range=[ANS]\nSample Variance=[ANS]", "answer_v2": [ "118", "2212.4" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A sample of prices for six cordless drills available at a local home improvement store is given below:\n248 \\quad 286 \\quad 242 \\quad 180 \\quad 200 \\quad 206 Find the range and variance for this sample. Sample Range=[ANS]\nSample Variance=[ANS]", "answer_v3": [ "106", "1505.2" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0082", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "statistics", "numerical", "descriptive statistics", "quartile" ], "problem_v1": "A sample of eight math SAT scores is given below:\n600 \\quad 640 \\quad 640 \\quad 560 \\quad 680 \\quad 610 \\quad 540 \\quad 720 Find the first and third quartiles and the interquartile range for this sample. First Quartile=[ANS]\nThird Quartile=[ANS]\nInterquartile Range=[ANS]", "answer_v1": [ "580", "660", "80" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A sample of eight math SAT scores is given below:\n650 \\quad 480 \\quad 650 \\quad 580 \\quad 720 \\quad 540 \\quad 580 \\quad 680 Find the first and third quartiles and the interquartile range for this sample. First Quartile=[ANS]\nThird Quartile=[ANS]\nInterquartile Range=[ANS]", "answer_v2": [ "560", "665", "105" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A sample of eight math SAT scores is given below:\n690 \\quad 550 \\quad 590 \\quad 590 \\quad 520 \\quad 650 \\quad 740 \\quad 640 Find the first and third quartiles and the interquartile range for this sample. First Quartile=[ANS]\nThird Quartile=[ANS]\nInterquartile Range=[ANS]", "answer_v3": [ "570", "670", "100" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Statistics_0083", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "statistics", "numerical", "descriptive statistics", "mean", "median", "mode" ], "problem_v1": "A sample of prices for eight television sets available at a local electronics store is given below:\n242 \\quad 202 \\quad 216 \\quad 250 \\quad 188 \\quad 250 \\quad 178 \\quad 298 Find the mean, median, and mode for this sample. Sample\nMean=[ANS]\nSample Median=[ANS]\nSample Mode=[ANS]", "answer_v1": [ "228", "229", "250" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A sample of prices for eight television sets available at a local electronics store is given below:\n196 \\quad 250 \\quad 150 \\quad 258 \\quad 200 \\quad 288 \\quad 176 \\quad 258 Find the mean, median, and mode for this sample. Sample\nMean=[ANS]\nSample Median=[ANS]\nSample Mode=[ANS]", "answer_v2": [ "222", "225", "258" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A sample of prices for eight television sets available at a local electronics store is given below:\n308 \\quad 250 \\quad 180 \\quad 200 \\quad 206 \\quad 170 \\quad 250 \\quad 242 Find the mean, median, and mode for this sample. Sample\nMean=[ANS]\nSample Median=[ANS]\nSample Mode=[ANS]", "answer_v3": [ "225.75", "224", "250" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Statistics_0084", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "statistics", "numerical", "descriptive statistics", "mean", "median" ], "problem_v1": "Monthly rent data in dollars for a sample of one-bedroom apartments in a mid-size Virginia college town is given below:\n485 \\quad 515 \\quad 520 \\quad 455 \\quad 550 \\quad 500 \\quad 445 \\quad 580 Find the mean and median for this sample. Sample\nMean=[ANS]\nSample Median=[ANS]", "answer_v1": [ "506.25", "507.5" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Monthly rent data in dollars for a sample of one-bedroom apartments in a mid-size Virginia college town is given below:\n525 \\quad 410 \\quad 530 \\quad 485 \\quad 580 \\quad 455 \\quad 485 \\quad 550 Find the mean and median for this sample. Sample\nMean=[ANS]\nSample Median=[ANS]", "answer_v2": [ "502.5", "505" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Monthly rent data in dollars for a sample of one-bedroom apartments in a mid-size Virginia college town is given below:\n565 \\quad 455 \\quad 485 \\quad 490 \\quad 435 \\quad 525 \\quad 595 \\quad 520 Find the mean and median for this sample. Sample\nMean=[ANS]\nSample Median=[ANS]", "answer_v3": [ "508.75", "505" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0085", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "3", "keywords": [ "statistics", "numerical", "descriptive statistics", "mean", "median", "mode", "variance", "standard deviation" ], "problem_v1": "Below is a sample of share prices (in dollars) for a particular stock, selected at random over several years:\n\\begin{array}{ccccccccc} 233 & 233 & 234 & 246 & 221 & 226 & 245 & 249 & 262\\\\ 249 & 230 & 251 & 265 & 263 & 223 & 221 & 233 & 258 \\end{array}\nUse Excel (or other form of electronic assistance) to find the mean, median, mode, variance, standard deviation, and coefficient of variation for this sample.\nMean=[ANS]\nMedian=[ANS]\nMode=[ANS]\nVariance=[ANS]\nStandard Deviation=[ANS]\nCoefficient of Variation=[ANS]", "answer_v1": [ "241.222222222222", "239.5", "233", "221.712418300654", "14.8900106883996", "0.0617273589109149" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Below is a sample of share prices (in dollars) for a particular stock, selected at random over several years:\n\\begin{array}{ccccccccc} 165 & 170 & 177 & 189 & 163 & 178 & 189 & 162 & 177\\\\ 162 & 183 & 198 & 177 & 206 & 192 & 202 & 203 & 183 \\end{array}\nUse Excel (or other form of electronic assistance) to find the mean, median, mode, variance, standard deviation, and coefficient of variation for this sample.\nMean=[ANS]\nMedian=[ANS]\nMode=[ANS]\nVariance=[ANS]\nStandard Deviation=[ANS]\nCoefficient of Variation=[ANS]", "answer_v2": [ "182", "180.5", "177", "209.294117647059", "14.4670009900829", "0.0794890164290268" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Below is a sample of share prices (in dollars) for a particular stock, selected at random over several years:\n\\begin{array}{ccccccccc} 208 & 223 & 214 & 206 & 181 & 190 & 209 & 209 & 193\\\\ 193 & 222 & 218 & 221 & 184 & 193 & 194 & 182 & 181 \\end{array}\nUse Excel (or other form of electronic assistance) to find the mean, median, mode, variance, standard deviation, and coefficient of variation for this sample.\nMean=[ANS]\nMedian=[ANS]\nMode=[ANS]\nVariance=[ANS]\nStandard Deviation=[ANS]\nCoefficient of Variation=[ANS]", "answer_v3": [ "201.166666666667", "200", "193", "223.323529411765", "14.9440131628611", "0.0742867265759456" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Statistics_0086", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "statistics", "numerical", "descriptive statistics", "range", "standard deviation" ], "problem_v1": "A sample of weights (in pounds) for six 12-year-olds is given below:\n93 \\quad 84 \\quad 81 \\quad 94 \\quad 99 \\quad 75 Find the range and standard deviation for this sample. Sample Range=[ANS]\nSample Standard Deviation=[ANS]", "answer_v1": [ "24", "9.114091653405" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A sample of weights (in pounds) for six 12-year-olds is given below:\n80 \\quad 80 \\quad 100 \\quad 73 \\quad 101 \\quad 95 Find the range and standard deviation for this sample. Sample Range=[ANS]\nSample Standard Deviation=[ANS]", "answer_v2": [ "28", "11.9568669251885" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A sample of weights (in pounds) for six 12-year-olds is given below:\n94 \\quad 100 \\quad 93 \\quad 73 \\quad 80 \\quad 81 Find the range and standard deviation for this sample. Sample Range=[ANS]\nSample Standard Deviation=[ANS]", "answer_v3": [ "27", "10.3424690798023" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0087", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "3", "keywords": [ "statistics", "descriptive statistics", "percentile" ], "problem_v1": "Consider the following data set: \\begin{array}{ccccccccc} 53 & 65 & 43 & 46 & 54 & 69 & 82 & 66 & 53\\\\ 69 & 85 & 83 & 41 & 41 & 53 & 71 & 50 & 78 \\end{array} Find the 16th and 86th percentiles for this data. 16th percentile=[ANS]\n86th percentile=[ANS]", "answer_v1": [ "43.12", "82.34" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Consider the following data set: \\begin{array}{ccccccccc} 20 & 33 & 15 & 27 & 39 & 12 & 13 & 27 & 33\\\\ 42 & 12 & 56 & 28 & 39 & 52 & 53 & 27 & 48 \\end{array} Find the 15th and 85th percentiles for this data. 15th percentile=[ANS]\n85th percentile=[ANS]", "answer_v2": [ "12.85", "52.15" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Consider the following data set: \\begin{array}{ccccccccc} 58 & 48 & 21 & 33 & 49 & 54 & 34 & 33 & 62\\\\ 61 & 49 & 30 & 33 & 62 & 46 & 21 & 24 & 22 \\end{array} Find the 17th and 89th percentiles for this data. 17th percentile=[ANS]\n89th percentile=[ANS]", "answer_v3": [ "22.46", "61.91" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0088", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "prealgebra", "common core", "mean", "median", "mode" ], "problem_v1": "Your teacher is trying to encourage her students to read more books. For the current 9-weeks period, student in her class read the following (sorted) numbers of books: 2 2 3 3 3 4 4 4 4 4 5 5 5 6 6 6 6 6 7 2 2 3 3 3 4 4 4 4 4 5 5 5 6 6 6 6 6 7 For this data, the mean number of books read is [ANS] and the median number read is [ANS]\nExpress answers that are not integers to two decimal places.", "answer_v1": [ "4.47368", "4" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Your teacher is trying to encourage her students to read more books. For the current 9-weeks period, student in her class read the following (sorted) numbers of books: 1 2 2 2 2 3 3 3 4 4 6 6 8 9 9 1 2 2 2 2 3 3 3 4 4 6 6 8 9 9 For this data, the mean number of books read is [ANS] and the median number read is [ANS]\nExpress answers that are not integers to two decimal places.", "answer_v2": [ "4.26667", "3" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Your teacher is trying to encourage her students to read more books. For the current 9-weeks period, student in her class read the following (sorted) numbers of books: 1 2 2 3 3 3 4 5 6 6 6 8 8 8 9 9 1 2 2 3 3 3 4 5 6 6 6 8 8 8 9 9 For this data, the mean number of books read is [ANS] and the median number read is [ANS]\nExpress answers that are not integers to two decimal places.", "answer_v3": [ "5.1875", "5.5" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0089", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "prealgebra", "common core", "mean", "median", "mode" ], "problem_v1": "Your school basketball team posted the following (sorted) scores in 16 recent games: 55 56 56 57 60 60 60 60 60 60 61 62 62 63 63 65 55 56 56 57 60 60 60 60 60 60 61 62 62 63 63 65 The average game score is [ANS] while the median score is [ANS]. The modal score is [ANS]. Express answers that are not integers to two decimal places.", "answer_v1": [ "60", "60", "60" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Your school basketball team posted the following (sorted) scores in 13 recent games: 51 51 51 51 51 53 53 56 56 57 61 69 69 51 51 51 51 51 53 53 56 56 57 61 69 69 The average game score is [ANS] while the median score is [ANS]. The modal score is [ANS]. Express answers that are not integers to two decimal places.", "answer_v2": [ "56.0769", "53", "51" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Your school basketball team posted the following (sorted) scores in 14 recent games: 54 54 55 56 56 56 56 56 57 61 62 66 68 69 54 54 55 56 56 56 56 56 57 61 62 66 68 69 The average game score is [ANS] while the median score is [ANS]. The modal score is [ANS]. Express answers that are not integers to two decimal places.", "answer_v3": [ "59", "56", "56" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Statistics_0090", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "prealgebra", "common core" ], "problem_v1": "The following deposits were made into a bank account last month: \\$155 \\$120 \\$130 \\$150 \\$65 The total of all deposits=[ANS]\nThe average amount of the deposits=[ANS]", "answer_v1": [ "155+120+130+150+65", "(155+120+130+150+65)/5" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The following deposits were made into a bank account last month: \\$25 \\$190 \\$35 \\$75 \\$190 The total of all deposits=[ANS]\nThe average amount of the deposits=[ANS]", "answer_v2": [ "25+190+35+75+190", "(25+190+35+75+190)/5" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The following deposits were made into a bank account last month: \\$70 \\$125 \\$60 \\$115 \\$50 The total of all deposits=[ANS]\nThe average amount of the deposits=[ANS]", "answer_v3": [ "70+125+60+115+50", "(70+125+60+115+50)/5" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0091", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summary statistics", "level": "2", "keywords": [ "prealgebra", "common core" ], "problem_v1": "The average of $10^\\circ,-11^\\circ, 4^\\circ, 7^\\circ,-7^\\circ,-6^\\circ$=[ANS] $^\\circ$", "answer_v1": [ "(10+-11+4+7+-7+-6)/6" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The average of $-17^\\circ,-5^\\circ,-12^\\circ,-5^\\circ, 15^\\circ,-6^\\circ$=[ANS] $^\\circ$", "answer_v2": [ "(-17+-5+-12+-5+15+-6)/6" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The average of $-8^\\circ,-10^\\circ,-7^\\circ, 2^\\circ,-10^\\circ,-5^\\circ$=[ANS] $^\\circ$", "answer_v3": [ "(-8+-10+-7+2+-10+-5)/6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0092", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Graphical representations", "level": "1", "keywords": [ "statistics", "quantitative data", "histogram" ], "problem_v1": "Select the best response.\n(a) In drawing a histogram, which of the following suggestions should be followed? [ANS] A. The heights of bars should equal the class frequency. B. The scale of the vertical axis should be that of the variable whose distribution you are displaying. C. Generally, bars should be square so that both the height and width equal the class count. D. Leave large gaps between bars. This allows room for comments.\n(b) When drawing a histogram it is important to [ANS] A. make certain the mean and median are contained in the same class interval, so that the correct type of skewness can be identified. B. have a separate class interval for each observation to get the most informative plot. C. label the vertical axis so the reader can determine the counts or percent in each class interval. D. make sure the heights of the bars exceed the widths of the class intervals so that the bars are true rectangles. E. None of the above.", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Select the best response.\n(a) When drawing a histogram it is important to [ANS] A. make certain the mean and median are contained in the same class interval, so that the correct type of skewness can be identified. B. have a separate class interval for each observation to get the most informative plot. C. make sure the heights of the bars exceed the widths of the class intervals so that the bars are true rectangles. D. label the vertical axis so the reader can determine the counts or percent in each class interval. E. None of the above.\n(b) In drawing a histogram, which of the following suggestions should be followed? [ANS] A. Generally, bars should be square so that both the height and width equal the class count. B. The scale of the vertical axis should be that of the variable whose distribution you are displaying. C. Leave large gaps between bars. This allows room for comments. D. The heights of bars should equal the class frequency.", "answer_v2": [ "D", "D" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Select the best response.\n(a) When drawing a histogram it is important to [ANS] A. make certain the mean and median are contained in the same class interval, so that the correct type of skewness can be identified. B. label the vertical axis so the reader can determine the counts or percent in each class interval. C. make sure the heights of the bars exceed the widths of the class intervals so that the bars are true rectangles. D. have a separate class interval for each observation to get the most informative plot. E. None of the above.\n(b) In drawing a histogram, which of the following suggestions should be followed? [ANS] A. The heights of bars should equal the class frequency. B. Leave large gaps between bars. This allows room for comments. C. Generally, bars should be square so that both the height and width equal the class count. D. The scale of the vertical axis should be that of the variable whose distribution you are displaying.", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0093", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Graphical representations", "level": "3", "keywords": [], "problem_v1": "Three students collected data for a school project. Anita asked her classmates which season they liked best. Kevin measured and recorded the height of the snow in his backyard every 15 minutes during a 2 hour snowstorm. Rachel recorded the total snowfall in 3 different towns. Who should use a line graph to display their data? [ANS] A. Anita and Rachel only B. Rachel only C. Kevin only D. Kevin and Rachel only", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "Three students collected data for a school project. Anita asked her classmates which season they liked best. Kevin measured and recorded the height of the snow in his backyard every 15 minutes during a 2 hour snowstorm. Rachel recorded the total snowfall in 3 different towns. Who should use a line graph to display their data? [ANS] A. Kevin only B. Rachel only C. Anita and Rachel only D. Kevin and Rachel only", "answer_v2": [ "A" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "Three students collected data for a school project. Anita asked her classmates which season they liked best. Kevin measured and recorded the height of the snow in his backyard every 15 minutes during a 2 hour snowstorm. Rachel recorded the total snowfall in 3 different towns. Who should use a line graph to display their data? [ANS] A. Anita and Rachel only B. Kevin only C. Rachel only D. Kevin and Rachel only", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0094", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Graphical representations", "level": "2", "keywords": [ "Statistics" ], "problem_v1": "What is the difference between a frequency histogram and a relative frequency histogram? [ANS] A. There is no difference between the two B. A frequency histogram displays class frequencies on the horizontal axis while a relative-frequency histogram displays class relative frequencies on the horizontal axis C. A frequency histogram displays class frequencies on the vertical axis while a relative-frequency historgram displays class relative frequencies on the vertical axis D. None of the above", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "What is the difference between a frequency histogram and a relative frequency histogram? [ANS] A. A frequency histogram displays class frequencies on the vertical axis while a relative-frequency historgram displays class relative frequencies on the vertical axis B. A frequency histogram displays class frequencies on the horizontal axis while a relative-frequency histogram displays class relative frequencies on the horizontal axis C. There is no difference between the two D. None of the above", "answer_v2": [ "A" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "What is the difference between a frequency histogram and a relative frequency histogram? [ANS] A. There is no difference between the two B. A frequency histogram displays class frequencies on the vertical axis while a relative-frequency historgram displays class relative frequencies on the vertical axis C. A frequency histogram displays class frequencies on the horizontal axis while a relative-frequency histogram displays class relative frequencies on the horizontal axis D. None of the above", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0095", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Graphical representations", "level": "2", "keywords": [ "Statistics" ], "problem_v1": "Which of the following is useful for qualitative data? [ANS] A. Dot plot B. Histogram C. Bar graph D. None of the above", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "Which of the following is useful for qualitative data? [ANS] A. Bar graph B. Histogram C. Dot plot D. None of the above", "answer_v2": [ "A" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "Which of the following is useful for qualitative data? [ANS] A. Dot plot B. Bar graph C. Histogram D. None of the above", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0097", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Graphical representations", "level": "1", "keywords": [ "statistics", "multiple choice", "display methods" ], "problem_v1": "You obtain starting salaries of ten UBC graduates and fifteen graduates from another university. Which of the following display methods would be the best for comparing the starting salary distributions of the two universities? [ANS] A. side-by-side histograms B. side-by-side bar graphs C. side-by-side box plots D. none of the above, because the size of the data sets are different", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "You obtain starting salaries of ten UBC graduates and fifteen graduates from another university. Which of the following display methods would be the best for comparing the starting salary distributions of the two universities? [ANS] A. side-by-side box plots B. side-by-side bar graphs C. side-by-side histograms D. none of the above, because the size of the data sets are different", "answer_v2": [ "A" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "You obtain starting salaries of ten UBC graduates and fifteen graduates from another university. Which of the following display methods would be the best for comparing the starting salary distributions of the two universities? [ANS] A. side-by-side histograms B. side-by-side box plots C. side-by-side bar graphs D. none of the above, because the size of the data sets are different", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0098", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Graphical representations", "level": "2", "keywords": [ "statistics", "descriptive statistics", "stem and leaf" ], "problem_v1": "Consider the following data set:\n\\begin{array}{ccccccccc} 53 & 53 & 54 & 66 & 41 & 46 & 65 & 69 & 82\\\\ 69 & 50 & 71 & 85 & 83 & 43 & 41 & 53 & 78 \\end{array} Below is a partially completed stem-and-leaf diagram for this data set:\n$\\begin{array}{cc}\\hline 4 & [ANS] \\\\ \\hline [ANS] & [ANS] \\\\ \\hline 6 & [ANS] \\\\ \\hline [ANS] & 18 \\\\ \\hline [ANS] & [ANS] \\\\ \\hline \\end{array}$\nDetermine the correct values for the missing entries, and fill them in. If no value is needed for a given entry, then enter X.", "answer_v1": [ "1136", "5", "03334", "5699", "7", "8", "235" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "Consider the following data set:\n\\begin{array}{ccccccccc} 15 & 20 & 27 & 39 & 13 & 28 & 39 & 12 & 27\\\\ 12 & 33 & 48 & 27 & 56 & 42 & 52 & 53 & 33 \\end{array} Below is a partially completed stem-and-leaf diagram for this data set:\n$\\begin{array}{cc}\\hline 1 & [ANS] \\\\ \\hline [ANS] & [ANS] \\\\ \\hline 3 & [ANS] \\\\ \\hline [ANS] & 28 \\\\ \\hline [ANS] & [ANS] \\\\ \\hline \\end{array}$\nDetermine the correct values for the missing entries, and fill them in. If no value is needed for a given entry, then enter X.", "answer_v2": [ "2235", "2", "07778", "3399", "4", "5", "236" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "Consider the following data set:\n\\begin{array}{ccccccccc} 48 & 62 & 54 & 46 & 21 & 30 & 49 & 49 & 33\\\\ 33 & 62 & 58 & 61 & 24 & 33 & 34 & 22 & 21 \\end{array} Below is a partially completed stem-and-leaf diagram for this data set:\n$\\begin{array}{cc}\\hline 2 & [ANS] \\\\ \\hline [ANS] & [ANS] \\\\ \\hline 4 & [ANS] \\\\ \\hline [ANS] & 48 \\\\ \\hline [ANS] & [ANS] \\\\ \\hline \\end{array}$\nDetermine the correct values for the missing entries, and fill them in. If no value is needed for a given entry, then enter X.", "answer_v3": [ "1124", "3", "03334", "6899", "5", "6", "122" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Statistics_0099", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Graphical representations", "level": "2", "keywords": [ "statistics", "descriptive statistics", "chart" ], "problem_v1": "The best type of chart for comparing two sets of categorical data is: [ANS] A. a bar chart B. a histogram C. a pie chart D. a line chart\nThe total area of the six bars in a relative frequency histogram for which the width of each bar is five units is: [ANS] A. 1 B. 6 C. 5 D. 11", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "The best type of chart for comparing two sets of categorical data is: [ANS] A. a pie chart B. a histogram C. a line chart D. a bar chart\nThe most appropriate type of chart for determining the number of observations at or below a specific value is: [ANS] A. a cumulative frequency ogive B. a time-series chart C. a histogram D. a pie chart", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "The best type of chart for comparing two sets of categorical data is: [ANS] A. a pie chart B. a bar chart C. a histogram D. a line chart\nThe total area of the bars in a relative frequency histogram: [ANS] A. depends on the number of bars. B. depends on the width of each bar. C. depends on the sample size. D. depends on both the sample size and on the number of bars.", "answer_v3": [ "B", "B" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0100", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Graphical representations", "level": "3", "keywords": [ "statistics", "descriptive statistics", "mean", "median", "mode" ], "problem_v1": "Which measure of central location is meaningful when the data are nominal? [ANS] A. The mode B. The median C. The geometric mean D. The arithmetic mean\nWhich of the following are measures of the linear relationship between two variables? [ANS] A. The variance B. The covariance C. The coefficient of correlation D. Both the covariance and the coefficient of correlation", "answer_v1": [ "A", "D" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Which measure of central location is meaningful when the data are nominal? [ANS] A. The geometric mean B. The median C. The arithmetic mean D. The mode\nIn a histogram, the proportion of the total area which must be to the right of the mean is: [ANS] A. exactly 1.0 if the distribution is symmetric and bimodal B. more than 0.50 if the distribution is skewed to the right C. exactly 0.50 if the distribution is symmetric and unimodal D. exactly 0.50 E. less than 0.50 if the distribution is skewed to the left", "answer_v2": [ "D", "C" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Which measure of central location is meaningful when the data are nominal? [ANS] A. The geometric mean B. The mode C. The median D. The arithmetic mean\nGenerally speaking, if two variables are unrelated (as one increases, the other shows no pattern), the covariance will be: [ANS] A. a large negative number B. a positive or negative number close to zero C. a large positive number D. none of the above", "answer_v3": [ "B", "B" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0101", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Graphical representations", "level": "3", "keywords": [ "statistics", "descriptive statistics", "mean", "median" ], "problem_v1": "In a histogram, the proportion of the total area which must be to the left of the median is: [ANS] A. exactly 0.50 B. between 0.25 and 0.60 if the distribution is symmetric and unimodal C. more than 0.50 if the distribution is skewed to the right D. less than 0.50 if the distribution is skewed to the left\nWhich of the following statements is true? [ANS] A. The sum of the squared deviations from the arithmetic mean is always zero B. The distance between the first and third quartiles is twice the distance between the first quartile and the median C. The sum of the deviations from the arithmetic mean is always zero D. The standard deviation is always less than the variance E. The arithmetic mean is always less than the geometric mean", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "In a histogram, the proportion of the total area which must be to the left of the median is: [ANS] A. more than 0.50 if the distribution is skewed to the right B. between 0.25 and 0.60 if the distribution is symmetric and unimodal C. less than 0.50 if the distribution is skewed to the left D. exactly 0.50\nWhich of the following statements about the arithmetic mean is not always correct? [ANS] A. Half of the observations are on either side of the mean B. The value of the mean times the number of observations equals the sum of all of the observations C. The sum of the deviations from the mean is zero D. The mean is a measure of the middle (center) of a distribution", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "In a histogram, the proportion of the total area which must be to the left of the median is: [ANS] A. more than 0.50 if the distribution is skewed to the right B. exactly 0.50 C. between 0.25 and 0.60 if the distribution is symmetric and unimodal D. less than 0.50 if the distribution is skewed to the left\nWhich of the following statements is true? [ANS] A. When the distribution is symmetric and unimodal, mean $=$ median $=$ mode B. When the distribution is skewed to the right, mean $<$ median $<$ mode C. When the distribution is skewed to the left, mean $>$ median $>$ mode D. When the distribution is symmetric and bimodal, mean $=$ median $=$ mode", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0102", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Description of distributions", "level": "3", "keywords": [ "Probability", "Independence" ], "problem_v1": "A study revealed the following data on methods Canadians use to get to work, by residence.\n$\\begin{array}{cccc}\\hline & Urban & Rural & Total \\\\ \\hline Automobile & 450 & 150 & 600 \\\\ \\hline Public Transit & 650 & 50 & 700 \\\\ \\hline Total & 1100 & 200 & 1300 \\\\ \\hline \\end{array}$\nPart i A worker is selected at random. What is the probability that he lives in an urban area given that he drives to works? [ANS] A. 450/600 B. 600/1300 C. 450/1300 D. 450/1100\nPart ii Which of the following statements is true? [ANS] A. Mode of transportation and residence are independent. B. Mode of transportation and residence are dependent, but they would be independent if we changed the second row of counts from 650, 50, 700 to 650, 650, 1300. C. Mode of transportation and residence are dependent, but they would be independent if we changed the second row of counts from 650, 50, 700 to 450, 250, 700. D. Mode of transportation and residence are dependent, but they would be independent if we changed the second row of counts from 650, 50, 700 to 550, 150, 700. E. Mode of transportation and residence are dependent, but they would be independent if we changed the second row of counts from 650, 50, 700 to 525, 175, 700. F. Mode of transportation and residence are dependent, but they would be independent if we changed the second row of counts from 650, 50, 700 to 475, 225, 700.\nPart iii To encourage people to take public transit, the government runs a program which provides free transit passes for a month. The 600 residents who drive to work are invited to participate in the following experiment. Half of the urban residents who drive to work are randomized to receive the free pass for one month and half do not receive such free pass. The rural residents who drive to work are randomized in a similar fashion. After six months, the proportion of urban residents who switch to taking public transit is compared between the free-pass and no-free pass groups. The same comparison is also done among the rural residents.\nIn the experiment described above, area of residence (urban versus rural) [CHECK ALL THAT APPLY] [ANS] A. is the blocking variable. B. defines the experimental units. C. is the response variable. D. defines the treatments. E. none of the above.\nPart iv What is the purpose of performing the treatment randomization and comparison for the urban and rural residents separately? Choose the most appropriate answer. [ANS] A. To control for the effect of area of residence on the residents\u2019 willingness to switch to taking public transit. B. To evaluate the effect of area of residence on the residents\u2019 willingness to switch to taking public transit. C. To ensure that both urban and rural residents can participate in the study.\nPart v To display the data for the two variables: area of residence and whether residents switch to taking public transit after six months, what is the most appropriate display to use? [ANS] A. A contingency table. B. A stem-and-leaf display. C. A scatterplot. D. Side-by-side boxplots. E. A histogram.", "answer_v1": [ "A", "E", "A", "A", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "A study revealed the following data on methods Canadians use to get to work, by residence.\n$\\begin{array}{cccc}\\hline & Urban & Rural & Total \\\\ \\hline Automobile & 450 & 150 & 600 \\\\ \\hline Public Transit & 650 & 50 & 700 \\\\ \\hline Total & 1100 & 200 & 1300 \\\\ \\hline \\end{array}$\nPart i A worker is selected at random. What is the probability that he drives to work given that he lives in an urban area? [ANS] A. 450/1300 B. 600/1300 C. 450/600 D. 450/1100\nPart ii Which of the following statements is true? [ANS] A. Mode of transportation and residence are dependent, but they would be independent if we changed the second row of counts from 650, 50, 700 to 475, 225, 700. B. Mode of transportation and residence are dependent, but they would be independent if we changed the second row of counts from 650, 50, 700 to 550, 150, 700. C. Mode of transportation and residence are independent. D. Mode of transportation and residence are dependent, but they would be independent if we changed the second row of counts from 650, 50, 700 to 650, 650, 1300. E. Mode of transportation and residence are dependent, but they would be independent if we changed the second row of counts from 650, 50, 700 to 525, 175, 700. F. Mode of transportation and residence are dependent, but they would be independent if we changed the second row of counts from 650, 50, 700 to 450, 250, 700.\nPart iii To encourage people to take public transit, the government runs a program which provides free transit passes for a month. The 600 residents who drive to work are invited to participate in the following experiment. Half of the urban residents who drive to work are randomized to receive the free pass for one month and half do not receive such free pass. The rural residents who drive to work are randomized in a similar fashion. After six months, the proportion of urban residents who switch to taking public transit is compared between the free-pass and no-free pass groups. The same comparison is also done among the rural residents.\nIn the experiment described above, area of residence (urban versus rural) [CHECK ALL THAT APPLY] [ANS] A. is the response variable. B. defines the experimental units. C. defines the treatments. D. is the blocking variable. E. none of the above.\nPart iv What is the purpose of performing the treatment randomization and comparison for the urban and rural residents separately? Choose the most appropriate answer. [ANS] A. To ensure that both urban and rural residents can participate in the study. B. To evaluate the effect of area of residence on the residents\u2019 willingness to switch to taking public transit. C. To control for the effect of area of residence on the residents\u2019 willingness to switch to taking public transit.\nPart v To display the data for the two variables: area of residence and whether residents switch to taking public transit after six months, what is the most appropriate display to use? [ANS] A. A contingency table. B. A histogram. C. Side-by-side boxplots. D. A stem-and-leaf display. E. A scatterplot.", "answer_v2": [ "D", "E", "D", "C", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "A study revealed the following data on methods Canadians use to get to work, by residence.\n$\\begin{array}{cccc}\\hline & Urban & Rural & Total \\\\ \\hline Automobile & 450 & 150 & 600 \\\\ \\hline Public Transit & 650 & 50 & 700 \\\\ \\hline Total & 1100 & 200 & 1300 \\\\ \\hline \\end{array}$\nPart i A worker is selected at random. What is the probability that he drives to work given that he lives in an urban area? [ANS] A. 450/1300 B. 450/1100 C. 600/1300 D. 450/600\nPart ii Which of the following statements is true? [ANS] A. Mode of transportation and residence are dependent, but they would be independent if we changed the second row of counts from 650, 50, 700 to 550, 150, 700. B. Mode of transportation and residence are dependent, but they would be independent if we changed the second row of counts from 650, 50, 700 to 475, 225, 700. C. Mode of transportation and residence are dependent, but they would be independent if we changed the second row of counts from 650, 50, 700 to 525, 175, 700. D. Mode of transportation and residence are dependent, but they would be independent if we changed the second row of counts from 650, 50, 700 to 650, 650, 1300. E. Mode of transportation and residence are independent. F. Mode of transportation and residence are dependent, but they would be independent if we changed the second row of counts from 650, 50, 700 to 450, 250, 700.\nPart iii To encourage people to take public transit, the government runs a program which provides free transit passes for a month. The 600 residents who drive to work are invited to participate in the following experiment. Half of the urban residents who drive to work are randomized to receive the free pass for one month and half do not receive such free pass. The rural residents who drive to work are randomized in a similar fashion. After six months, the proportion of urban residents who switch to taking public transit is compared between the free-pass and no-free pass groups. The same comparison is also done among the rural residents.\nIn the experiment described above, area of residence (urban versus rural) [CHECK ALL THAT APPLY] [ANS] A. is the blocking variable. B. defines the treatments. C. is the response variable. D. defines the experimental units. E. none of the above.\nPart iv What is the purpose of performing the treatment randomization and comparison for the urban and rural residents separately? Choose the most appropriate answer. [ANS] A. To ensure that both urban and rural residents can participate in the study. B. To control for the effect of area of residence on the residents\u2019 willingness to switch to taking public transit. C. To evaluate the effect of area of residence on the residents\u2019 willingness to switch to taking public transit.\nPart v To display the data for the two variables: area of residence and whether residents switch to taking public transit after six months, what is the most appropriate display to use? [ANS] A. A contingency table. B. A scatterplot. C. A histogram. D. Side-by-side boxplots. E. A stem-and-leaf display.", "answer_v3": [ "B", "C", "A", "B", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Statistics_0103", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summarizing data in tables", "level": "2", "keywords": [ "Statistics" ], "problem_v1": "One study was performed in an attempt to determine the day of the week when road rage occurs most often. Complete the table below.\n$\\begin{array}{ccc}\\hline Day of the Week & Frequency & Relative Frequency \\\\ \\hline Sunday & 8 & [ANS] \\\\ \\hline Monday & 6 & [ANS] \\\\ \\hline Tuesday & 7 & [ANS] \\\\ \\hline Wednesday & [ANS] & 0.205882352941176 \\\\ \\hline Thursday & 11 & [ANS] \\\\ \\hline Friday & 16 & [ANS] \\\\ \\hline Saturday & 6 & [ANS] \\\\ \\hline total & 68 & 1 \\\\ \\hline \\end{array}$", "answer_v1": [ "0.117647058823529", "0.0882352941176471", "0.102941176470588", "14", "0.161764705882353", "0.235294117647059", "0.0882352941176471" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "One study was performed in an attempt to determine the day of the week when road rage occurs most often. Complete the table below.\n$\\begin{array}{ccc}\\hline Day of the Week & Frequency & Relative Frequency \\\\ \\hline Sunday & 1 & [ANS] \\\\ \\hline Monday & 10 & [ANS] \\\\ \\hline Tuesday & 2 & [ANS] \\\\ \\hline Wednesday & [ANS] & 0.206896551724138 \\\\ \\hline Thursday & 15 & [ANS] \\\\ \\hline Friday & 16 & [ANS] \\\\ \\hline Saturday & 2 & [ANS] \\\\ \\hline total & 58 & 1 \\\\ \\hline \\end{array}$", "answer_v2": [ "0.0172413793103448", "0.172413793103448", "0.0344827586206897", "12", "0.258620689655172", "0.275862068965517", "0.0344827586206897" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "One study was performed in an attempt to determine the day of the week when road rage occurs most often. Complete the table below.\n$\\begin{array}{ccc}\\hline Day of the Week & Frequency & Relative Frequency \\\\ \\hline Sunday & 4 & [ANS] \\\\ \\hline Monday & 7 & [ANS] \\\\ \\hline Tuesday & 3 & [ANS] \\\\ \\hline Wednesday & [ANS] & 0.203125 \\\\ \\hline Thursday & 11 & [ANS] \\\\ \\hline Friday & 17 & [ANS] \\\\ \\hline Saturday & 9 & [ANS] \\\\ \\hline total & 64 & 1 \\\\ \\hline \\end{array}$", "answer_v3": [ "0.0625", "0.109375", "0.046875", "13", "0.171875", "0.265625", "0.140625" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Statistics_0104", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summarizing data in tables", "level": "2", "keywords": [ "contingency table", "conditional distribution" ], "problem_v1": "A proficiency examination was given to 100 students. The breakdown of the exam results among male and female students is shown in the following table.\n$\\begin{array}{c|cc|c} & Male & Female & Total \\\\ \\hline Pass & & & \\\\ Fail & & & \\\\ \\hline Total & & & 100 \\end{array}$\nWhich of choices (A-D) below gives the conditional distribution of gender for students who failed the exam?\n[ANS]\nA $\\begin{array}{|c|c|c|} \\hline Male & Female & Total \\\\ \\hline ~ (\\%) & ~(\\%) & 0 ~ (100\\%) \\\\ \\hline \\end{array}$ B $\\begin{array}{|c|c|c|} \\hline Male & Female & Total \\\\ \\hline ~ (\\%) & ~ (\\%) & ~ (100\\%) \\\\ \\hline \\end{array}$ C $\\begin{array}{|c|c|c|} \\hline Male & Female & Total \\\\ \\hline ~(\\%) & ~(\\%) & ~ (100\\%) \\\\ \\hline \\end{array}$ D $\\begin{array}{|c|c|c|} \\hline Male & Female & Total \\\\ \\hline ~ (\\%) & ~ (\\%) & ~ (100\\%) \\\\ \\hline \\end{array}$", "answer_v1": [ "D" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "A proficiency examination was given to 100 students. The breakdown of the exam results among male and female students is shown in the following table.\n$\\begin{array}{c|cc|c} & Male & Female & Total \\\\ \\hline Pass & & & \\\\ Fail & & & \\\\ \\hline Total & & & 100 \\end{array}$\nWhich of choices (A-D) below gives the conditional distribution of gender for students who failed the exam?\n[ANS]\nA $\\begin{array}{|c|c|c|} \\hline Male & Female & Total \\\\ \\hline ~ (\\%) & ~ (\\%) & ~ (100\\%) \\\\ \\hline \\end{array}$ B $\\begin{array}{|c|c|c|} \\hline Male & Female & Total \\\\ \\hline ~ (\\%) & ~(\\%) & 0 ~ (100\\%) \\\\ \\hline \\end{array}$ C $\\begin{array}{|c|c|c|} \\hline Male & Female & Total \\\\ \\hline ~ (\\%) & ~ (\\%) & ~ (100\\%) \\\\ \\hline \\end{array}$ D $\\begin{array}{|c|c|c|} \\hline Male & Female & Total \\\\ \\hline ~(\\%) & ~(\\%) & ~ (100\\%) \\\\ \\hline \\end{array}$", "answer_v2": [ "A" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "A proficiency examination was given to 100 students. The breakdown of the exam results among male and female students is shown in the following table.\n$\\begin{array}{c|cc|c} & Male & Female & Total \\\\ \\hline Pass & & & \\\\ Fail & & & \\\\ \\hline Total & & & 100 \\end{array}$\nWhich of choices (A-D) below gives the conditional distribution of gender for students who failed the exam?\n[ANS]\nA $\\begin{array}{|c|c|c|} \\hline Male & Female & Total \\\\ \\hline ~ (\\%) & ~ (\\%) & ~ (100\\%) \\\\ \\hline \\end{array}$ B $\\begin{array}{|c|c|c|} \\hline Male & Female & Total \\\\ \\hline ~(\\%) & ~(\\%) & ~ (100\\%) \\\\ \\hline \\end{array}$ C $\\begin{array}{|c|c|c|} \\hline Male & Female & Total \\\\ \\hline ~ (\\%) & ~ (\\%) & ~ (100\\%) \\\\ \\hline \\end{array}$ D $\\begin{array}{|c|c|c|} \\hline Male & Female & Total \\\\ \\hline ~ (\\%) & ~(\\%) & 0 ~ (100\\%) \\\\ \\hline \\end{array}$", "answer_v3": [ "C" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0105", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summarizing data in tables", "level": "2", "keywords": [ "statistics", "descriptive statistics", "population frequency", "relative frequency" ], "problem_v1": "The percent frequency of a class is computed by: [ANS] A. multiplying the relative frequency by 100 B. adding 100 to the relative frequency C. dividing the relative frequency by 100 D. multiplying the relative frequency by 10\nThe sum of the frequencies for all classes will always equal: [ANS] A. a value between 0 and 1 B. 1 C. the number of elements in a data set D. the number of classes", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "The percent frequency of a class is computed by: [ANS] A. dividing the relative frequency by 100 B. adding 100 to the relative frequency C. multiplying the relative frequency by 10 D. multiplying the relative frequency by 100\nThe relative frequency of a class is computed by: [ANS] A. dividing the frequency of the class by the sample size B. dividing the sample size by the frequency of the class C. dividing the midpoint of the class by the sample size D. dividing the frequency of the class by the midpoint", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "The percent frequency of a class is computed by: [ANS] A. dividing the relative frequency by 100 B. multiplying the relative frequency by 100 C. adding 100 to the relative frequency D. multiplying the relative frequency by 10\nA researcher is gathering data from four geographical areas designated: South=1; North=2; East=3; West=4. The designated geographical regions represent: [ANS] A. label data B. qualitative data C. quantitative data D. either quantitative or qualitative data", "answer_v3": [ "B", "B" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0106", "subject": "Statistics", "topic": "Exploratory data analysis/descriptive statistics", "subtopic": "Summarizing data in tables", "level": "2", "keywords": [ "statistics", "descriptive statistics", "frequency distribution" ], "problem_v1": "A tabular summary of a set of data showing the fraction of the total number items in several classes is a: [ANS] A. relative frequency distribution B. cumulative frequency distribution C. frequency D. frequency distribution\nQualitative data can be graphically represented by using a(n): [ANS] A. ogive B. histogram C. bar graph D. frequency polygon", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "A tabular summary of a set of data showing the fraction of the total number items in several classes is a: [ANS] A. frequency B. cumulative frequency distribution C. frequency distribution D. relative frequency distribution\nA frequency distribution is: [ANS] A. a tabular summary of a set of data showing the frequency of items in each of several nonoverlapping classes B. a graphical device for presenting qualitative data C. a tabular summary of a set of data showing the relative frequency D. a graphical form of representing data", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "A tabular summary of a set of data showing the fraction of the total number items in several classes is a: [ANS] A. frequency B. relative frequency distribution C. cumulative frequency distribution D. frequency distribution\nThe sum of the relative frequencies for all classes will always equal: [ANS] A. one B. the number of classes C. the sample size D. any value larger than one", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0107", "subject": "Statistics", "topic": "Sampling distributions", "subtopic": "Sample mean", "level": "3", "keywords": [ "statistics", "inference", "hypothesis testing", "t score" ], "problem_v1": "An SRS of size 22 is drawn from a population that has a normal distribution. The sample has a mean of 135 and a standard deviation of 8.5. Give the standard error of the mean: [ANS]", "answer_v1": [ "8.5/[sqrt(22)]" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "An SRS of size 30 is drawn from a population that has a normal distribution. The sample has a mean of 116 and a standard deviation of 3.5. Give the standard error of the mean: [ANS]", "answer_v2": [ "3.5/[sqrt(30)]" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "An SRS of size 22 is drawn from a population that has a normal distribution. The sample has a mean of 121 and a standard deviation of 5. Give the standard error of the mean: [ANS]", "answer_v3": [ "5/[sqrt(22)]" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0108", "subject": "Statistics", "topic": "Sampling distributions", "subtopic": "Sample mean", "level": "1", "keywords": [ "statistic", "parameter", "sampling distribution", "sample" ], "problem_v1": "For each problem, select the best response.\n(a) The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 266 days and standard deviation 16 days. The probability that the average pregnancy length for 6 randomly chosen women exceeds 270 days is about [ANS] A. 0.07 B. 0.27 C. 0.4 D. None of the above.\n(b) Scores on the mathematics part of the SAT exam in a recent year were roughly Normal with mean 559 and standard deviation 106. You choose an SRS of 100 students and average their SAT math scores. If you do this many times, with a sample of size 100 each time, the standard deviation of the average scores will get close to [ANS] A. 106 B. 1.06 C. 10.6 D. None of the above.\n(c) A newborn baby has extremely low birth weight (ELBW) if it weighs less than 1000 grams. A study of the health of such children in later years examined a random sample of 219 children. Their mean weight at birth was $\\bar {x}=$ 810 grams. This sample mean is an unbiased estimator of the mean weight $\\mu$ in the population of all ELBW babies. This means that [ANS] A. the average sample mean $\\bar {x}$, over all possible samples, is equal to $\\mu$. B. the sample mean $\\bar {x}$ will have a distribution that is close to Normal. C. the sample mean $\\bar {x}$ is always equal to $\\mu$. D. None of the above.", "answer_v1": [ "B", "C", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "For each problem, select the best response.\n(a) Scores on the mathematics part of the SAT exam in a recent year were roughly Normal with mean 500 and standard deviation 119. You choose an SRS of 100 students and average their SAT math scores. If you do this many times, with a sample of size 100 each time, the mean of the average scores will get close to [ANS] A. 50 B. 5 C. 500 D. None of the above.\n(b) The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 266 days and standard deviation 16 days. The probability that the average pregnancy length for 6 randomly chosen women exceeds 270 days is about [ANS] A. 0.27 B. 0.07 C. 0.4 D. None of the above.\n(c) Scores on the mathematics part of the SAT exam in a recent year were roughly Normal with mean 500 and standard deviation 119. You choose an SRS of 100 students and average their SAT math scores. If you do this many times, with a sample of size 100 each time, the standard deviation of the average scores will get close to [ANS] A. 1.19 B. 119 C. 11.9 D. None of the above.", "answer_v2": [ "C", "A", "C" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "For each problem, select the best response.\n(a) Scores on the mathematics part of the SAT exam in a recent year were roughly Normal with mean 532 and standard deviation 104. You choose an SRS of 100 students and average their SAT math scores. If you do this many times, with a sample of size 100 each time, the standard deviation of the average scores will get close to [ANS] A. 1.04 B. 104 C. 10.4 D. None of the above.\n(b) A newborn baby has extremely low birth weight (ELBW) if it weighs less than 1000 grams. A study of the health of such children in later years examined a random sample of 219 children. Their mean weight at birth was $\\bar {x}=$ 810 grams. This sample mean is an unbiased estimator of the mean weight $\\mu$ in the population of all ELBW babies. This means that [ANS] A. the sample mean $\\bar {x}$ is always equal to $\\mu$. B. the average sample mean $\\bar {x}$, over all possible samples, is equal to $\\mu$. C. the sample mean $\\bar {x}$ will have a distribution that is close to Normal. D. None of the above.\n(c) Scores on the mathematics part of the SAT exam in a recent year were roughly Normal with mean 532 and standard deviation 104. You choose an SRS of 100 students and average their SAT math scores. If you do this many times, with a sample of size 100 each time, the mean of the average scores will get close to [ANS] A. 5.32 B. 53.2 C. 532 D. None of the above.", "answer_v3": [ "C", "B", "C" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0109", "subject": "Statistics", "topic": "Sampling distributions", "subtopic": "Sample mean", "level": "3", "keywords": [ "statistic", "parameter", "sampling distribution", "sample" ], "problem_v1": "The scores of students on the SAT college entrance examinations at a certain high school had a normal distribution with mean $\\mu=552.6$ and standard deviation $\\sigma=27.9$.\n(a) What is the probability that a single student randomly chosen from all those taking the test scores 558 or higher? ANSWER: [ANS]\nFor parts (b) through (d), consider a simple random sample (SRS) of 30 students who took the test. (b) What are the mean and standard deviation of the sample mean score $\\bar x$, of 30 students? The mean of the sampling distribution for $\\bar x$ is: [ANS]\nThe standard deviation of the sampling distribution for $\\bar x$ is: [ANS]\n(c) What z-score corresponds to the mean score $\\bar x$ of 558? ANSWER: [ANS]\n(d) What is the probability that the mean score $\\bar x$ of these students is 558 or higher? ANSWER: [ANS]", "answer_v1": [ "0.424655", "552.6", "5.09382", "1.06", "0.144572" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "The scores of students on the SAT college entrance examinations at a certain high school had a normal distribution with mean $\\mu=532.4$ and standard deviation $\\sigma=29.7$.\n(a) What is the probability that a single student randomly chosen from all those taking the test scores 537 or higher? ANSWER: [ANS]\nFor parts (b) through (d), consider a simple random sample (SRS) of 25 students who took the test. (b) What are the mean and standard deviation of the sample mean score $\\bar x$, of 25 students? The mean of the sampling distribution for $\\bar x$ is: [ANS]\nThe standard deviation of the sampling distribution for $\\bar x$ is: [ANS]\n(c) What z-score corresponds to the mean score $\\bar x$ of 537? ANSWER: [ANS]\n(d) What is the probability that the mean score $\\bar x$ of these students is 537 or higher? ANSWER: [ANS]", "answer_v2": [ "0.440382", "532.4", "5.94", "0.77", "0.22065" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "The scores of students on the SAT college entrance examinations at a certain high school had a normal distribution with mean $\\mu=539.4$ and standard deviation $\\sigma=28$.\n(a) What is the probability that a single student randomly chosen from all those taking the test scores 545 or higher? ANSWER: [ANS]\nFor parts (b) through (d), consider a simple random sample (SRS) of 25 students who took the test. (b) What are the mean and standard deviation of the sample mean score $\\bar x$, of 25 students? The mean of the sampling distribution for $\\bar x$ is: [ANS]\nThe standard deviation of the sampling distribution for $\\bar x$ is: [ANS]\n(c) What z-score corresponds to the mean score $\\bar x$ of 545? ANSWER: [ANS]\n(d) What is the probability that the mean score $\\bar x$ of these students is 545 or higher? ANSWER: [ANS]", "answer_v3": [ "0.42074", "539.4", "5.6", "1", "0.158655" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Statistics_0110", "subject": "Statistics", "topic": "Sampling distributions", "subtopic": "Sample mean", "level": "1", "keywords": [ "statistic", "parameter", "sampling distribution", "sample" ], "problem_v1": "Suppose a random variable $x$ is normally distributed with $\\mu=28$ and $\\sigma=5.8$. According to the Central Limit Theorem, for samples of size 12:\n(a) The mean of the sampling distribution for $\\bar x$ is: [ANS]\n(b) The standard deviation of the sampling distribution for $\\bar x$ is: [ANS]", "answer_v1": [ "28", "1.67432" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose a random variable $x$ is normally distributed with $\\mu=17$ and $\\sigma=6.8$. According to the Central Limit Theorem, for samples of size 8:\n(a) The mean of the sampling distribution for $\\bar x$ is: [ANS]\n(b) The standard deviation of the sampling distribution for $\\bar x$ is: [ANS]", "answer_v2": [ "17", "2.40416" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose a random variable $x$ is normally distributed with $\\mu=21$ and $\\sigma=5.8$. According to the Central Limit Theorem, for samples of size 9:\n(a) The mean of the sampling distribution for $\\bar x$ is: [ANS]\n(b) The standard deviation of the sampling distribution for $\\bar x$ is: [ANS]", "answer_v3": [ "21", "1.93333" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0111", "subject": "Statistics", "topic": "Sampling distributions", "subtopic": "Sample mean", "level": "1", "keywords": [ "statistic", "parameter", "sampling distribution", "sample" ], "problem_v1": "The level of nitrogen oxides (NOX) in the exhaust of a particular car model varies with mean 0.9 grams per mile and standard deviation 0.18 grams per mile.\n(a) What sample size is needed so that the standard deviation of the sampling distribution is 0.01 grams per mile? ANSWER: [ANS]\n(b) If a smaller sample is considered, the standard deviation for $\\bar x$ would be [ANS]. (NOTE: Enter ''SMALLER'',''LARGER'' or ''THE SAME'' without the quotes.)", "answer_v1": [ "324", "LARGER" ], "answer_type_v1": [ "NV", "MCS" ], "options_v1": [ [], [ "SMALLER", "LARGER", "THE SAME" ] ], "problem_v2": "The level of nitrogen oxides (NOX) in the exhaust of a particular car model varies with mean 0.7 grams per mile and standard deviation 0.2 grams per mile.\n(a) What sample size is needed so that the standard deviation of the sampling distribution is 0.01 grams per mile? ANSWER: [ANS]\n(b) If a larger sample is considered, the standard deviation for $\\bar x$ would be [ANS]. (NOTE: Enter ''SMALLER'',''LARGER'' or ''THE SAME'' without the quotes.)", "answer_v2": [ "400", "SMALLER" ], "answer_type_v2": [ "NV", "MCS" ], "options_v2": [ [], [ "SMALLER", "LARGER", "THE SAME" ] ], "problem_v3": "The level of nitrogen oxides (NOX) in the exhaust of a particular car model varies with mean 0.7 grams per mile and standard deviation 0.18 grams per mile.\n(a) What sample size is needed so that the standard deviation of the sampling distribution is 0.01 grams per mile? ANSWER: [ANS]\n(b) If a larger sample is considered, the standard deviation for $\\bar x$ would be [ANS]. (NOTE: Enter ''SMALLER'',''LARGER'' or ''THE SAME'' without the quotes.)", "answer_v3": [ "324", "SMALLER" ], "answer_type_v3": [ "NV", "MCS" ], "options_v3": [ [], [ "SMALLER", "LARGER", "THE SAME" ] ] }, { "id": "Statistics_0112", "subject": "Statistics", "topic": "Sampling distributions", "subtopic": "Sample mean", "level": "3", "keywords": [ "statistic", "parameter", "sampling distribution", "sample" ], "problem_v1": "Consider the population of four juvenile condors. Their weights in pounds are: 5, 7, 9, 13\n(a) Let $x$ be the weight of a juvenile condor. Write the possible unique values for $x$: (NOTE: Separate each value in the list with a comma.) [ANS]. (b) Find the mean of the population: [ANS]\n(c) Let $\\bar x$ be the average weight from a sample of two juvenile condors. List all possible outcomes for $\\bar x$. (If a value occurs twice, make sure to list it twice.) This is the sampling distribution for samples of size 2: (NOTE: Separate each value in the list with a comma.) [ANS]. (d) Find the mean of the sampling distribution: [ANS]", "answer_v1": [ "(5, 7, 9, 13)", "8.5", "(6, 7, 9, 8, 10, 11)", "8.5" ], "answer_type_v1": [ "UOL", "NV", "UOL", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Consider the population of four juvenile condors. Their weights in pounds are: 2, 4, 5, 9\n(a) Let $x$ be the weight of a juvenile condor. Write the possible unique values for $x$: (NOTE: Separate each value in the list with a comma.) [ANS]. (b) Find the mean of the population: [ANS]\n(c) Let $\\bar x$ be the average weight from a sample of two juvenile condors. List all possible outcomes for $\\bar x$. (If a value occurs twice, make sure to list it twice.) This is the sampling distribution for samples of size 2: (NOTE: Separate each value in the list with a comma.) [ANS]. (d) Find the mean of the sampling distribution: [ANS]", "answer_v2": [ "(2, 4, 5, 9)", "5", "(3, 3.5, 5.5, 4.5, 6.5, 7)", "5" ], "answer_type_v2": [ "UOL", "NV", "UOL", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Consider the population of four juvenile condors. Their weights in pounds are: 3, 5, 6, 10\n(a) Let $x$ be the weight of a juvenile condor. Write the possible unique values for $x$: (NOTE: Separate each value in the list with a comma.) [ANS]. (b) Find the mean of the population: [ANS]\n(c) Let $\\bar x$ be the average weight from a sample of two juvenile condors. List all possible outcomes for $\\bar x$. (If a value occurs twice, make sure to list it twice.) This is the sampling distribution for samples of size 2: (NOTE: Separate each value in the list with a comma.) [ANS]. (d) Find the mean of the sampling distribution: [ANS]", "answer_v3": [ "(3, 5, 6, 10)", "6", "(4, 4.5, 6.5, 5.5, 7.5, 8)", "6" ], "answer_type_v3": [ "UOL", "NV", "UOL", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Statistics_0113", "subject": "Statistics", "topic": "Sampling distributions", "subtopic": "Sample mean", "level": "1", "keywords": [ "statistics", "Central Limit Theorem", "Sample Mean" ], "problem_v1": "Altough in general you cannot know the sampling distribution of the sample mean exactly, by what distribution can you often approximate it? [ANS] A. reverse-J-shaped distribution B. uniform distribution C. normal distribution D. none of the above", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "Altough in general you cannot know the sampling distribution of the sample mean exactly, by what distribution can you often approximate it? [ANS] A. normal distribution B. uniform distribution C. reverse-J-shaped distribution D. none of the above", "answer_v2": [ "A" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "Altough in general you cannot know the sampling distribution of the sample mean exactly, by what distribution can you often approximate it? [ANS] A. reverse-J-shaped distribution B. normal distribution C. uniform distribution D. none of the above", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0114", "subject": "Statistics", "topic": "Sampling distributions", "subtopic": "Sample mean", "level": "2", "keywords": [ "mean", "Statistics", "Sample Mean" ], "problem_v1": "The following table provides the starting players of a basketball team and their heights\n$\\begin{array}{cccccc}\\hline Player & A & B & C & D & E \\\\ \\hline Height (in.) & 75 & 77 & 79 & 82 & 87 \\\\ \\hline \\end{array}$\na. The population mean height of the five players is [ANS]. b. Find the sample means for samples of size 2. A, B: $\\bar{x}$=[ANS]. A, C: $\\bar{x}$=[ANS]. A, D: $\\bar{x}$=[ANS]. A, E: $\\bar{x}$=[ANS]. B, C: $\\bar{x}$=[ANS]. B, D: $\\bar{x}$=[ANS]. B, E: $\\bar{x}$=[ANS]. C, D: $\\bar{x}$=[ANS]. C, E: $\\bar{x}$=[ANS]. D, E: $\\bar{x}$=[ANS]. c. Find the mean of all sample means from above: $\\bar{x}$=[ANS]. The answers from parts\n(a) and (c) [ANS] A. should always be equal B. are not equal C. if they are equal it is only a coincidence.", "answer_v1": [ "80", "76", "77", "78.5", "81", "78", "79.5", "82", "80.5", "83", "84.5", "80", "A" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [ "A", "B", "C" ] ], "problem_v2": "The following table provides the starting players of a basketball team and their heights\n$\\begin{array}{cccccc}\\hline Player & A & B & C & D & E \\\\ \\hline Height (in.) & 75 & 76 & 78 & 80 & 84 \\\\ \\hline \\end{array}$\na. The population mean height of the five players is [ANS]. b. Find the sample means for samples of size 2. A, B: $\\bar{x}$=[ANS]. A, C: $\\bar{x}$=[ANS]. A, D: $\\bar{x}$=[ANS]. A, E: $\\bar{x}$=[ANS]. B, C: $\\bar{x}$=[ANS]. B, D: $\\bar{x}$=[ANS]. B, E: $\\bar{x}$=[ANS]. C, D: $\\bar{x}$=[ANS]. C, E: $\\bar{x}$=[ANS]. D, E: $\\bar{x}$=[ANS]. c. Find the mean of all sample means from above: $\\bar{x}$=[ANS]. The answers from parts\n(a) and (c) [ANS] A. if they are equal it is only a coincidence. B. are not equal C. should always be equal", "answer_v2": [ "78.6", "75.5", "76.5", "77.5", "79.5", "77", "78", "80", "79", "81", "82", "78.6", "C" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [ "A", "B", "C" ] ], "problem_v3": "The following table provides the starting players of a basketball team and their heights\n$\\begin{array}{cccccc}\\hline Player & A & B & C & D & E \\\\ \\hline Height (in.) & 75 & 76 & 78 & 80 & 84 \\\\ \\hline \\end{array}$\na. The population mean height of the five players is [ANS]. b. Find the sample means for samples of size 2. A, B: $\\bar{x}$=[ANS]. A, C: $\\bar{x}$=[ANS]. A, D: $\\bar{x}$=[ANS]. A, E: $\\bar{x}$=[ANS]. B, C: $\\bar{x}$=[ANS]. B, D: $\\bar{x}$=[ANS]. B, E: $\\bar{x}$=[ANS]. C, D: $\\bar{x}$=[ANS]. C, E: $\\bar{x}$=[ANS]. D, E: $\\bar{x}$=[ANS]. c. Find the mean of all sample means from above: $\\bar{x}$=[ANS]. The answers from parts\n(a) and (c) [ANS] A. if they are equal it is only a coincidence. B. should always be equal C. are not equal", "answer_v3": [ "78.6", "75.5", "76.5", "77.5", "79.5", "77", "78", "80", "79", "81", "82", "78.6", "B" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [ "A", "B", "C" ] ] }, { "id": "Statistics_0115", "subject": "Statistics", "topic": "Sampling distributions", "subtopic": "Sample mean", "level": "2", "keywords": [ "percent" ], "problem_v1": "A variable of a population has a mean of $\\mu=250$ and a standard deviation of $\\sigma=54$. The sampling distribution of the sample mean for samples of size 81 is approximately normally distributed with mean [ANS] and standard deviation [ANS].", "answer_v1": [ "250", "6" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A variable of a population has a mean of $\\mu=100$ and a standard deviation of $\\sigma=72$. The sampling distribution of the sample mean for samples of size 81 is approximately normally distributed with mean [ANS] and standard deviation [ANS].", "answer_v2": [ "100", "8" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A variable of a population has a mean of $\\mu=150$ and a standard deviation of $\\sigma=54$. The sampling distribution of the sample mean for samples of size 81 is approximately normally distributed with mean [ANS] and standard deviation [ANS].", "answer_v3": [ "150", "6" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0116", "subject": "Statistics", "topic": "Sampling distributions", "subtopic": "Sample mean", "level": "2", "keywords": [ "central limit" ], "problem_v1": "Fluorescent lighbulbs have lifetimes that follow a normal distribution, with an average life of 1,645 days and a standard deviation of 1,582 hours. In the production process the manufacturer draws random samples of 348 lightbulbs and determines the mean lifetime of the sample. What is the standard deviation, in hours, of this sample mean? Answer: [ANS]", "answer_v1": [ "84.8041015056696" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Fluorescent lighbulbs have lifetimes that follow a normal distribution, with an average life of 1,155 days and a standard deviation of 1,932 hours. In the production process the manufacturer draws random samples of 159 lightbulbs and determines the mean lifetime of the sample. What is the standard deviation, in hours, of this sample mean? Answer: [ANS]", "answer_v2": [ "153.217566360745" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Fluorescent lighbulbs have lifetimes that follow a normal distribution, with an average life of 1,324 days and a standard deviation of 1,606 hours. In the production process the manufacturer draws random samples of 211 lightbulbs and determines the mean lifetime of the sample. What is the standard deviation, in hours, of this sample mean? Answer: [ANS]", "answer_v3": [ "110.561599565935" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0117", "subject": "Statistics", "topic": "Sampling distributions", "subtopic": "Sample mean", "level": "3", "keywords": [], "problem_v1": "Consider measuring the time until death after diagnosis of a particular disease, and suppose that we have complete follow up, so that all death times are observed. We will assume the following to be known values for the population. We know that the mean time until death after diagnosis is 6.5 years. We also know that the standard deviation of time until death is 4.1 years. Further, we know that the distribution of time until death is very strongly skewed to the right (positively skewed) for this population. Use this to answer the following.\nPart (a) Suppose that we take a simple random sample of 15 people from this population. What would we 'expect' our sample mean to be for our sample of 15 individuals? That is, what is the expected value of the sample mean? [ANS] A. Exactly equal to 6.5 B. It depends on the sample size C. Less than 6.5 D. Greater than 6.5\nPart (b) Suppose that we take a simple random sample of 15 people from this population. What would we 'expect' our sample standard deviation to be for our sample of 15 individuals? That is, what is the expected value of the sample standard deviation? [ANS] A. Exactly equal to 4.1 B. It depends on the sample size C. Greater than 4.1 D. Less than 4.1\nPart (c) If we made a histogram of these 10 observations, what shape would we 'expect' this histogram to have? [ANS] A. Skewed to the left B. It depends on the sample size C. Skewed to the right D. Normally distributed E. Symmetric\nPart (d) Suppose that we take a simple random sample of 150 people from this population. What would we 'expect' our sample mean to be for our sample of 150 individuals? That is, what is the expected value of the sample mean? [ANS] A. Exactly equal to 6.5 B. It depends on the sample size C. Greater than 6.5 D. Less than 6.5\nPart (e) Suppose that we take a simple random sample of 150 people from this population. What would we 'expect' our sample standard deviation to be for our sample of 150 individuals? That is, what is the expected value of the sample standard deviation? [ANS] A. It depends on the sample size B. Less than 4.1 C. Greater than 4.1 D. Exactly equal to 4.1\nPart (f) If we made a histogram of these 150 observations, what shape would we 'expect' this histogram to have? [ANS] A. Skewed to the right B. Symmetric C. Normally distributed D. It depends on the sample size E. Skewed to the left\nPart (g) (On-Paper) Again, consider taking a sample of 150 observations from this population. In a few sentences (you are encouraged to use pictures or notation if this helps), explain what is mean by the term \"The Sampling Distribution of The Sample Mean\", in the context of this example. In your explanation, imagine that you are explaining to a friend with limited understanding of statistics, and avoid the use of technical terms as much as possible. Also, please state any assumptions that are relevant to your description of the sampling distribution.", "answer_v1": [ "A", "A", "C", "A", "D", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Consider measuring the time until death after diagnosis of a particular disease, and suppose that we have complete follow up, so that all death times are observed. We will assume the following to be known values for the population. We know that the mean time until death after diagnosis is 6.5 years. We also know that the standard deviation of time until death is 4.1 years. Further, we know that the distribution of time until death is very strongly skewed to the right (positively skewed) for this population. Use this to answer the following.\nPart (a) Suppose that we take a simple random sample of 15 people from this population. What would we 'expect' our sample mean to be for our sample of 15 individuals? That is, what is the expected value of the sample mean? [ANS] A. Exactly equal to 6.5 B. It depends on the sample size C. Less than 6.5 D. Greater than 6.5\nPart (b) Suppose that we take a simple random sample of 15 people from this population. What would we 'expect' our sample standard deviation to be for our sample of 15 individuals? That is, what is the expected value of the sample standard deviation? [ANS] A. Greater than 4.1 B. It depends on the sample size C. Less than 4.1 D. Exactly equal to 4.1\nPart (c) If we made a histogram of these 10 observations, what shape would we 'expect' this histogram to have? [ANS] A. It depends on the sample size B. Symmetric C. Normally distributed D. Skewed to the right E. Skewed to the left\nPart (d) Suppose that we take a simple random sample of 150 people from this population. What would we 'expect' our sample mean to be for our sample of 150 individuals? That is, what is the expected value of the sample mean? [ANS] A. Greater than 6.5 B. It depends on the sample size C. Less than 6.5 D. Exactly equal to 6.5\nPart (e) Suppose that we take a simple random sample of 150 people from this population. What would we 'expect' our sample standard deviation to be for our sample of 150 individuals? That is, what is the expected value of the sample standard deviation? [ANS] A. Exactly equal to 4.1 B. Greater than 4.1 C. It depends on the sample size D. Less than 4.1\nPart (f) If we made a histogram of these 150 observations, what shape would we 'expect' this histogram to have? [ANS] A. Skewed to the right B. Skewed to the left C. It depends on the sample size D. Symmetric E. Normally distributed\nPart (g) (On-Paper) Again, consider taking a sample of 150 observations from this population. In a few sentences (you are encouraged to use pictures or notation if this helps), explain what is mean by the term \"The Sampling Distribution of The Sample Mean\", in the context of this example. In your explanation, imagine that you are explaining to a friend with limited understanding of statistics, and avoid the use of technical terms as much as possible. Also, please state any assumptions that are relevant to your description of the sampling distribution.", "answer_v2": [ "A", "D", "D", "D", "A", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Consider measuring the time until death after diagnosis of a particular disease, and suppose that we have complete follow up, so that all death times are observed. We will assume the following to be known values for the population. We know that the mean time until death after diagnosis is 6.5 years. We also know that the standard deviation of time until death is 4.1 years. Further, we know that the distribution of time until death is very strongly skewed to the right (positively skewed) for this population. Use this to answer the following.\nPart (a) Suppose that we take a simple random sample of 15 people from this population. What would we 'expect' our sample mean to be for our sample of 15 individuals? That is, what is the expected value of the sample mean? [ANS] A. Exactly equal to 6.5 B. Less than 6.5 C. It depends on the sample size D. Greater than 6.5\nPart (b) Suppose that we take a simple random sample of 15 people from this population. What would we 'expect' our sample standard deviation to be for our sample of 15 individuals? That is, what is the expected value of the sample standard deviation? [ANS] A. Greater than 4.1 B. Exactly equal to 4.1 C. It depends on the sample size D. Less than 4.1\nPart (c) If we made a histogram of these 10 observations, what shape would we 'expect' this histogram to have? [ANS] A. It depends on the sample size B. Skewed to the right C. Skewed to the left D. Normally distributed E. Symmetric\nPart (d) Suppose that we take a simple random sample of 150 people from this population. What would we 'expect' our sample mean to be for our sample of 150 individuals? That is, what is the expected value of the sample mean? [ANS] A. Exactly equal to 6.5 B. Less than 6.5 C. Greater than 6.5 D. It depends on the sample size\nPart (e) Suppose that we take a simple random sample of 150 people from this population. What would we 'expect' our sample standard deviation to be for our sample of 150 individuals? That is, what is the expected value of the sample standard deviation? [ANS] A. Exactly equal to 4.1 B. Greater than 4.1 C. Less than 4.1 D. It depends on the sample size\nPart (f) If we made a histogram of these 150 observations, what shape would we 'expect' this histogram to have? [ANS] A. Skewed to the right B. Normally distributed C. Skewed to the left D. It depends on the sample size E. Symmetric\nPart (g) (On-Paper) Again, consider taking a sample of 150 observations from this population. In a few sentences (you are encouraged to use pictures or notation if this helps), explain what is mean by the term \"The Sampling Distribution of The Sample Mean\", in the context of this example. In your explanation, imagine that you are explaining to a friend with limited understanding of statistics, and avoid the use of technical terms as much as possible. Also, please state any assumptions that are relevant to your description of the sampling distribution.", "answer_v3": [ "A", "B", "B", "A", "A", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Statistics_0118", "subject": "Statistics", "topic": "Sampling distributions", "subtopic": "Sample mean", "level": "3", "keywords": [ "statistics", "sampling distributions", "normal sampling distributions" ], "problem_v1": "An HP laser printer is advertised to print text documents at a speed of 18 ppm (pages per minute). The manufacturer tells you that the printing speed is actually a Normal random variable with a mean of 17.67 ppm and a standard deviation of 4.25 ppm. Suppose that you draw a random sample of 13 printers.\nPart i) Using the information about the distribution of the printing speeds given by the manufacturer, find the probability that the mean printing speed of the sample is greater than 18.47 ppm. (Please carry answers to at least six decimal places in intermediate steps. Give your final answer to the nearest three decimal places). Probability (as a proportion) [ANS]\nPart ii) Use normal approximation to find the probability that more than 48.9\\% of the sampled printers operate at the advertised speed (i.e. the printing speed is equal to or greater than 18 ppm) [ANS] A. 0.5574 B. 0.5044 C. 0.4956 D. 0.4426", "answer_v1": [ "0.248666515567938", "D" ], "answer_type_v1": [ "NV", "MCS" ], "options_v1": [ [], [ "A", "B", "C", "D" ] ], "problem_v2": "An HP laser printer is advertised to print text documents at a speed of 18 ppm (pages per minute). The manufacturer tells you that the printing speed is actually a Normal random variable with a mean of 17.07 ppm and a standard deviation of 5 ppm. Suppose that you draw a random sample of 10 printers.\nPart i) Using the information about the distribution of the printing speeds given by the manufacturer, find the probability that the mean printing speed of the sample is greater than 17.47 ppm. (Please carry answers to at least six decimal places in intermediate steps. Give your final answer to the nearest three decimal places). Probability (as a proportion) [ANS]\nPart ii) Use normal approximation to find the probability that more than 47.6\\% of the sampled printers operate at the advertised speed (i.e. the printing speed is equal to or greater than 18 ppm) [ANS] A. 0.6254 B. 0.3746 C. 0.4872 D. 0.5128", "answer_v2": [ "0.400140978177843", "B" ], "answer_type_v2": [ "NV", "MCS" ], "options_v2": [ [], [ "A", "B", "C", "D" ] ], "problem_v3": "An HP laser printer is advertised to print text documents at a speed of 18 ppm (pages per minute). The manufacturer tells you that the printing speed is actually a Normal random variable with a mean of 17.28 ppm and a standard deviation of 4.25 ppm. Suppose that you draw a random sample of 11 printers.\nPart i) Using the information about the distribution of the printing speeds given by the manufacturer, find the probability that the mean printing speed of the sample is greater than 17.88 ppm. (Please carry answers to at least six decimal places in intermediate steps. Give your final answer to the nearest three decimal places). Probability (as a proportion) [ANS]\nPart ii) Use normal approximation to find the probability that more than 45.3\\% of the sampled printers operate at the advertised speed (i.e. the printing speed is equal to or greater than 18 ppm) [ANS] A. 0.5049 B. 0.4467 C. 0.4951 D. 0.5533", "answer_v3": [ "0.31981028025327", "B" ], "answer_type_v3": [ "NV", "MCS" ], "options_v3": [ [], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0119", "subject": "Statistics", "topic": "Sampling distributions", "subtopic": "Sample mean", "level": "2", "keywords": [ "statistics", "numerical", "descriptive statistics", "mean", "standard deviation" ], "problem_v1": "A sample of 12 measurements has a mean of 35 and a standard deviation of 3.75. Suppose that the sample is enlarged to 14 measurements, by including two additional measurements having a common value of 35 each. A. Find the mean of the sample of 14 measurements.\nMean=[ANS]\nB. Find the standard deviation of the sample of 14 measurements. Standard Deviation=[ANS]", "answer_v1": [ "35", "3.44949829127925" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A sample of 12 measurements has a mean of 21 and a standard deviation of 5. Suppose that the sample is enlarged to 14 measurements, by including two additional measurements having a common value of 21 each. A. Find the mean of the sample of 14 measurements.\nMean=[ANS]\nB. Find the standard deviation of the sample of 14 measurements. Standard Deviation=[ANS]", "answer_v2": [ "21", "4.599331055039" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A sample of 12 measurements has a mean of 26 and a standard deviation of 3.75. Suppose that the sample is enlarged to 14 measurements, by including two additional measurements having a common value of 26 each. A. Find the mean of the sample of 14 measurements.\nMean=[ANS]\nB. Find the standard deviation of the sample of 14 measurements. Standard Deviation=[ANS]", "answer_v3": [ "26", "3.44949829127925" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0120", "subject": "Statistics", "topic": "Sampling distributions", "subtopic": "Sample proportion", "level": "3", "keywords": [ "central limit" ], "problem_v1": "A survey of 88 students found that 38\\% were in favor of raising tuition to build a new recreation center. The standard deviation of the sample proportion is 8.6\\%. How large a sample (to the nearest person) would be required to reduce this standard deviation to 3.5\\%? Answer: [ANS]", "answer_v1": [ "531" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A survey of 54 students found that 48\\% were in favor of raising tuition to pave new parking lots. The standard deviation of the sample proportion is 6.6\\%. How large a sample (to the nearest person) would be required to reduce this standard deviation to 2.9\\%? Answer: [ANS]", "answer_v2": [ "280" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A survey of 65 students found that 38\\% were in favor of raising tuition to build a new math building. The standard deviation of the sample proportion is 7.7\\%. How large a sample (to the nearest person) would be required to reduce this standard deviation to 3\\%? Answer: [ANS]", "answer_v3": [ "428" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0121", "subject": "Statistics", "topic": "Sampling distributions", "subtopic": "Sample proportion", "level": "2", "keywords": [], "problem_v1": "The proportion of adults who own a cell phone in a certain Canadian city is believed to be 85\\%. Thirty adults are to be selected at random from the city. Let $X$ be the number in the sample who own a cell phone. Under the assumptions given, the distribution of $X$ is exactly [ANS] A. $Bin(30,25.5)$ B. $N(30,4.5)$ C. $Bin(30,0.85)$ D. $N(25.5,4.5)$", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "The proportion of adults who own a cell phone in a certain Canadian city is believed to be 60\\%. Fifty adults are to be selected at random from the city. Let $X$ be the number in the sample who own a cell phone. Under the assumptions given, the distribution of $X$ is exactly [ANS] A. $Bin(50,30)$ B. $N(50,20)$ C. $N(30,20)$ D. $Bin(50,0.6)$", "answer_v2": [ "D" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "The proportion of adults who own a cell phone in a certain Canadian city is believed to be 70\\%. Forty adults are to be selected at random from the city. Let $X$ be the number in the sample who own a cell phone. Under the assumptions given, the distribution of $X$ is exactly [ANS] A. $Bin(40,28)$ B. $Bin(40,0.7)$ C. $N(28,12)$ D. $N(40,12)$", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0122", "subject": "Statistics", "topic": "Sampling distributions", "subtopic": "Sample proportion", "level": "3", "keywords": [ "Statistical inference", "Point estimation", "biasedness", "Binomial data", "determine the bias for an estimator of a population proportion" ], "problem_v1": "In a study to estimate the proportion of residents in a city that support the construction of a new bypass road in the vicinity, a random sample of $2025$ residents were polled. Let $X$ denote the number in the sample who supported the proposal. To estimate the true proportion in support of the plan, we can compute $\\hat{p}=\\frac{X+\\sqrt{2025}/2}{2025}.$ The estimator $\\hat{p}$ has bias [ANS] A. $1/(2 \\times 45)$. B. 0, so unbiased. C. $1/45$. D. $1/2025$. E. $45/2$.", "answer_v1": [ "A" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ] ], "problem_v2": "In a study to estimate the proportion of residents in a city that support the construction of a new bypass road in the vicinity, a random sample of $625$ residents were polled. Let $X$ denote the number in the sample who supported the proposal. To estimate the true proportion in support of the plan, we can compute $\\hat{p}=\\frac{X+\\sqrt{625}/2}{625}.$ The estimator $\\hat{p}$ has bias [ANS] A. 0, so unbiased. B. $1/25$. C. $1/625$. D. $1/(2 \\times 25)$. E. $25/2$.", "answer_v2": [ "D" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ] ], "problem_v3": "In a study to estimate the proportion of residents in a city that support the construction of a new bypass road in the vicinity, a random sample of $900$ residents were polled. Let $X$ denote the number in the sample who supported the proposal. To estimate the true proportion in support of the plan, we can compute $\\hat{p}=\\frac{X+\\sqrt{900}/2}{900}.$ The estimator $\\hat{p}$ has bias [ANS] A. $30/2$. B. 0, so unbiased. C. $1/900$. D. $1/30$. E. $1/(2 \\times 30)$.", "answer_v3": [ "E" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ] ] }, { "id": "Statistics_0123", "subject": "Statistics", "topic": "Sampling distributions", "subtopic": "General", "level": "1", "keywords": [ "statistic", "parameter", "sampling distribution", "sample" ], "problem_v1": "For the following problems, select the best response:\n(a) Sampling variation is caused by [ANS] A. changes in a population parameter from sample to sample. B. systematic errors in our procedure. C. changes in a population parameter that cannot be predicted. D. random selection of a sample.\n(b) A statistic is said to be unbiased if [ANS] A. it is used for only honest purposes. B. the survey used to obtain the statistic was designed so as to avoid even the hint of racial or sexual prejudice. C. both the person who calculated the statistic and the subjects whose responses make up the statistic were truthful. D. the mean of its sampling distribution is equal to the true value of the parameter being estimated.\n(c) The sampling distribution of a statistic is [ANS] A. the distribution of values taken by a statistic in all possible samples of the same size from the same population. B. the extent to which the sample results differ systematically from the truth. C. the mechanism that determines whether or not randomization was effective. D. the probability that we obtain the statistic in repeated random samples.", "answer_v1": [ "D", "D", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "For the following problems, select the best response:\n(a) The sampling distribution of a statistic is [ANS] A. the mechanism that determines whether or not randomization was effective. B. the extent to which the sample results differ systematically from the truth. C. the probability that we obtain the statistic in repeated random samples. D. the distribution of values taken by a statistic in all possible samples of the same size from the same population.\n(b) Sampling variation is caused by [ANS] A. changes in a population parameter from sample to sample. B. random selection of a sample. C. changes in a population parameter that cannot be predicted. D. systematic errors in our procedure.\n(c) A statistic is said to be unbiased if [ANS] A. the mean of its sampling distribution is equal to the true value of the parameter being estimated. B. both the person who calculated the statistic and the subjects whose responses make up the statistic were truthful. C. it is used for only honest purposes. D. the survey used to obtain the statistic was designed so as to avoid even the hint of racial or sexual prejudice.", "answer_v2": [ "D", "B", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "For the following problems, select the best response:\n(a) The sampling distribution of a statistic is [ANS] A. the distribution of values taken by a statistic in all possible samples of the same size from the same population. B. the probability that we obtain the statistic in repeated random samples. C. the mechanism that determines whether or not randomization was effective. D. the extent to which the sample results differ systematically from the truth.\n(b) Sampling variation is caused by [ANS] A. systematic errors in our procedure. B. random selection of a sample. C. changes in a population parameter that cannot be predicted. D. changes in a population parameter from sample to sample.\n(c) A statistic is said to be unbiased if [ANS] A. the mean of its sampling distribution is equal to the true value of the parameter being estimated. B. both the person who calculated the statistic and the subjects whose responses make up the statistic were truthful. C. the survey used to obtain the statistic was designed so as to avoid even the hint of racial or sexual prejudice. D. it is used for only honest purposes.", "answer_v3": [ "A", "B", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0124", "subject": "Statistics", "topic": "Sampling distributions", "subtopic": "General", "level": "3", "keywords": [ "probability", "normal distribution", "finite population", "sample size" ], "problem_v1": "If two populations are normally distributed, the sampling distribution of the sample mean difference $\\bar{X_1}-\\bar{X_2}$ will be: [ANS] A. normally distributed B. normally distributed only if both population sizes are greater than 30 C. normally distributed only if both sample sizes are greater than 30 D. approximately normally distributed\nGiven a binomial distribution with $n$ trials and probability $p$ of success on any trial, a conventional rule of thumb is that the normal distribution will provide an adequate approximation of the binomial distribution if [ANS] A. $np \\leq 5$ and $n(1-p) \\geq 5$ B. $np \\leq 5$ and $n(1-p) \\leq 5$ C. $np \\geq 5$ and $n(1-p) \\geq 5$ D. $np \\geq 5$ and $n(1-p) \\leq 5$", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "If two populations are normally distributed, the sampling distribution of the sample mean difference $\\bar{X_1}-\\bar{X_2}$ will be: [ANS] A. normally distributed only if both sample sizes are greater than 30 B. normally distributed only if both population sizes are greater than 30 C. approximately normally distributed D. normally distributed\nThe finite population correction factor should not be used when: [ANS] A. we are sampling from an infinite population B. we are sampling from a finite population C. sample size is greater than 1\\% of the population size D. None of the above statements is correct", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "If two populations are normally distributed, the sampling distribution of the sample mean difference $\\bar{X_1}-\\bar{X_2}$ will be: [ANS] A. normally distributed only if both sample sizes are greater than 30 B. normally distributed C. normally distributed only if both population sizes are greater than 30 D. approximately normally distributed\nIf two random samples of sizes $n_1$ and $n_2$ are selected independently from two populations with variances $\\sigma_1^2$ and $\\sigma_2^2$, then the standard error of the sampling distribution of the sample mean difference, $\\bar{X_1}-\\bar{X_2}$, equals [ANS] A. $ \\sqrt{\\frac{\\sigma_1^2}{n_1}+\\frac{\\sigma_2^2}{n_2}}$ B. $ \\sqrt{\\frac{\\sigma_1^2+\\sigma_2^2}{n_1 n_2}}$ C. $ \\sqrt{\\frac{\\sigma_1^2-\\sigma_2^2}{n_1 n_2}}$ D. $ \\sqrt{\\frac{\\sigma_1^2}{n_1}-\\frac{\\sigma_2^2}{n_2}}$", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0125", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "Concepts", "level": "2", "keywords": [ "independent sampling", "difference", "inferences" ], "problem_v1": "The purpose of this question is to compare the variability of $\\overline{x}_1$ and $\\overline{x}_2$ with the variability of $(\\overline{x}_1-\\overline{x}_2)$.\n(a) $\\ $ Suppose the first sample of $100$ observations is selected from a population with mean $\\mu_1=180$ and variance $\\sigma_1^2=1400$. Construct an interval extending 2 standard deviations of $\\overline{x}_1$ on each side of $\\mu_1$. [ANS] $\\leq \\mu_1 \\leq$ [ANS]\n(b) $\\ $ Suppose the second sample of $100$ observations is selected from a population with mean $\\mu_2=180$ and variance $\\sigma_2^2=1270$. Construct an interval extending 2 standard deviations of $\\overline{x}_2$ on each side of $\\mu_2$. [ANS] $\\leq \\mu_2 \\leq$ [ANS]\n(c) $\\ $ Consider the difference between the two sample means $(\\overline{x}_1-\\overline{x}_2)$. Compute the mean and the standard deviation of the sampling distribution of $(\\overline{x}_1-\\overline{x}_2)$. mean=[ANS]\nstandard deviation=[ANS]\n(d) $\\ $ Based on $100$ observations, construct an interval extending 2 standard deviations of $(\\overline{x}_1-\\overline{x}_2)$ on each side of $(\\mu_1-\\mu_2)$ [ANS] $\\leq (\\mu_1-\\mu_2) \\leq$ [ANS]", "answer_v1": [ "172.516685226452", "187.483314773548", "172.872588127518", "187.127411872482", "0", "5.16720427310552", "-10.334408546211", "10.334408546211" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "The purpose of this question is to compare the variability of $\\overline{x}_1$ and $\\overline{x}_2$ with the variability of $(\\overline{x}_1-\\overline{x}_2)$.\n(a) $\\ $ Suppose the first sample of $100$ observations is selected from a population with mean $\\mu_1=150$ and variance $\\sigma_1^2=860$. Construct an interval extending 2 standard deviations of $\\overline{x}_1$ on each side of $\\mu_1$. [ANS] $\\leq \\mu_1 \\leq$ [ANS]\n(b) $\\ $ Suppose the second sample of $100$ observations is selected from a population with mean $\\mu_2=150$ and variance $\\sigma_2^2=1550$. Construct an interval extending 2 standard deviations of $\\overline{x}_2$ on each side of $\\mu_2$. [ANS] $\\leq \\mu_2 \\leq$ [ANS]\n(c) $\\ $ Consider the difference between the two sample means $(\\overline{x}_1-\\overline{x}_2)$. Compute the mean and the standard deviation of the sampling distribution of $(\\overline{x}_1-\\overline{x}_2)$. mean=[ANS]\nstandard deviation=[ANS]\n(d) $\\ $ Based on $100$ observations, construct an interval extending 2 standard deviations of $(\\overline{x}_1-\\overline{x}_2)$ on each side of $(\\mu_1-\\mu_2)$ [ANS] $\\leq (\\mu_1-\\mu_2) \\leq$ [ANS]", "answer_v2": [ "144.134848680554", "155.865151319446", "142.125992125988", "157.874007874012", "0", "4.90917508345343", "-9.81835016690686", "9.81835016690686" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "The purpose of this question is to compare the variability of $\\overline{x}_1$ and $\\overline{x}_2$ with the variability of $(\\overline{x}_1-\\overline{x}_2)$.\n(a) $\\ $ Suppose the first sample of $100$ observations is selected from a population with mean $\\mu_1=160$ and variance $\\sigma_1^2=1050$. Construct an interval extending 2 standard deviations of $\\overline{x}_1$ on each side of $\\mu_1$. [ANS] $\\leq \\mu_1 \\leq$ [ANS]\n(b) $\\ $ Suppose the second sample of $100$ observations is selected from a population with mean $\\mu_2=160$ and variance $\\sigma_2^2=1290$. Construct an interval extending 2 standard deviations of $\\overline{x}_2$ on each side of $\\mu_2$. [ANS] $\\leq \\mu_2 \\leq$ [ANS]\n(c) $\\ $ Consider the difference between the two sample means $(\\overline{x}_1-\\overline{x}_2)$. Compute the mean and the standard deviation of the sampling distribution of $(\\overline{x}_1-\\overline{x}_2)$. mean=[ANS]\nstandard deviation=[ANS]\n(d) $\\ $ Based on $100$ observations, construct an interval extending 2 standard deviations of $(\\overline{x}_1-\\overline{x}_2)$ on each side of $(\\mu_1-\\mu_2)$ [ANS] $\\leq (\\mu_1-\\mu_2) \\leq$ [ANS]", "answer_v3": [ "153.519259301592", "166.480740698408", "152.816686001573", "167.183313998427", "0", "4.83735464897913", "-9.67470929795826", "9.67470929795826" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Statistics_0126", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "Concepts", "level": "1", "keywords": [ "small sample", "confidence interval", "statistics", "estimates" ], "problem_v1": "Suppose you have selected a random sample of $n=12$ measurements from a normal distribution. Compare the standard normal $z$ values with the corresponding $t$ values if you were forming the following confidence intervals.\n(a) $\\ $ $95$ \\% confidence interval $z=$ [ANS]\n$t=$ [ANS]\n(b) $\\ $ $98$ \\% confidence interval $z=$ [ANS]\n$t=$ [ANS]\n(c) $\\ $ $99$ \\% confidence interval $z=$ [ANS]\n$t=$ [ANS]", "answer_v1": [ "1.95996398465615", "2.20098", "2.32634787539286", "2.71808", "2.57582930805671", "3.10580" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Suppose you have selected a random sample of $n=4$ measurements from a normal distribution. Compare the standard normal $z$ values with the corresponding $t$ values if you were forming the following confidence intervals.\n(a) $\\ $ $99$ \\% confidence interval $z=$ [ANS]\n$t=$ [ANS]\n(b) $\\ $ $80$ \\% confidence interval $z=$ [ANS]\n$t=$ [ANS]\n(c) $\\ $ $95$ \\% confidence interval $z=$ [ANS]\n$t=$ [ANS]", "answer_v2": [ "2.57582930805671", "5.84091", "1.28155156554455", "1.63774", "1.95996398465615", "3.18245" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Suppose you have selected a random sample of $n=7$ measurements from a normal distribution. Compare the standard normal $z$ values with the corresponding $t$ values if you were forming the following confidence intervals.\n(a) $\\ $ $98$ \\% confidence interval $z=$ [ANS]\n$t=$ [ANS]\n(b) $\\ $ $90$ \\% confidence interval $z=$ [ANS]\n$t=$ [ANS]\n(c) $\\ $ $95$ \\% confidence interval $z=$ [ANS]\n$t=$ [ANS]", "answer_v3": [ "2.32634787539286", "3.14261", "1.64485362695934", "1.94318", "1.95996398465615", "2.44690" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Statistics_0127", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "Concepts", "level": "2", "keywords": [ "statistics", "inference" ], "problem_v1": "A report in a research journal states that the average weight loss of people on a certain drug is $27$ lbs with a margin of error of $\\pm4$ lbs with confidence level C=95\\%.\n(a) According to this information, the mean weight loss of people on this drug, $\\mu$, could be as low as [ANS] lbs. (b) If the study is repeated, how large should the sample size be so that the margin of error would be less than 2 lbs? (Assume $\\sigma=\\ 9$ lbs.) ANSWER: [ANS]", "answer_v1": [ "23", "78" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A report in a research journal states that the average weight loss of people on a certain drug is $34$ lbs with a margin of error of $\\pm2$ lbs with confidence level C=95\\%.\n(a) According to this information, the mean weight loss of people on this drug, $\\mu$, could be as low as [ANS] lbs. (b) If the study is repeated, how large should the sample size be so that the margin of error would be less than 1 lbs? (Assume $\\sigma=\\ 5$ lbs.) ANSWER: [ANS]", "answer_v2": [ "32", "97" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A report in a research journal states that the average weight loss of people on a certain drug is $27$ lbs with a margin of error of $\\pm3$ lbs with confidence level C=95\\%.\n(a) According to this information, the mean weight loss of people on this drug, $\\mu$, could be as low as [ANS] lbs. (b) If the study is repeated, how large should the sample size be so that the margin of error would be less than 1.5 lbs? (Assume $\\sigma=\\ 6$ lbs.) ANSWER: [ANS]", "answer_v3": [ "24", "62" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0128", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "Concepts", "level": "2", "keywords": [ "statistics", "inference", "confidence interval" ], "problem_v1": "For each problem, select the best response.\n(a) A 95\\% confidence interval for the mean $\\mu$ of a population is computed from a random sample and found to be $10 \\pm 4$. We may conclude [ANS] A. that if we took many, many additional samples and from each computed a 95\\% confidence interval for $\\mu$, approximately 95\\% of these intervals would contain $\\mu$. B. 95\\% of the population is between 6 and 14. C. that there is a 95\\% probability that $\\mu$ is between 6 and 14. D. that there is a 95\\% probability that the true mean is 10 and a 95\\% chance the true margin of error is 4. E. All of the above.\n(b) Suppose you collect a SRS of size n from a population and from the data collected you computed a 95\\% confidence interval for the mean of the population. Which of the following would produce a new confidence interval with larger width (larger margin of error) based on these same data? [ANS] A. Nothing can guarantee absolutely that you will get a larger interval. One can only say the chance of obtaining a larger interval is 0.05. B. Use a smaller confidence level. C. Use a larger confidence level. D. Use the same confidence level, but compute the interval n times. Approximately 5\\% of these intervals will be larger. E. None of the above.", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "For each problem, select the best response.\n(a) Suppose you collect a SRS of size n from a population and from the data collected you computed a 95\\% confidence interval for the mean of the population. Which of the following would produce a new confidence interval with larger width (larger margin of error) based on these same data? [ANS] A. Nothing can guarantee absolutely that you will get a larger interval. One can only say the chance of obtaining a larger interval is 0.05. B. Use a smaller confidence level. C. Use the same confidence level, but compute the interval n times. Approximately 5\\% of these intervals will be larger. D. Use a larger confidence level. E. None of the above.\n(b) A 95\\% confidence interval for the mean $\\mu$ of a population is computed from a random sample and found to be $10 \\pm 4$. We may conclude [ANS] A. that there is a 95\\% probability that $\\mu$ is between 6 and 14. B. 95\\% of the population is between 6 and 14. C. that there is a 95\\% probability that the true mean is 10 and a 95\\% chance the true margin of error is 4. D. that if we took many, many additional samples and from each computed a 95\\% confidence interval for $\\mu$, approximately 95\\% of these intervals would contain $\\mu$. E. All of the above.", "answer_v2": [ "D", "D" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "For each problem, select the best response.\n(a) Suppose you collect a SRS of size n from a population and from the data collected you computed a 95\\% confidence interval for the mean of the population. Which of the following would produce a new confidence interval with larger width (larger margin of error) based on these same data? [ANS] A. Nothing can guarantee absolutely that you will get a larger interval. One can only say the chance of obtaining a larger interval is 0.05. B. Use a larger confidence level. C. Use the same confidence level, but compute the interval n times. Approximately 5\\% of these intervals will be larger. D. Use a smaller confidence level. E. None of the above.\n(b) A 95\\% confidence interval for the mean $\\mu$ of a population is computed from a random sample and found to be $10 \\pm 4$. We may conclude [ANS] A. that if we took many, many additional samples and from each computed a 95\\% confidence interval for $\\mu$, approximately 95\\% of these intervals would contain $\\mu$. B. that there is a 95\\% probability that the true mean is 10 and a 95\\% chance the true margin of error is 4. C. that there is a 95\\% probability that $\\mu$ is between 6 and 14. D. 95\\% of the population is between 6 and 14. E. All of the above.", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Statistics_0129", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "Concepts", "level": "3", "keywords": [ "Statistics" ], "problem_v1": "A random sample of 20 U.S. weddings yielded the following data on wedding costs in dollars: 27576, 22442, 23615, 26675, 14044, 14928, 22017, 22221, 16118, 19824, 24584, 13092, 20215, 17818, 16900, 18039, 20419, 9029, 9743, 16815 a) Use the data to obtain a point estimate for the population mean wedding cost, $\\mu$, of all recent U.S. weddings. Note: The sum of the data is 376114. $\\bar{x}=$ [ANS]\nb) Is your point estimate in part (a) likely equal to $\\mu$ exactly (yes or no)? [ANS]", "answer_v1": [ "18805.7", "NO" ], "answer_type_v1": [ "NV", "TF" ], "options_v1": [ [], [] ], "problem_v2": "A random sample of 20 U.S. weddings yielded the following data on wedding costs in dollars: 7482, 32948, 9471, 15013, 33407, 14486, 10491, 14843, 21915, 7040, 24227, 18226, 29327, 10780, 10678, 13109, 22204, 10627, 14673, 21694 a) Use the data to obtain a point estimate for the population mean wedding cost, $\\mu$, of all recent U.S. weddings. Note: The sum of the data is 342641. $\\bar{x}=$ [ANS]\nb) Is your point estimate in part (a) likely equal to $\\mu$ exactly (yes or no)? [ANS]", "answer_v2": [ "17132.05", "NO" ], "answer_type_v2": [ "NV", "TF" ], "options_v2": [ [], [] ], "problem_v3": "A random sample of 20 U.S. weddings yielded the following data on wedding costs in dollars: 14401, 23156, 13347, 21456, 11171, 15370, 29259, 32336, 31180, 11001, 13687, 12531, 6275, 22170, 34376, 28749, 23428, 8162, 13772, 24227 a) Use the data to obtain a point estimate for the population mean wedding cost, $\\mu$, of all recent U.S. weddings. Note: The sum of the data is 390054. $\\bar{x}=$ [ANS]\nb) Is your point estimate in part (a) likely equal to $\\mu$ exactly (yes or no)? [ANS]", "answer_v3": [ "19502.7", "NO" ], "answer_type_v3": [ "NV", "TF" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0130", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "Concepts", "level": "2", "keywords": [ "Statistics" ], "problem_v1": "What is the relationship between confidence and precision? [ANS] A. For a fixed sample size, decreasing the confidence level has no effect on the precision B. For a fixed sample size, decreasing the confidence level decreases the precision C. For a fixed sample size, decreasing the confidence level increases the precision D. None of the above", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "What is the relationship between confidence and precision? [ANS] A. For a fixed sample size, decreasing the confidence level increases the precision B. For a fixed sample size, decreasing the confidence level decreases the precision C. For a fixed sample size, decreasing the confidence level has no effect on the precision D. None of the above", "answer_v2": [ "A" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "What is the relationship between confidence and precision? [ANS] A. For a fixed sample size, decreasing the confidence level has no effect on the precision B. For a fixed sample size, decreasing the confidence level increases the precision C. For a fixed sample size, decreasing the confidence level decreases the precision D. None of the above", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0131", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "Concepts", "level": "3", "keywords": [ "Statistics" ], "problem_v1": "Suppose you calculate a 99\\% confidence interval of 14.51 to 21.16. Determine the margin of error. [ANS]", "answer_v1": [ "3.325" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose you calculate a 99\\% confidence interval of 13.16 to 21.87. Determine the margin of error. [ANS]", "answer_v2": [ "4.355" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose you calculate a 99\\% confidence interval of 13.63 to 21.21. Determine the margin of error. [ANS]", "answer_v3": [ "3.79" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0132", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "Concepts", "level": "3", "keywords": [ "Statistics" ], "problem_v1": "A confidence interval for a population mean has length 26. a) Determine the margin of error. [ANS]\nb) If the sample mean is 60.8, obtain the confidence interval. Confidence interval: ([ANS], [ANS]).", "answer_v1": [ "13", "47.8", "73.8" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A confidence interval for a population mean has length 20. a) Determine the margin of error. [ANS]\nb) If the sample mean is 64.4, obtain the confidence interval. Confidence interval: ([ANS], [ANS]).", "answer_v2": [ "10", "54.4", "74.4" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A confidence interval for a population mean has length 22. a) Determine the margin of error. [ANS]\nb) If the sample mean is 61.1, obtain the confidence interval. Confidence interval: ([ANS], [ANS]).", "answer_v3": [ "11", "50.1", "72.1" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Statistics_0133", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "Concepts", "level": "2", "keywords": [ "statistics", "multiple choice", "display methods" ], "problem_v1": "Based on a random sample of 50, a 95\\% confidence interval for the population proportion was computed. Holding everything else constant, which of the following will reduce the length of the confidence interval by half? (CHECK ALL THAT APPLY): [ANS] A. Quadruple the sample size. B. Decrease the sample proportion by half. C. Change the confidence level to 99.7\\%. D. Double the sample size. E. Change the confidence level to 68\\%.", "answer_v1": [ "AE" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Based on a random sample of 50, a 95\\% confidence interval for the population proportion was computed. Holding everything else constant, which of the following will reduce the length of the confidence interval by half? (CHECK ALL THAT APPLY): [ANS] A. Change the confidence level to 68\\%. B. Decrease the sample proportion by half. C. Change the confidence level to 99.7\\%. D. Quadruple the sample size. E. Double the sample size.", "answer_v2": [ "AD" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Based on a random sample of 50, a 95\\% confidence interval for the population proportion was computed. Holding everything else constant, which of the following will reduce the length of the confidence interval by half? (CHECK ALL THAT APPLY): [ANS] A. Double the sample size. B. Change the confidence level to 68\\%. C. Decrease the sample proportion by half. D. Change the confidence level to 99.7\\%. E. Quadruple the sample size.", "answer_v3": [ "BE" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E" ] ] }, { "id": "Statistics_0134", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "Concepts", "level": "3", "keywords": [ "statistics", "Introduction to Estimation" ], "problem_v1": "In developing an interval estimate for a population mean, a sample of 50 observations was used. The interval estimate was $19.76 \\pm 1.32$. Had the sample size been 200 instead of 50, the interval estimate would have been [ANS] A. $19.76 \\pm 0.66$ B. $4.94 \\pm 1.32$ C. $9.88 \\pm 1.32$ D. $19.76 \\pm 0.33$\nThe minimum sample size needed to estimate a population mean within 2 units with a 95\\% confidence when the population standard deviation equals 8 is [ANS] A. 8 B. 9 C. 62 D. 61", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "In developing an interval estimate for a population mean, a sample of 50 observations was used. The interval estimate was $19.76 \\pm 1.32$. Had the sample size been 200 instead of 50, the interval estimate would have been [ANS] A. $9.88 \\pm 1.32$ B. $4.94 \\pm 1.32$ C. $19.76 \\pm 0.33$ D. $19.76 \\pm 0.66$\nThe $z$ value for a 96.6\\% confidence interval estimate is [ANS] A. 2.12 B. 1.96 C. 1.82 D. 2.00", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "In developing an interval estimate for a population mean, a sample of 50 observations was used. The interval estimate was $19.76 \\pm 1.32$. Had the sample size been 200 instead of 50, the interval estimate would have been [ANS] A. $9.88 \\pm 1.32$ B. $19.76 \\pm 0.66$ C. $4.94 \\pm 1.32$ D. $19.76 \\pm 0.33$\nIn developing an interval estimate for a population mean, the interval estimate was 62.84 to 69.46. The population standard deviation was assumed to be 6.50, and a sample of 100 observations was used. The mean of the sample was [ANS] A. 66.15 B. 62.96 C. 56.34 D. 13.24", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0135", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "Concepts", "level": "3", "keywords": [ "statistics", "descriptive statistics" ], "problem_v1": "Select True or False, depending on whether the corresponding statement is true or false.\n[ANS] 1. Estimating characteristics of a population is the main goal of descriptive statistics. [ANS] 2. Statistical inference is the process of making an estimate, prediction, or decision about a population based on sample data. [ANS] 3. In a sample of 500 students at a university, 12\\% of them are accounting majors. The 12\\% is an example of statistical inference. [ANS] 4. Conclusions and estimates about a population based on sample data are not always going to be correct. For this reason measures of reliability, such a significance level and confidence level, should be built into the statistical inference.", "answer_v1": [ "F", "T", "F", "T" ], "answer_type_v1": [ "TF", "TF", "TF", "TF" ], "options_v1": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v2": "Select True or False, depending on whether the corresponding statement is true or false.\n[ANS] 1. Statistical inference is the process of making an estimate, prediction, or decision about a population based on sample data. [ANS] 2. Conclusions and estimates about a population based on sample data are not always going to be correct. For this reason measures of reliability, such a significance level and confidence level, should be built into the statistical inference. [ANS] 3. Estimating characteristics of a population is the main goal of descriptive statistics. [ANS] 4. A local cable system using a sample of 500 subscribers estimates that forty percent of its subscribers watch a premium channel at least once per day. This is an example of statistical inference as opposed to descriptive statistics.", "answer_v2": [ "T", "T", "F", "T" ], "answer_type_v2": [ "TF", "TF", "TF", "TF" ], "options_v2": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v3": "Select True or False, depending on whether the corresponding statement is true or false.\n[ANS] 1. A summary measure that is computed from a population is called a parameter. [ANS] 2. A local cable system using a sample of 500 subscribers estimates that forty percent of its subscribers watch a premium channel at least once per day. This is an example of statistical inference as opposed to descriptive statistics. [ANS] 3. Estimating characteristics of a population is the main goal of descriptive statistics. [ANS] 4. The confidence level is the proportion of times that an estimating procedure will be wrong.", "answer_v3": [ "T", "T", "F", "F" ], "answer_type_v3": [ "TF", "TF", "TF", "TF" ], "options_v3": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ] }, { "id": "Statistics_0136", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "Concepts", "level": "4", "keywords": [ "statistics", "Inference", "inference about a population" ], "problem_v1": "Based on the sample data, the 90\\% confidence interval limits for the population mean are LCL=170.86, UCL=195.42. If the 10\\% level of significance were used in testing the hypotheses $H_0: \\mu=201$ vs. $H_1: \\mu \\not=201$, the null hypothesis: [ANS] A. would be rejected B. would have to be revised C. would not be rejected D. None of the above\nA random sample of 25 observations is selected from a normally distributed population. The sample variance is 10. In the 95\\% confidence interval for the population variance, the upper limit will be: [ANS] A. 17.331 B. 17.110 C. 19.353 D. 6.097", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Based on the sample data, the 90\\% confidence interval limits for the population mean are LCL=170.86, UCL=195.42. If the 10\\% level of significance were used in testing the hypotheses $H_0: \\mu=201$ vs. $H_1: \\mu \\not=201$, the null hypothesis: [ANS] A. would be rejected B. would not be rejected C. would have to be revised D. None of the above\nIn testing the hypotheses $H_0: p=0.40$ vs. $H_1: p > 0.40$ at the 5\\% level, if the sample proportion is 0.45 and the standard error of the sample proportion is 0.035, the appropriate conclusion would be: [ANS] A. not to reject $H_0$ B. to reject both $H_0$ and $H_1$ C. to reject $H_0$ D. to reject $H_1$", "answer_v2": [ "A", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Based on the sample data, the 90\\% confidence interval limits for the population mean are LCL=170.86, UCL=195.42. If the 10\\% level of significance were used in testing the hypotheses $H_0: \\mu=201$ vs. $H_1: \\mu \\not=201$, the null hypothesis: [ANS] A. would have to be revised B. would be rejected C. would not be rejected D. None of the above\nIn a hypothesis test for the population variance, the hypotheses are $H_0: \\sigma^2=100$ vs. $H_1: \\sigma^2 \\not=100$. If the sample size is 15 and the test is carried out at the 10\\% level of significance, the rejection region is [ANS] A. $\\chi^2 < 6.571$ or $\\chi^2 > 23.685$ B. $\\chi^2 < 8.547$ or $\\chi^2 > 22.307$ C. $\\chi^2 < 7.790$ or $\\chi^2 > 21.064$ D. $\\chi^2 < 7.261$ or $\\chi^2 < 24.996$", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0137", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "2", "keywords": [], "problem_v1": "The EPA wants to test a randomly selected sample of $n$ water specimens and estimate the mean daily rate of pollution produced by a mining operation. If the EPA wants a $95$ \\% confidence interval with a bound of error of $2$ milligram per liter (mg/L), how many water specimens are required in the sample? Assume prior knowledge indicates that pollution readings in water samples taken during a day have been approximately normally distributed with a standard deviation of $4.9$ (mg/L). $n=$ [ANS]", "answer_v1": [ "23.0583565739482" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The EPA wants to test a randomly selected sample of $n$ water specimens and estimate the mean daily rate of pollution produced by a mining operation. If the EPA wants a $99$ \\% confidence interval with a bound of error of $0.5$ milligram per liter (mg/L), how many water specimens are required in the sample? Assume prior knowledge indicates that pollution readings in water samples taken during a day have been approximately normally distributed with a standard deviation of $3.4$ (mg/L). $n=$ [ANS]", "answer_v2": [ "306.797619905039" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The EPA wants to test a randomly selected sample of $n$ water specimens and estimate the mean daily rate of pollution produced by a mining operation. If the EPA wants a $98$ \\% confidence interval with a bound of error of $1$ milligram per liter (mg/L), how many water specimens are required in the sample? Assume prior knowledge indicates that pollution readings in water samples taken during a day have been approximately normally distributed with a standard deviation of $3.8$ (mg/L). $n=$ [ANS]", "answer_v3": [ "78.1477556752601" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0138", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "2", "keywords": [ "Confidence Interval", "Mean", "statistics", "estimates", "population", "samples" ], "problem_v1": "Listed below are the lenths (in minutes) of randomly selected music CDs. Construct a 93\\% confidence interval for the mean length of all such CDs. $\\begin{array}{lllllll} 58.06& 52.42& 48.16& 55.15& 56.49& 47.9& 53.56 \\cr 51.03& 59.45& 66.46& 71.17& 72.53& 53.69& 74.84 \\cr 35.15& 70.04& 57.73& 47.68& 46.98& 53.13& 56.35 \\cr 50.39& 55.29& 46.48& 46.14& 53.41& 51.62& 53.55 \\cr 45.42& 57.42& 52.88& 60.06& 54.89& 70.56& 55.49 \\cr 52.94& 35.41& 33.95& 36.2& 56.63 \\cr \\end{array}$ [ANS] $< \\mu <$ [ANS]", "answer_v1": [ "51.1276636268752", "56.7073363731248" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Listed below are the lenths (in minutes) of randomly selected music CDs. Construct a 91\\% confidence interval for the mean length of all such CDs. $\\begin{array}{lllllll} 51.68& 48.79& 62.05& 47.99& 46.36& 53.82& 63.94 \\cr 49.41& 81.68& 56.13& 47.42& 50.27& 78.49& 63.64 \\cr 61.22& 48.85& 38.69& 46.66& 51.33& 57.25& 51.45 \\cr 69.74& 47.16& 50.41& 57.82& 52.35& 48.66& 51.56 \\cr 70.84& 50.07& 56.47& 50.59& 38.88& 39.52& 54.43 \\cr 29.2& 81.11& 39.98& 48.75& 56.28 \\cr \\end{array}$ [ANS] $< \\mu <$ [ANS]", "answer_v2": [ "50.7641144158702", "56.7828855841298" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Listed below are the lenths (in minutes) of randomly selected music CDs. Construct a 91\\% confidence interval for the mean length of all such CDs. $\\begin{array}{lllllll} 63.23& 64.59& 51.77& 57.11& 50.57& 81.05& 62.46 \\cr 56.45& 55.97& 57.82& 63.14& 34.28& 71.88& 65.62 \\cr 49& 51.27& 57.47& 58.27& 47.11& 55.92& 34.57 \\cr 49.12& 51.92& 63.32& 50.8& 50.02& 61.18& 37.3 \\cr 51.92& 50.86& 68.77& 73.3& 45.85& 57.74& 60.84 \\cr 57.29& 52.67& 55.65& 66.64& 70.24 \\cr \\end{array}$ [ANS] $< \\mu <$ [ANS]", "answer_v3": [ "53.9656147833269", "59.2833852166731" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0139", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "2", "keywords": [ "Sample Size", "Standard Deviation", "Rule of Thumb", "Confidence", "Mean" ], "problem_v1": "Suppose that the minimum and maximum ages for typical textbooks currently used in college courses are $0$ and $8$ years. Use the range rule of thumb to estimate the standard deviation. Standard deviation=[ANS] Find the size of the sample required to estimage the mean age of textbooks currently used in college courses. Assume that you want $97$ \\% confidence that the sample mean is within $0.25$ year of the population mean.\nRequired sample size=[ANS]", "answer_v1": [ "2", "302" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose that the minimum and maximum ages for typical textbooks currently used in college courses are $0$ and $8$ years. Use the range rule of thumb to estimate the standard deviation. Standard deviation=[ANS] Find the size of the sample required to estimage the mean age of textbooks currently used in college courses. Assume that you want $90$ \\% confidence that the sample mean is within $0.4$ year of the population mean.\nRequired sample size=[ANS]", "answer_v2": [ "2", "68" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose that the minimum and maximum ages for typical textbooks currently used in college courses are $0$ and $8$ years. Use the range rule of thumb to estimate the standard deviation. Standard deviation=[ANS] Find the size of the sample required to estimage the mean age of textbooks currently used in college courses. Assume that you want $93$ \\% confidence that the sample mean is within $0.25$ year of the population mean.\nRequired sample size=[ANS]", "answer_v3": [ "2", "211" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0140", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "2", "keywords": [], "problem_v1": "According to the Food and Drug Administration (FDA), a cup of coffee contains on average $115$ miligrams (mg) of caffeine, with the amount per cup ranging from $60$ to $180$ mg. Suppose you want to repeat the FDA experiment to obtain an estimate of the mean caffeine content in a cup of coffee correct to witin $5.3$ mg with $95$ \\% confidence. How many cups of coffee would have to be included in your sample? $n=$ [ANS]", "answer_v1": [ "123.079848310228" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "According to the Food and Drug Administration (FDA), a cup of coffee contains on average $115$ miligrams (mg) of caffeine, with the amount per cup ranging from $60$ to $180$ mg. Suppose you want to repeat the FDA experiment to obtain an estimate of the mean caffeine content in a cup of coffee correct to witin $3.2$ mg with $99$ \\% confidence. How many cups of coffee would have to be included in your sample? $n=$ [ANS]", "answer_v2": [ "583.145211115189" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "According to the Food and Drug Administration (FDA), a cup of coffee contains on average $115$ miligrams (mg) of caffeine, with the amount per cup ranging from $60$ to $180$ mg. Suppose you want to repeat the FDA experiment to obtain an estimate of the mean caffeine content in a cup of coffee correct to witin $3.9$ mg with $98$ \\% confidence. How many cups of coffee would have to be included in your sample? $n=$ [ANS]", "answer_v3": [ "320.230440079579" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0141", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "2", "keywords": [ "Confidence Interval", "Mean", "Standard Deviation", "Margin of Error", "statistics", "estimates", "population", "samples" ], "problem_v1": "Starting salaries of 125 college graduates who have taken a statistics course have a mean of \\$43,658 and a standard deviation of \\$10,371. Using a 0.93 degree of confidence, find both of the following: A. $\\ $ The margin of error $E$ [ANS]\nB. $\\ $ The confidence interval for the mean $\\mu$: [ANS] $< \\mu <$ [ANS]", "answer_v1": [ "1680.74725622325", "41977.2527437767", "45338.7472562233" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Starting salaries of 55 college graduates who have taken a statistics course have a mean of \\$44,234 and a standard deviation of \\$9,952. Using a 0.93 degree of confidence, find both of the following: A. $\\ $ The margin of error $E$ [ANS]\nB. $\\ $ The confidence interval for the mean $\\mu$: [ANS] $< \\mu <$ [ANS]", "answer_v2": [ "2431.45258975358", "41802.5474102464", "46665.4525897536" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Starting salaries of 80 college graduates who have taken a statistics course have a mean of \\$43,350 and a standard deviation of \\$9,285. Using a 0.93 degree of confidence, find both of the following: A. $\\ $ The margin of error $E$ [ANS]\nB. $\\ $ The confidence interval for the mean $\\mu$: [ANS] $< \\mu <$ [ANS]", "answer_v3": [ "1880.93461021513", "41469.0653897849", "45230.9346102151" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Statistics_0142", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "1", "keywords": [ "Sample Size", "Mean", "Standard Deviation", "statistics", "estimates", "population", "samples" ], "problem_v1": "The standard IQ test is designed so that the mean is $100$ and the standard deviation is $15$ for the population of all adults. We wish to find the sample size necessary to estimate the mean IQ score of statistics students. Suppose we want to be $97$ \\% confident that our sample mean is within $1.5$ IQ points of the true mean. The mean for this population is clearly greater than $100$. The standard deviation for this population is probably less than $15$ because it is a group with less variation than a group randomly selected from the general population; therefore, if we use $\\sigma=15,$ we are being conservative by using a value that will make the sample size at least as large as necessary. Assume then that $\\sigma=15$ and determine the required sample size. Answer: [ANS]", "answer_v1": [ "471" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The standard IQ test is designed so that the mean is $100$ and the standard deviation is $15$ for the population of all adults. We wish to find the sample size necessary to estimate the mean IQ score of statistics students. Suppose we want to be $90$ \\% confident that our sample mean is within $2$ IQ points of the true mean. The mean for this population is clearly greater than $100$. The standard deviation for this population is probably less than $15$ because it is a group with less variation than a group randomly selected from the general population; therefore, if we use $\\sigma=15,$ we are being conservative by using a value that will make the sample size at least as large as necessary. Assume then that $\\sigma=15$ and determine the required sample size. Answer: [ANS]", "answer_v2": [ "153" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The standard IQ test is designed so that the mean is $100$ and the standard deviation is $15$ for the population of all adults. We wish to find the sample size necessary to estimate the mean IQ score of statistics students. Suppose we want to be $93$ \\% confident that our sample mean is within $1.5$ IQ points of the true mean. The mean for this population is clearly greater than $100$. The standard deviation for this population is probably less than $15$ because it is a group with less variation than a group randomly selected from the general population; therefore, if we use $\\sigma=15,$ we are being conservative by using a value that will make the sample size at least as large as necessary. Assume then that $\\sigma=15$ and determine the required sample size. Answer: [ANS]", "answer_v3": [ "329" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0143", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "1", "keywords": [ "large sample", "confidence interval", "statistics" ], "problem_v1": "A random sample of $n$ measurements was selected from a population with unknown mean $\\mu$ and standard deviation $\\sigma$. Calculate a 99\\% confidence interval for $\\mu$ for each of the following situations:\n(a) $\\ $ $n=100, \\ \\overline{x}=65.6, \\ s=4.17$ [ANS] $\\leq \\mu \\leq$ [ANS]\n(b) $\\ $ $n=85, \\ \\overline{x}=35.4, \\ s=3.7$ [ANS] $\\leq \\mu \\leq$ [ANS]\n(c) $\\ $ $n=100, \\ \\overline{x}=39.6, \\ s=3.48$ [ANS] $\\leq \\mu \\leq$ [ANS]\n(d) $\\ $ $n=105, \\ \\overline{x}=29, \\ s=3.52$ [ANS] $\\leq \\mu \\leq$ [ANS]", "answer_v1": [ "64.50478703", "66.69521297", "34.3422654182999", "36.4577345817001", "38.68600932", "40.51399068", "28.098635895599", "29.901364104401" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "A random sample of $n$ measurements was selected from a population with unknown mean $\\mu$ and standard deviation $\\sigma$. Calculate a 90\\% confidence interval for $\\mu$ for each of the following situations:\n(a) $\\ $ $n=120, \\ \\overline{x}=16.5, \\ s=3$ [ANS] $\\leq \\mu \\leq$ [ANS]\n(b) $\\ $ $n=120, \\ \\overline{x}=33.9, \\ s=2.55$ [ANS] $\\leq \\mu \\leq$ [ANS]\n(c) $\\ $ $n=85, \\ \\overline{x}=59.7, \\ s=2.2$ [ANS] $\\leq \\mu \\leq$ [ANS]\n(d) $\\ $ $n=105, \\ \\overline{x}=46.9, \\ s=4.44$ [ANS] $\\leq \\mu \\leq$ [ANS]", "answer_v2": [ "16.0460037265351", "16.9539962734649", "33.5141031675549", "34.2858968324451", "59.3031213888116", "60.0968786111884", "46.1808785981017", "47.6191214018983" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "A random sample of $n$ measurements was selected from a population with unknown mean $\\mu$ and standard deviation $\\sigma$. Calculate a 90\\% confidence interval for $\\mu$ for each of the following situations:\n(a) $\\ $ $n=100, \\ \\overline{x}=30, \\ s=3.65$ [ANS] $\\leq \\mu \\leq$ [ANS]\n(b) $\\ $ $n=80, \\ \\overline{x}=37, \\ s=4.43$ [ANS] $\\leq \\mu \\leq$ [ANS]\n(c) $\\ $ $n=120, \\ \\overline{x}=91.8, \\ s=2.6$ [ANS] $\\leq \\mu \\leq$ [ANS]\n(d) $\\ $ $n=85, \\ \\overline{x}=27.1, \\ s=2.12$ [ANS] $\\leq \\mu \\leq$ [ANS]", "answer_v3": [ "29.39395765", "30.60604235", "36.1756557521739", "37.8243442478261", "91.4065365629971", "92.1934634370029", "26.7175533383094", "27.4824466616906" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Statistics_0144", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "1", "keywords": [ "large sample", "estimates", "confidence interval" ], "problem_v1": "Studies have suggusted that twins, in their early years, tend to have lower IQs and pick up language more slowly than nontwins. The slower intellectual growth might be caused by benign parental neglect. Suppose it is desired to estimate the mean attention time given to twins per week by their parents. A sample of $70$ sets of 2 year old boys is taken, and after 1 week the attention time received was recorded. The data (in hours) calculated the mean at $23$ and the standard deviation at $11.6$. Use this information to contruct a $98$ \\% confidence interval for the mean attention time given to al twin boys by their parents. [ANS] $< \\mu <$ [ANS]", "answer_v1": [ "19.7745967126229", "26.2254032873771" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Studies have suggusted that twins, in their early years, tend to have lower IQs and pick up language more slowly than nontwins. The slower intellectual growth might be caused by benign parental neglect. Suppose it is desired to estimate the mean attention time given to twins per week by their parents. A sample of $40$ sets of 2 year old boys is taken, and after 1 week the attention time received was recorded. The data (in hours) calculated the mean at $29.5$ and the standard deviation at $11.6$. Use this information to contruct a $90$ \\% confidence interval for the mean attention time given to al twin boys by their parents. [ANS] $< \\mu <$ [ANS]", "answer_v2": [ "26.4831460025898", "32.5168539974102" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Studies have suggusted that twins, in their early years, tend to have lower IQs and pick up language more slowly than nontwins. The slower intellectual growth might be caused by benign parental neglect. Suppose it is desired to estimate the mean attention time given to twins per week by their parents. A sample of $50$ sets of 2 year old boys is taken, and after 1 week the attention time received was recorded. The data (in hours) calculated the mean at $22$ and the standard deviation at $11.7$. Use this information to contruct a $95$ \\% confidence interval for the mean attention time given to al twin boys by their parents. [ANS] $< \\mu <$ [ANS]", "answer_v3": [ "18.7569916439607", "25.2430083560393" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0145", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "2", "keywords": [ "statistics", "inference" ], "problem_v1": "Suppose we want a 95\\% confidence interval for the average amount spent on books by freshmen in their first year at college. The amount spent has a normal distribution with standard deviation \\$30.\n(a) How large should the sample be if the margin of error is to be less than \\$4? ANSWER: [ANS]\n(b) If we wanted a smaller margin of error, we would need a [ANS] sample. (Enter: ''LARGER'', ''SMALLER'' or ''SAME SIZE'', without the quotes.)", "answer_v1": [ "217", "LARGER" ], "answer_type_v1": [ "NV", "MCS" ], "options_v1": [ [], [ "LARGER", "SMALLER", "SAME SIZE" ] ], "problem_v2": "Suppose we want a 99\\% confidence interval for the average amount spent on books by freshmen in their first year at college. The amount spent has a normal distribution with standard deviation \\$16.\n(a) How large should the sample be if the margin of error is to be less than \\$2? ANSWER: [ANS]\n(b) If we wanted a smaller margin of error, we would need a [ANS] sample. (Enter: ''LARGER'', ''SMALLER'' or ''SAME SIZE'', without the quotes.)", "answer_v2": [ "425", "LARGER" ], "answer_type_v2": [ "NV", "MCS" ], "options_v2": [ [], [ "LARGER", "SMALLER", "SAME SIZE" ] ], "problem_v3": "Suppose we want a 95\\% confidence interval for the average amount spent on books by freshmen in their first year at college. The amount spent has a normal distribution with standard deviation \\$21.\n(a) How large should the sample be if the margin of error is to be less than \\$3? ANSWER: [ANS]\n(b) If we wanted a smaller margin of error, we would need a [ANS] sample. (Enter: ''LARGER'', ''SMALLER'' or ''SAME SIZE'', without the quotes.)", "answer_v3": [ "189", "LARGER" ], "answer_type_v3": [ "NV", "MCS" ], "options_v3": [ [], [ "LARGER", "SMALLER", "SAME SIZE" ] ] }, { "id": "Statistics_0146", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "2", "keywords": [ "statistics", "inference", "confidence interval" ], "problem_v1": "A study of the career paths of hotel general managers sent questionnaires to an SRS of 280 hotels belonging to major U.S. hotel chains. There were 165 responses. The average time these 165 general managers had spent with their current company was 11.73 years. (Take it as known that the standard deviation of time with the company for all general managers is 3.6 years.)\n(a) Find the margin of error for an 80\\% confidence interval to estimate the mean time a general manager had spent with their current company: [ANS] years (b) Find the margin of error for a 99\\% confidence interval to estimate the mean time a general manager had spent with their current company: [ANS] years (c) In general, increasing the confidence level [ANS] the margin of error (width) of the confidence interval. (Enter: ''DECREASES'', ''DOES NOT CHANGE'' or ''INCREASES'', without the quotes.)", "answer_v1": [ "0.359293", "0.721669", "INCREASES" ], "answer_type_v1": [ "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "DECREASES", "DOES NOT CHANGE", "INCREASES" ] ], "problem_v2": "A study of the career paths of hotel general managers sent questionnaires to an SRS of 200 hotels belonging to major U.S. hotel chains. There were 76 responses. The average time these 76 general managers had spent with their current company was 8.89 years. (Take it as known that the standard deviation of time with the company for all general managers is 2.4 years.)\n(a) Find the margin of error for a 90\\% confidence interval to estimate the mean time a general manager had spent with their current company: [ANS] years (b) Find the margin of error for a 99\\% confidence interval to estimate the mean time a general manager had spent with their current company: [ANS] years (c) In general, increasing the confidence level [ANS] the margin of error (width) of the confidence interval. (Enter: ''DECREASES'', ''DOES NOT CHANGE'' or ''INCREASES'', without the quotes.)", "answer_v2": [ "0.452867", "0.708895", "INCREASES" ], "answer_type_v2": [ "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "DECREASES", "DOES NOT CHANGE", "INCREASES" ] ], "problem_v3": "A study of the career paths of hotel general managers sent questionnaires to an SRS of 230 hotels belonging to major U.S. hotel chains. There were 115 responses. The average time these 115 general managers had spent with their current company was 9.67 years. (Take it as known that the standard deviation of time with the company for all general managers is 3.1 years.)\n(a) Find the margin of error for an 80\\% confidence interval to estimate the mean time a general manager had spent with their current company: [ANS] years (b) Find the margin of error for a 99\\% confidence interval to estimate the mean time a general manager had spent with their current company: [ANS] years (c) In general, increasing the confidence level [ANS] the margin of error (width) of the confidence interval. (Enter: ''DECREASES'', ''DOES NOT CHANGE'' or ''INCREASES'', without the quotes.)", "answer_v3": [ "0.370596", "0.744372", "INCREASES" ], "answer_type_v3": [ "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "DECREASES", "DOES NOT CHANGE", "INCREASES" ] ] }, { "id": "Statistics_0147", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "2", "keywords": [ "Statistics", "Confidence Intervals" ], "problem_v1": "a) For 30 randomly selected Rolling Stones concerts, the mean gross earnings is 2.76 million dollars. Assuming a population standard deviation gross earnings of 0.51 million dollars, obtain a 99\\% confidence interval for the mean gross earnings of all Rolling Stones concerts (in millions). Confidence interval: ([ANS], [ANS]).\nb) Which of the following is the correct interpretation for your answer in part (a)? [ANS] A. We can be 99\\% confident that the mean gross earnings of all Rolling Stones concerts lies in the interval B. We can be 99\\% confident that the mean gross earnings for this sample of 30 Rolling Stones concerts lies in the interval C. There is a 99\\% chance that the mean gross earnings of all Rolling Stones concerts lies in the interval D. None of the above", "answer_v1": [ "2.52014133761734", "2.99985866238266", "A" ], "answer_type_v1": [ "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "A", "B", "C", "D" ] ], "problem_v2": "a) For 30 randomly selected Rolling Stones concerts, the mean gross earnings is 2.08 million dollars. Assuming a population standard deviation gross earnings of 0.55 million dollars, obtain a 99\\% confidence interval for the mean gross earnings of all Rolling Stones concerts (in millions). Confidence interval: ([ANS], [ANS]).\nb) Which of the following is the correct interpretation for your answer in part (a)? [ANS] A. There is a 99\\% chance that the mean gross earnings of all Rolling Stones concerts lies in the interval B. We can be 99\\% confident that the mean gross earnings of all Rolling Stones concerts lies in the interval C. We can be 99\\% confident that the mean gross earnings for this sample of 30 Rolling Stones concerts lies in the interval D. None of the above", "answer_v2": [ "1.82132889350889", "2.33867110649111", "B" ], "answer_type_v2": [ "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "A", "B", "C", "D" ] ], "problem_v3": "a) For 30 randomly selected Rolling Stones concerts, the mean gross earnings is 2.31 million dollars. Assuming a population standard deviation gross earnings of 0.51 million dollars, obtain a 99\\% confidence interval for the mean gross earnings of all Rolling Stones concerts (in millions). Confidence interval: ([ANS], [ANS]).\nb) Which of the following is the correct interpretation for your answer in part (a)? [ANS] A. We can be 99\\% confident that the mean gross earnings of all Rolling Stones concerts lies in the interval B. There is a 99\\% chance that the mean gross earnings of all Rolling Stones concerts lies in the interval C. We can be 99\\% confident that the mean gross earnings for this sample of 30 Rolling Stones concerts lies in the interval D. None of the above", "answer_v3": [ "2.07014133761734", "2.54985866238266", "A" ], "answer_type_v3": [ "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0148", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "2", "keywords": [ "Statistics" ], "problem_v1": "A random sample of 49 female cottonmouth snakes yielded a sample mean of 8.31 snakes per litter. Assume that $\\sigma=2.4$. a) Use the data provided above to obtain an approximate 95.44\\% confidence interval for the mean number of young per litter of all female cottonmouths. Confidence interval: ([ANS], [ANS]).\nb) How confident are you that your interval from part (a) contains the mean number of young per litter of all female cottonmouths? [ANS] \\%.", "answer_v1": [ "7.62428571428571", "8.99571428571429", "95.44" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A random sample of 41 female cottonmouth snakes yielded a sample mean of 9.74 snakes per litter. Assume that $\\sigma=2.4$. a) Use the data provided above to obtain an approximate 95.44\\% confidence interval for the mean number of young per litter of all female cottonmouths. Confidence interval: ([ANS], [ANS]).\nb) How confident are you that your interval from part (a) contains the mean number of young per litter of all female cottonmouths? [ANS] \\%.", "answer_v2": [ "8.99036594293469", "10.4896340570653", "95.44" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A random sample of 44 female cottonmouth snakes yielded a sample mean of 8.42 snakes per litter. Assume that $\\sigma=2.4$. a) Use the data provided above to obtain an approximate 95.44\\% confidence interval for the mean number of young per litter of all female cottonmouths. Confidence interval: ([ANS], [ANS]).\nb) How confident are you that your interval from part (a) contains the mean number of young per litter of all female cottonmouths? [ANS] \\%.", "answer_v3": [ "7.69637277301337", "9.14362722698663", "95.44" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Statistics_0149", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "2", "keywords": [ "Statistics" ], "problem_v1": "Following are the number of miles traveled for 30 randomly selected business flights within the United States during 1999. 1707, 1435, 1486, 1656, 976, 1027, 1401, 1418, 1095, 1282, 1554, 925, 1316, 1180, 1129, 1197, 1316, 704, 755, 1129, 1316, 1758, 1231, 1010, 1248, 1928, 2098, 942, 534, 908 a) Use the data to obtain a point estimate for the population mean number of miles traveled per business flight, $\\mu$, in 1999. Note: The sum of the data is 37661. $\\bar{x}=$ [ANS]\nb) Determine a 95.44\\% confidence interval for the population mean number of miles traveled per business flight in 1999. Assume that $\\sigma=450$ miles. Confidence interval: ([ANS], [ANS]). c) Must the number of miles traveled per business flight in 1999 be exactly normally distributed for the confidence interval that you obtained in part (b) to be approximately correct? [ANS]\nd) What theorem helped you answer part (c)? [ANS]", "answer_v1": [ "1255.36666666667", "1091.04989941512", "1419.68343391822", "no", "central limit theorem" ], "answer_type_v1": [ "NV", "NV", "NV", "TF", "OE" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Following are the number of miles traveled for 30 randomly selected business flights within the United States during 1999. 619, 1996, 738, 1027, 2013, 1010, 789, 1027, 1401, 602, 1520, 1197, 1809, 806, 806, 925, 1418, 789, 1010, 1384, 670, 687, 653, 993, 1282, 636, 1945, 976, 1129, 1588 a) Use the data to obtain a point estimate for the population mean number of miles traveled per business flight, $\\mu$, in 1999. Note: The sum of the data is 33445. $\\bar{x}=$ [ANS]\nb) Determine a 95.44\\% confidence interval for the population mean number of miles traveled per business flight in 1999. Assume that $\\sigma=450$ miles. Confidence interval: ([ANS], [ANS]). c) Must the number of miles traveled per business flight in 1999 be exactly normally distributed for the confidence interval that you obtained in part (b) to be approximately correct? [ANS]\nd) What theorem helped you answer part (c)? [ANS]", "answer_v2": [ "1114.83333333333", "950.516566081783", "1279.15010058488", "no", "central limit theorem" ], "answer_type_v2": [ "NV", "NV", "NV", "TF", "OE" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Following are the number of miles traveled for 30 randomly selected business flights within the United States during 1999. 993, 1469, 942, 1384, 823, 1044, 1792, 1962, 1894, 823, 959, 891, 568, 1418, 2081, 1775, 1486, 670, 959, 1520, 1622, 1333, 1163, 1979, 1775, 1860, 1401, 1435, 2030, 1724 a) Use the data to obtain a point estimate for the population mean number of miles traveled per business flight, $\\mu$, in 1999. Note: The sum of the data is 41775. $\\bar{x}=$ [ANS]\nb) Determine a 95.44\\% confidence interval for the population mean number of miles traveled per business flight in 1999. Assume that $\\sigma=450$ miles. Confidence interval: ([ANS], [ANS]). c) Must the number of miles traveled per business flight in 1999 be exactly normally distributed for the confidence interval that you obtained in part (b) to be approximately correct? [ANS]\nd) What theorem helped you answer part (c)? [ANS]", "answer_v3": [ "1392.5", "1228.18323274845", "1556.81676725155", "no", "central limit theorem" ], "answer_type_v3": [ "NV", "NV", "NV", "TF", "OE" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Statistics_0150", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "2", "keywords": [ "Statistics", "Confidence Intervals" ], "problem_v1": "The ability of lizards to recognize their predators via tongue flicks can often mean life or death for lizards. Seventeen juvenile common lizards were exposed to the chemical cues of the viper snake. Their responses, in number of tongue flicks per 20 minutes, are presented below. 659, 540, 574, 642, 370, 387, 540, 540, 421, 489, 591, 353, 506, 455, 438, 455, 506 a) Preliminary data analyses indicate that you can reasonably apply the z-interval procedure. Find a 90\\% confidence interval for the mean number of tongue flicks per 20 minutes for all juvenile common lizards. Assume a population standard deviation of 190.0. Note: The sum of the data is 8466. Confidence interval: ([ANS], [ANS]).\nb) Which of the following is the correct interpretation for your answer in part (a)? [ANS] A. We can be 90\\% confident that the mean number of tongue flicks per 20 minutes for this sample of 17 lizards lies in the interval B. There is a 90\\% chance that the mean number of tongue flicks per 20 minutes for all juvenile common lizards lies in the interval C. We can be 90\\% confident that the mean number of tongue flicks per 20 minutes for all juvenile common lizards lies in the interval D. None of the above", "answer_v1": [ "422.195490394894", "573.804509605106", "C" ], "answer_type_v1": [ "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "A", "B", "C", "D" ] ], "problem_v2": "The ability of lizards to recognize their predators via tongue flicks can often mean life or death for lizards. Seventeen juvenile common lizards were exposed to the chemical cues of the viper snake. Their responses, in number of tongue flicks per 20 minutes, are presented below. 234, 761, 285, 404, 778, 387, 302, 387, 540, 234, 591, 455, 693, 302, 302, 353, 540 a) Preliminary data analyses indicate that you can reasonably apply the z-interval procedure. Find a 90\\% confidence interval for the mean number of tongue flicks per 20 minutes for all juvenile common lizards. Assume a population standard deviation of 190.0. Note: The sum of the data is 7548. Confidence interval: ([ANS], [ANS]).\nb) Which of the following is the correct interpretation for your answer in part (a)? [ANS] A. We can be 90\\% confident that the mean number of tongue flicks per 20 minutes for all juvenile common lizards lies in the interval B. There is a 90\\% chance that the mean number of tongue flicks per 20 minutes for all juvenile common lizards lies in the interval C. We can be 90\\% confident that the mean number of tongue flicks per 20 minutes for this sample of 17 lizards lies in the interval D. None of the above", "answer_v2": [ "368.195490394894", "519.804509605106", "A" ], "answer_type_v2": [ "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "A", "B", "C", "D" ] ], "problem_v3": "The ability of lizards to recognize their predators via tongue flicks can often mean life or death for lizards. Seventeen juvenile common lizards were exposed to the chemical cues of the viper snake. Their responses, in number of tongue flicks per 20 minutes, are presented below. 387, 557, 370, 523, 319, 404, 693, 744, 727, 319, 370, 353, 217, 540, 795, 676, 574 a) Preliminary data analyses indicate that you can reasonably apply the z-interval procedure. Find a 90\\% confidence interval for the mean number of tongue flicks per 20 minutes for all juvenile common lizards. Assume a population standard deviation of 190.0. Note: The sum of the data is 8568. Confidence interval: ([ANS], [ANS]).\nb) Which of the following is the correct interpretation for your answer in part (a)? [ANS] A. We can be 90\\% confident that the mean number of tongue flicks per 20 minutes for this sample of 17 lizards lies in the interval B. There is a 90\\% chance that the mean number of tongue flicks per 20 minutes for all juvenile common lizards lies in the interval C. We can be 90\\% confident that the mean number of tongue flicks per 20 minutes for all juvenile common lizards lies in the interval D. None of the above", "answer_v3": [ "428.195490394894", "579.804509605106", "C" ], "answer_type_v3": [ "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0151", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "3", "keywords": [ "Statistics" ], "problem_v1": "In estimating the mean monthly fuel expenditure, $\\mu$, per household vehicle, the U.S. Energy Information Administration takes a sample of size 6377. Assuming that $\\sigma$=20.58 dollars, determine the margin of error in estimating $\\mu$ at the 95\\% level of confidence. [ANS] dollars.", "answer_v1": [ "0.505118451639423" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "In estimating the mean monthly fuel expenditure, $\\mu$, per household vehicle, the U.S. Energy Information Administration takes a sample of size 6041. Assuming that $\\sigma$=20.94 dollars, determine the margin of error in estimating $\\mu$ at the 95\\% level of confidence. [ANS] dollars.", "answer_v2": [ "0.528053992815058" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "In estimating the mean monthly fuel expenditure, $\\mu$, per household vehicle, the U.S. Energy Information Administration takes a sample of size 6157. Assuming that $\\sigma$=20.61 dollars, determine the margin of error in estimating $\\mu$ at the 95\\% level of confidence. [ANS] dollars.", "answer_v3": [ "0.514812977730704" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0152", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "3", "keywords": [ "Confidence Intervals" ], "problem_v1": "The standard deviation of the length of skateboards in a board shop is 0.56 in. A random sample of 24 skateboards is selected and found to have a mean length of 31.2 in. Find a 95\\% confidence interval for the true mean length of skateboards in the board shop. Your results should be accurate to the nearest hundredth cm (at least). Answer: [ANS] to [ANS] in", "answer_v1": [ "30.9713809573402", "31.4286190426598" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The standard deviation of the length of snowboards in a board shop is 0.54 cm. A random sample of 17 snowboards is selected and found to have a mean length of 151 cm. Find a 95\\% confidence interval for the true mean length of snowboards in the board shop. Your results should be accurate to the nearest hundredth cm (at least). Answer: [ANS] to [ANS] cm", "answer_v2": [ "150.738061524961", "151.261938475039" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The standard deviation of the length of surfboards in a board shop is 0.58 in. A random sample of 27 surfboards is selected and found to have a mean length of 72.9 in. Find a 95\\% confidence interval for the true mean length of surfboards in the board shop. Your results should be accurate to the nearest hundredth cm (at least). Answer: [ANS] to [ANS] in", "answer_v3": [ "72.6767578959134", "73.1232421040867" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0153", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "3", "keywords": [ "statistics", "statistical inference", "confidence interval for mean", "calculation", "interpretation" ], "problem_v1": "An online used car company sells second-hand cars. For 30 randomly selected transactions, the mean price is 2800 dollars. Part a) Assuming a population standard deviation transaction prices of 220 dollars, obtain a 99\\% confidence interval for the mean price of all transactions. Please carry at least three decimal places in intermediate steps. Give your final answer to the nearest two decimal places.\nConfidence interval: ([ANS], [ANS]).\nPart b) Which of the following is a correct interpretation for your answer in part (a)? Select ALL the correct answers, there may be more than one. [ANS] A. We can be 99\\% confident that the mean price for this sample of 30 transactions lies in the interval. B. 99\\% of their mean sales price lies inside this interval. C. 99\\% of the cars they sell have a price that lies inside this interval. D. If we repeat the study many times, approximately 99\\% of the calculated confidence intervals will contain the mean price of all transactions. E. There is a 99\\% chance that the mean price of all transactions lies in the interval. F. We can be 99\\% confident that all of the cars they sell have a price inside this interval. G. None of the above.", "answer_v1": [ "2696.53", "2903.47", "DE" ], "answer_type_v1": [ "NV", "NV", "MCM" ], "options_v1": [ [], [], [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v2": "An online used car company sells second-hand cars. For 30 randomly selected transactions, the mean price is 2000 dollars. Part a) Assuming a population standard deviation transaction prices of 290 dollars, obtain a 99\\% confidence interval for the mean price of all transactions. Please carry at least three decimal places in intermediate steps. Give your final answer to the nearest two decimal places.\nConfidence interval: ([ANS], [ANS]).\nPart b) Which of the following is a correct interpretation for your answer in part (a)? Select ALL the correct answers, there may be more than one. [ANS] A. We can be 99\\% confident that the mean price for this sample of 30 transactions lies in the interval. B. If we repeat the study many times, approximately 99\\% of the calculated confidence intervals will contain the mean price of all transactions. C. 99\\% of their mean sales price lies inside this interval. D. 99\\% of the cars they sell have a price that lies inside this interval. E. There is a 99\\% chance that the mean price of all transactions lies in the interval. F. We can be 99\\% confident that all of the cars they sell have a price inside this interval. G. None of the above.", "answer_v2": [ "1863.61", "2136.39", "BE" ], "answer_type_v2": [ "NV", "NV", "MCM" ], "options_v2": [ [], [], [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v3": "An online used car company sells second-hand cars. For 30 randomly selected transactions, the mean price is 2300 dollars. Part a) Assuming a population standard deviation transaction prices of 220 dollars, obtain a 99\\% confidence interval for the mean price of all transactions. Please carry at least three decimal places in intermediate steps. Give your final answer to the nearest two decimal places.\nConfidence interval: ([ANS], [ANS]).\nPart b) Which of the following is a correct interpretation for your answer in part (a)? Select ALL the correct answers, there may be more than one. [ANS] A. We can be 99\\% confident that all of the cars they sell have a price inside this interval. B. If we repeat the study many times, approximately 99\\% of the calculated confidence intervals will contain the mean price of all transactions. C. 99\\% of the cars they sell have a price that lies inside this interval. D. There is a 99\\% chance that the mean price of all transactions lies in the interval. E. 99\\% of their mean sales price lies inside this interval. F. We can be 99\\% confident that the mean price for this sample of 30 transactions lies in the interval. G. None of the above.", "answer_v3": [ "2196.53", "2403.47", "BD" ], "answer_type_v3": [ "NV", "NV", "MCM" ], "options_v3": [ [], [], [ "A", "B", "C", "D", "E", "F", "G" ] ] }, { "id": "Statistics_0154", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "3", "keywords": [ "statistics", "continuous random variables", "expectation" ], "problem_v1": "The time in hours for a worker to repair an electrical instrument is a Normally distributed random variable with a mean of $\\mu$ and a standard deviation of 50. The repair times for 12 such instruments chosen at random are as follows: $\\begin{array}{cccccccccccc}\\hline 183 & 222 & 303 & 262 & 178 & 232 & 268 & 201 & 244 & 183 & 201 & 140 \\\\ \\hline \\end{array}$\nPart a) Find a 95\\% confidence interval for $\\mu$. For both sides of the bound, leave your answer with 1 decimal place. ([ANS], [ANS]). Part b) Find the least number of repair times needed to be sampled in order to reduce the width of the confidence interval to below 30 hours. [ANS]", "answer_v1": [ "189.793", "246.373", "43" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The time in hours for a worker to repair an electrical instrument is a Normally distributed random variable with a mean of $\\mu$ and a standard deviation of 50. The repair times for 12 such instruments chosen at random are as follows: $\\begin{array}{cccccccccccc}\\hline 183 & 222 & 303 & 262 & 178 & 232 & 268 & 201 & 244 & 183 & 201 & 140 \\\\ \\hline \\end{array}$\nPart a) Find a 95\\% confidence interval for $\\mu$. For both sides of the bound, leave your answer with 1 decimal place. ([ANS], [ANS]). Part b) Find the least number of repair times needed to be sampled in order to reduce the width of the confidence interval to below 25 hours. [ANS]", "answer_v2": [ "189.793", "246.373", "62" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The time in hours for a worker to repair an electrical instrument is a Normally distributed random variable with a mean of $\\mu$ and a standard deviation of 50. The repair times for 12 such instruments chosen at random are as follows: $\\begin{array}{cccccccccccc}\\hline 183 & 222 & 303 & 262 & 178 & 232 & 268 & 201 & 244 & 183 & 201 & 140 \\\\ \\hline \\end{array}$\nPart a) Find a 95\\% confidence interval for $\\mu$. For both sides of the bound, leave your answer with 1 decimal place. ([ANS], [ANS]). Part b) Find the least number of repair times needed to be sampled in order to reduce the width of the confidence interval to below 26 hours. [ANS]", "answer_v3": [ "189.793", "246.373", "57" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Statistics_0155", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "3", "keywords": [ "statistics", "probability" ], "problem_v1": "A medical statistician wants to estimate the average weight loss of people who are on a new diet plan. In a preliminary study, he guesses that the standard deviation of the population of weight loss is about 11 pounds. How large a sample should he take to estimate the mean weight loss to within 4 pounds, with 96\\% confidence? Sample Size=[ANS]", "answer_v1": [ "32" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A medical statistician wants to estimate the average weight loss of people who are on a new diet plan. In a preliminary study, he guesses that the standard deviation of the population of weight loss is about 8 pounds. How large a sample should he take to estimate the mean weight loss to within 5 pounds, with 91\\% confidence? Sample Size=[ANS]", "answer_v2": [ "8" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A medical statistician wants to estimate the average weight loss of people who are on a new diet plan. In a preliminary study, he guesses that the standard deviation of the population of weight loss is about 9 pounds. How large a sample should he take to estimate the mean weight loss to within 4 pounds, with 92\\% confidence? Sample Size=[ANS]", "answer_v3": [ "16" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0156", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "3", "keywords": [ "statistics", "probability" ], "problem_v1": "Determine the sample size required to estimate a population mean to within 11 units given that the population standard deviation is 55. A confidence level of 90\\% is judged to be appropriate. Sample Size=[ANS]\nDetermine the sample size required to estimate a population mean to within 11 units given that the population standard deviation is 105. Use a confidence level of 90\\%. Sample Size=[ANS]\nDetermine the sample size required to estimate a population mean to within 11 units given that the population standard deviation is 55. Use a confidence level of 95\\%. Sample Size=[ANS]\nDetermine the sample size required to estimate a population mean to within 22 units given that the population standard deviation is 55. A confidence level of 90\\% is judged to be appropriate. Sample Size=[ANS]", "answer_v1": [ "68", "247", "97", "17" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Determine the sample size required to estimate a population mean to within 8 units given that the population standard deviation is 65. A confidence level of 90\\% is judged to be appropriate. Sample Size=[ANS]\nDetermine the sample size required to estimate a population mean to within 8 units given that the population standard deviation is 90. Use a confidence level of 90\\%. Sample Size=[ANS]\nDetermine the sample size required to estimate a population mean to within 8 units given that the population standard deviation is 65. Use a confidence level of 95\\%. Sample Size=[ANS]\nDetermine the sample size required to estimate a population mean to within 18 units given that the population standard deviation is 65. A confidence level of 90\\% is judged to be appropriate. Sample Size=[ANS]", "answer_v2": [ "179", "343", "254", "36" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Determine the sample size required to estimate a population mean to within 9 units given that the population standard deviation is 60. A confidence level of 90\\% is judged to be appropriate. Sample Size=[ANS]\nDetermine the sample size required to estimate a population mean to within 9 units given that the population standard deviation is 90. Use a confidence level of 90\\%. Sample Size=[ANS]\nDetermine the sample size required to estimate a population mean to within 9 units given that the population standard deviation is 60. Use a confidence level of 95\\%. Sample Size=[ANS]\nDetermine the sample size required to estimate a population mean to within 21 units given that the population standard deviation is 60. A confidence level of 90\\% is judged to be appropriate. Sample Size=[ANS]", "answer_v3": [ "121", "271", "171", "23" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Statistics_0157", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "3", "keywords": [ "statistics", "probability" ], "problem_v1": "A statistics practitioner took a random sample of 53 observations from a population whose standard deviation is 33 and computed the sample mean to be 104. Note: For each confidence interval, enter your answer in the form (LCL, UCL). You must include the parentheses and the comma between the confidence limits.\nA. Estimate the population mean with 95\\% confidence. Confidence Interval=[ANS]\nB. Estimate the population mean with 90\\% confidence. Confidence Interval=[ANS]\nC. Estimate the population mean with 99\\% confidence. Confidence Interval=[ANS]", "answer_v1": [ "(95.1156874065548,112.884312593445)", "(96.5440459644814,111.455954035519)", "(92.3240177075953,115.675982292405)" ], "answer_type_v1": [ "INT", "INT", "INT" ], "options_v1": [ [], [], [] ], "problem_v2": "A statistics practitioner took a random sample of 43 observations from a population whose standard deviation is 25 and computed the sample mean to be 109. Note: For each confidence interval, enter your answer in the form (LCL, UCL). You must include the parentheses and the comma between the confidence limits.\nA. Estimate the population mean with 95\\% confidence. Confidence Interval=[ANS]\nB. Estimate the population mean with 90\\% confidence. Confidence Interval=[ANS]\nC. Estimate the population mean with 99\\% confidence. Confidence Interval=[ANS]", "answer_v2": [ "(101.527711926935,116.472288073065)", "(102.729054912585,115.270945087415)", "(99.179735088427,118.820264911573)" ], "answer_type_v2": [ "INT", "INT", "INT" ], "options_v2": [ [], [], [] ], "problem_v3": "A statistics practitioner took a random sample of 45 observations from a population whose standard deviation is 28 and computed the sample mean to be 104. Note: For each confidence interval, enter your answer in the form (LCL, UCL). You must include the parentheses and the comma between the confidence limits.\nA. Estimate the population mean with 95\\% confidence. Confidence Interval=[ANS]\nB. Estimate the population mean with 90\\% confidence. Confidence Interval=[ANS]\nC. Estimate the population mean with 99\\% confidence. Confidence Interval=[ANS]", "answer_v3": [ "(95.819127953143,112.180872046857)", "(97.1343921873227,110.865607812677)", "(93.2484965886945,114.751503411306)" ], "answer_type_v3": [ "INT", "INT", "INT" ], "options_v3": [ [], [], [] ] }, { "id": "Statistics_0158", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "3", "keywords": [ "statistics", "probability" ], "problem_v1": "Among the most exciting aspects of a university professor's life are the departmental meetings where such critical issues as the color the walls will be painted and who gets a new desk are decided. A sample of 20 professors was asked how many hours per year are devoted to such meetings. The responses are listed below. Assuming that the variable is normally distributed with a standard deviation of 8 hours, estimate the mean number of hours spent at departmental meetings by all professors. Use a confidence level of 96\\%.\n\\begin{array}{cccccccccc} 8, & 12, & 16, & 3, & 16, & 4, & 20, & 15, & 16, & 19, \\\\ 13, & 8, & 7, & 8, & 15, & 2, & 12, & 16, & 3, & 14 \\end{array} Note: For each confidence interval, enter your answer in the form (LCL, UCL). You must include the parentheses and the comma between the confidence limits.\nConfidence Interval=[ANS]", "answer_v1": [ "(7.67614031296785,15.0238596870322)" ], "answer_type_v1": [ "INT" ], "options_v1": [ [] ], "problem_v2": "Among the most exciting aspects of a university professor's life are the departmental meetings where such critical issues as the color the walls will be painted and who gets a new desk are decided. A sample of 20 professors was asked how many hours per year are devoted to such meetings. The responses are listed below. Assuming that the variable is normally distributed with a standard deviation of 6 hours, estimate the mean number of hours spent at departmental meetings by all professors. Use a confidence level of 90\\%.\n\\begin{array}{cccccccccc} 5, & 14, & 13, & 2, & 19, & 4, & 18, & 14, & 17, & 15, \\\\ 13, & 9, & 8, & 7, & 14, & 2, & 12, & 16, & 4, & 15 \\end{array} Note: For each confidence interval, enter your answer in the form (LCL, UCL). You must include the parentheses and the comma between the confidence limits.\nConfidence Interval=[ANS]", "answer_v2": [ "(8.84320215232568,13.2567978476743)" ], "answer_type_v2": [ "INT" ], "options_v2": [ [] ], "problem_v3": "Among the most exciting aspects of a university professor's life are the departmental meetings where such critical issues as the color the walls will be painted and who gets a new desk are decided. A sample of 20 professors was asked how many hours per year are devoted to such meetings. The responses are listed below. Assuming that the variable is normally distributed with a standard deviation of 9 hours, estimate the mean number of hours spent at departmental meetings by all professors. Use a confidence level of 94\\%.\n\\begin{array}{cccccccccc} 6, & 12, & 14, & 3, & 16, & 4, & 22, & 16, & 19, & 16, \\\\ 11, & 8, & 5, & 10, & 18, & 4, & 12, & 15, & 4, & 15 \\end{array} Note: For each confidence interval, enter your answer in the form (LCL, UCL). You must include the parentheses and the comma between the confidence limits.\nConfidence Interval=[ANS]", "answer_v3": [ "(7.71498313773835,15.2850168622616)" ], "answer_type_v3": [ "INT" ], "options_v3": [ [] ] }, { "id": "Statistics_0159", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "3", "keywords": [ "statistics", "probability" ], "problem_v1": "One of the few negative side effects of quitting smoking is weight gain. Suppose that the weight gain in the 12 months following a cessation in smoking is normally distributed with a standard deviation of 8 pounds. To estimate the mean weight gain, a random sample of 55 quitters was drawn and the sample mean was found to be 26 pounds. Determine the 97\\% confidence interval estimate of the mean 12-month weight gain for all quitters. Note: For each confidence interval, enter your answer in the form (LCL, UCL). You must include the parentheses and the comma between the confidence limits.\nConfidence Interval=[ANS]", "answer_v1": [ "(23.6590809927474,28.3409190072526)" ], "answer_type_v1": [ "INT" ], "options_v1": [ [] ], "problem_v2": "One of the few negative side effects of quitting smoking is weight gain. Suppose that the weight gain in the 12 months following a cessation in smoking is normally distributed with a standard deviation of 5 pounds. To estimate the mean weight gain, a random sample of 65 quitters was drawn and the sample mean was found to be 21 pounds. Determine the 93\\% confidence interval estimate of the mean 12-month weight gain for all quitters. Note: For each confidence interval, enter your answer in the form (LCL, UCL). You must include the parentheses and the comma between the confidence limits.\nConfidence Interval=[ANS]", "answer_v2": [ "(19.8763011202523,22.1236988797477)" ], "answer_type_v2": [ "INT" ], "options_v2": [ [] ], "problem_v3": "One of the few negative side effects of quitting smoking is weight gain. Suppose that the weight gain in the 12 months following a cessation in smoking is normally distributed with a standard deviation of 6 pounds. To estimate the mean weight gain, a random sample of 60 quitters was drawn and the sample mean was found to be 23 pounds. Determine the 95\\% confidence interval estimate of the mean 12-month weight gain for all quitters. Note: For each confidence interval, enter your answer in the form (LCL, UCL). You must include the parentheses and the comma between the confidence limits.\nConfidence Interval=[ANS]", "answer_v3": [ "(21.4818215121535,24.5181784878465)" ], "answer_type_v3": [ "INT" ], "options_v3": [ [] ] }, { "id": "Statistics_0160", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "3", "keywords": [ "statistics", "probability" ], "problem_v1": "The operations manager of a large production plant would like to estimate the average amount of time workers take to assemble a new electronic component. After observing a number of workers assembling similar devices, she guesses that the standard deviation is 9 minutes. How large a sample of workers should she take if she wishes to estimate the mean assembly time to within 21 seconds. Assume the confidence level to be 96\\%. Sample Size=[ANS]", "answer_v1": [ "2789" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The operations manager of a large production plant would like to estimate the average amount of time workers take to assemble a new electronic component. After observing a number of workers assembling similar devices, she guesses that the standard deviation is 5 minutes. How large a sample of workers should she take if she wishes to estimate the mean assembly time to within 25 seconds. Assume the confidence level to be 91\\%. Sample Size=[ANS]", "answer_v2": [ "414" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The operations manager of a large production plant would like to estimate the average amount of time workers take to assemble a new electronic component. After observing a number of workers assembling similar devices, she guesses that the standard deviation is 6 minutes. How large a sample of workers should she take if she wishes to estimate the mean assembly time to within 21 seconds. Assume the confidence level to be 92\\%. Sample Size=[ANS]", "answer_v3": [ "901" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0161", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "3", "keywords": [ "statistics", "probability" ], "problem_v1": "The number of cars sold annually by used car salespeople is normally distributed with a standard deviation of 17. A random sample of 420 salespeople was taken and the mean number of cars sold annually was found to be 78. Find the 95\\% confidence interval estimate of the population mean. Note: For each confidence interval, enter your answer in the form (LCL, UCL). You must include the parentheses and the comma between the confidence limits.\nConfidence Interval=[ANS]", "answer_v1": [ "(76.3741836590702,79.6258163409298)" ], "answer_type_v1": [ "INT" ], "options_v1": [ [] ], "problem_v2": "The number of cars sold annually by used car salespeople is normally distributed with a standard deviation of 13. A random sample of 490 salespeople was taken and the mean number of cars sold annually was found to be 68. Find the 92\\% confidence interval estimate of the population mean. Note: For each confidence interval, enter your answer in the form (LCL, UCL). You must include the parentheses and the comma between the confidence limits.\nConfidence Interval=[ANS]", "answer_v2": [ "(66.9718545371508,69.0281454628492)" ], "answer_type_v2": [ "INT" ], "options_v2": [ [] ], "problem_v3": "The number of cars sold annually by used car salespeople is normally distributed with a standard deviation of 14. A random sample of 420 salespeople was taken and the mean number of cars sold annually was found to be 70. Find the 94\\% confidence interval estimate of the population mean. Note: For each confidence interval, enter your answer in the form (LCL, UCL). You must include the parentheses and the comma between the confidence limits.\nConfidence Interval=[ANS]", "answer_v3": [ "(68.7151758312594,71.2848241687406)" ], "answer_type_v3": [ "INT" ], "options_v3": [ [] ] }, { "id": "Statistics_0162", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "3", "keywords": [ "statistics", "probability" ], "problem_v1": "It is known that the amount of time needed to change the oil in a car is normally distributed with a standard deviation of 7 minutes. A random sample of 105 oil changes yielded a sample mean of 26 minutes. Compute the 97\\% confidence interval estimate for the population mean. Note: For each confidence interval, enter your answer in the form (LCL, UCL). You must include the parentheses and the comma between the confidence limits.\nConfidence Interval=[ANS]", "answer_v1": [ "(24.5175463074866,27.4824536925134)" ], "answer_type_v1": [ "INT" ], "options_v1": [ [] ], "problem_v2": "It is known that the amount of time needed to change the oil in a car is normally distributed with a standard deviation of 3 minutes. A random sample of 125 oil changes yielded a sample mean of 21 minutes. Compute the 93\\% confidence interval estimate for the population mean. Note: For each confidence interval, enter your answer in the form (LCL, UCL). You must include the parentheses and the comma between the confidence limits.\nConfidence Interval=[ANS]", "answer_v2": [ "(20.5138135285066,21.4861864714934)" ], "answer_type_v2": [ "INT" ], "options_v2": [ [] ], "problem_v3": "It is known that the amount of time needed to change the oil in a car is normally distributed with a standard deviation of 4 minutes. A random sample of 105 oil changes yielded a sample mean of 23 minutes. Compute the 95\\% confidence interval estimate for the population mean. Note: For each confidence interval, enter your answer in the form (LCL, UCL). You must include the parentheses and the comma between the confidence limits.\nConfidence Interval=[ANS]", "answer_v3": [ "(22.2349099572095,23.7650900427905)" ], "answer_type_v3": [ "INT" ], "options_v3": [ [] ] }, { "id": "Statistics_0163", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "3", "keywords": [ "statistics", "probability" ], "problem_v1": "A medical researcher wants to investigate the amount of time it takes for patients' headache pain to be relieved after taking a new prescription painkiller. She plans to use statistical methods to estimate the mean of the population of relief times. She believes that the population is normally distributed with a standard deviation of 23 minutes. How large a sample should she take to estimate the mean time to within 4 minutes with 96\\% confidence? Sample Size=[ANS]", "answer_v1": [ "140" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A medical researcher wants to investigate the amount of time it takes for patients' headache pain to be relieved after taking a new prescription painkiller. She plans to use statistical methods to estimate the mean of the population of relief times. She believes that the population is normally distributed with a standard deviation of 15 minutes. How large a sample should she take to estimate the mean time to within 5 minutes with 91\\% confidence? Sample Size=[ANS]", "answer_v2": [ "26" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A medical researcher wants to investigate the amount of time it takes for patients' headache pain to be relieved after taking a new prescription painkiller. She plans to use statistical methods to estimate the mean of the population of relief times. She believes that the population is normally distributed with a standard deviation of 18 minutes. How large a sample should she take to estimate the mean time to within 4 minutes with 92\\% confidence? Sample Size=[ANS]", "answer_v3": [ "63" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0164", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "3", "keywords": [ "statistics", "probability" ], "problem_v1": "How many rounds of golf do those physicians who play golf play per year? A survey of 12 physicians revealed the following numbers:\n8, \\quad 42, \\quad 16, \\quad 3, \\quad 32, \\quad 37, \\quad 20, \\quad 15, \\quad 16, \\quad 29, \\quad 13, \\quad 49 Estimate with 93\\% confidence the mean number of rounds played per year by physicians, assuming that the population is normally distributed with a standard deviation of 7. Note: For each confidence interval, enter your answer in the form (LCL, UCL). You must include the parentheses and the comma between the confidence limits.\nConfidence Interval=[ANS]", "answer_v1": [ "(19.6719597914675,26.9947068751991)" ], "answer_type_v1": [ "INT" ], "options_v1": [ [] ], "problem_v2": "How many rounds of golf do those physicians who play golf play per year? A survey of 12 physicians revealed the following numbers:\n5, \\quad 44, \\quad 13, \\quad 2, \\quad 36, \\quad 37, \\quad 18, \\quad 14, \\quad 17, \\quad 25, \\quad 13, \\quad 50 Estimate with 91\\% confidence the mean number of rounds played per year by physicians, assuming that the population is normally distributed with a standard deviation of 8. Note: For each confidence interval, enter your answer in the form (LCL, UCL). You must include the parentheses and the comma between the confidence limits.\nConfidence Interval=[ANS]", "answer_v2": [ "(18.917974747797,26.7486919188697)" ], "answer_type_v2": [ "INT" ], "options_v2": [ [] ], "problem_v3": "How many rounds of golf do those physicians who play golf play per year? A survey of 12 physicians revealed the following numbers:\n6, \\quad 42, \\quad 14, \\quad 3, \\quad 31, \\quad 38, \\quad 22, \\quad 16, \\quad 19, \\quad 26, \\quad 11, \\quad 49 Estimate with 94\\% confidence the mean number of rounds played per year by physicians, assuming that the population is normally distributed with a standard deviation of 6. Note: For each confidence interval, enter your answer in the form (LCL, UCL). You must include the parentheses and the comma between the confidence limits.\nConfidence Interval=[ANS]", "answer_v3": [ "(19.8257094949659,26.3409571717008)" ], "answer_type_v3": [ "INT" ], "options_v3": [ [] ] }, { "id": "Statistics_0165", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - z", "level": "3", "keywords": [ "statistics", "probability" ], "problem_v1": "A statistics professor wants to compare today's students with those 25 years ago. All of his current students' marks are stored on a computer so that he can easily determine the population mean. However, the marks 25 years ago reside only in his musty files. He does not want to retrieve all the marks and will be satisfied with a 96\\% confidence interval estimate of the mean mark 25 years ago. If he assumes that the population standard deviation is 14, how large a sample should he take to estimate the mean to within 4 marks? Sample Size=[ANS]", "answer_v1": [ "52" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A statistics professor wants to compare today's students with those 25 years ago. All of his current students' marks are stored on a computer so that he can easily determine the population mean. However, the marks 25 years ago reside only in his musty files. He does not want to retrieve all the marks and will be satisfied with a 91\\% confidence interval estimate of the mean mark 25 years ago. If he assumes that the population standard deviation is 10, how large a sample should he take to estimate the mean to within 5 marks? Sample Size=[ANS]", "answer_v2": [ "12" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A statistics professor wants to compare today's students with those 25 years ago. All of his current students' marks are stored on a computer so that he can easily determine the population mean. However, the marks 25 years ago reside only in his musty files. He does not want to retrieve all the marks and will be satisfied with a 92\\% confidence interval estimate of the mean mark 25 years ago. If he assumes that the population standard deviation is 11, how large a sample should he take to estimate the mean to within 4 marks? Sample Size=[ANS]", "answer_v3": [ "24" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0166", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - t", "level": "2", "keywords": [ "Confidence Interval", "Mean", "Normal Distribution", "Standard Deviation", "statistics", "estimates", "population", "samples" ], "problem_v1": "Use the given data to find the 95\\% confidence interval estimate of the population mean $\\mu$. Assume that the population has a normal distribution. IQ scores of professional athletes: Sample size $n=25$ Mean $\\overline{x}=105$ Standard deviation $s=13$ [ANS] $< \\mu <$ [ANS]", "answer_v1": [ "99.63386", "110.36614" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Use the given data to find the 95\\% confidence interval estimate of the population mean $\\mu$. Assume that the population has a normal distribution. IQ scores of professional athletes: Sample size $n=10$ Mean $\\overline{x}=106$ Standard deviation $s=10$ [ANS] $< \\mu <$ [ANS]", "answer_v2": [ "98.8464219682735", "113.153578031726" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Use the given data to find the 95\\% confidence interval estimate of the population mean $\\mu$. Assume that the population has a normal distribution. IQ scores of professional athletes: Sample size $n=15$ Mean $\\overline{x}=105$ Standard deviation $s=10$ [ANS] $< \\mu <$ [ANS]", "answer_v3": [ "99.4621760325919", "110.537823967408" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0167", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - t", "level": "2", "keywords": [ "Confidence Interval", "t Distribution", "statistics", "estimates", "population", "samples" ], "problem_v1": "Weights of 10 red and 36 brown randomly chosen M{\\&}M plain candies are listed below. Red: $\\begin{array}{lllll} 0.897& 0.924& 0.874& 0.912& 0.909 \\cr 0.913& 0.891& 0.907& 0.913& 0.92 \\end{array}$ Brown: $\\begin{array}{llllll} 0.898& 0.856& 0.9& 0.867& 0.93& 0.955 \\cr 0.875& 0.913& 0.93& 0.923& 0.966& 0.915 \\cr 0.928& 0.866& 0.889& 0.872& 0.857& 0.985 \\cr 0.931& 0.876& 0.902& 0.904& 0.936& 0.92 \\cr 0.912& 0.909& 0.871& 0.902& 0.905& 0.914 \\cr 0.858& 1.001& 0.92& 0.988& 0.921& 0.931 \\end{array}$ 1. $\\ $ To construct a 90\\% confidence interval for the mean weight of red M{\\&}M plain candies, you have to use [ANS] A. The t distribution with 9 degrees of freedom B. The t distribution with 10 degrees of freedom C. The normal distribution D. The t distribution with 11 degrees of freedom E. None of the above 2. $\\ $ A 90\\% confidence interval for the mean weight of red M{\\&}M plain candies is [ANS] $< \\mu <$ [ANS] 3. $\\ $ To construct a 90\\% confidence interval for the mean weight of brown M{\\&}M plain candies, you have to use [ANS] A. The t distribution with 35 degrees of freedom B. The t distribution with 37 degrees of freedom C. The normal distribution D. The t distribution with 36 degrees of freedom E. None of the above 4. $\\ $ A 90\\% confidence interval for the mean weight of brown M{\\&}M plain candies is [ANS] $< \\mu <$ [ANS]", "answer_v1": [ "A", "0.895352836509192", "0.916647163490808", "C", "0.89990204375043", "0.923764622916237" ], "answer_type_v1": [ "MCS", "NV", "NV", "MCS", "NV", "NV" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [], [], [ "A", "B", "C", "D", "E" ], [], [] ], "problem_v2": "Weights of 10 red and 36 brown randomly chosen M{\\&}M plain candies are listed below. Red: $\\begin{array}{lllll} 0.933& 0.898& 0.908& 0.92& 0.882 \\cr 0.983& 0.913& 0.936& 0.907& 0.871 \\end{array}$ Brown: $\\begin{array}{llllll} 0.898& 0.867& 0.915& 0.871& 1.001& 0.866 \\cr 0.909& 0.93& 0.936& 0.966& 0.914& 0.904 \\cr 0.913& 0.923& 0.875& 0.918& 0.986& 0.955 \\cr 0.9& 0.931& 0.877& 0.889& 0.897& 0.929 \\cr 0.86& 0.902& 0.928& 0.92& 0.872& 0.93 \\cr 0.985& 0.988& 0.861& 0.931& 0.856& 0.921 \\end{array}$ 1. $\\ $ To construct a 90\\% confidence interval for the mean weight of red M{\\&}M plain candies, you have to use [ANS] A. The t distribution with 10 degrees of freedom B. The t distribution with 11 degrees of freedom C. The t distribution with 9 degrees of freedom D. The normal distribution E. None of the above 2. $\\ $ A 90\\% confidence interval for the mean weight of red M{\\&}M plain candies is [ANS] $< \\mu <$ [ANS] 3. $\\ $ To construct a 90\\% confidence interval for the mean weight of brown M{\\&}M plain candies, you have to use [ANS] A. The t distribution with 35 degrees of freedom B. The normal distribution C. The t distribution with 36 degrees of freedom D. The t distribution with 37 degrees of freedom E. None of the above 4. $\\ $ A 90\\% confidence interval for the mean weight of brown M{\\&}M plain candies is [ANS] $< \\mu <$ [ANS]", "answer_v2": [ "C", "0.892675756047527", "0.937524243952473", "B", "0.902017233538142", "0.927093877572968" ], "answer_type_v2": [ "MCS", "NV", "NV", "MCS", "NV", "NV" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [], [], [ "A", "B", "C", "D", "E" ], [], [] ], "problem_v3": "Weights of 10 red and 36 brown randomly chosen M{\\&}M plain candies are listed below. Red: $\\begin{array}{lllll} 0.913& 0.907& 0.909& 0.924& 0.908 \\cr 0.92& 0.877& 0.898& 0.898& 0.952 \\end{array}$ Brown: $\\begin{array}{llllll} 0.866& 0.876& 0.86& 0.966& 0.857& 0.861 \\cr 0.921& 0.913& 0.988& 0.897& 0.92& 0.9 \\cr 0.867& 0.872& 0.92& 0.912& 0.928& 0.889 \\cr 0.986& 0.909& 0.904& 0.955& 0.929& 0.898 \\cr 0.871& 0.875& 0.936& 0.914& 0.923& 0.858 \\cr 0.93& 0.856& 0.909& 0.931& 0.918& 0.905 \\end{array}$ 1. $\\ $ To construct a 90\\% confidence interval for the mean weight of red M{\\&}M plain candies, you have to use [ANS] A. The normal distribution B. The t distribution with 9 degrees of freedom C. The t distribution with 10 degrees of freedom D. The t distribution with 11 degrees of freedom E. None of the above 2. $\\ $ A 90\\% confidence interval for the mean weight of red M{\\&}M plain candies is [ANS] $< \\mu <$ [ANS] 3. $\\ $ To construct a 90\\% confidence interval for the mean weight of brown M{\\&}M plain candies, you have to use [ANS] A. The normal distribution B. The t distribution with 37 degrees of freedom C. The t distribution with 36 degrees of freedom D. The t distribution with 35 degrees of freedom E. None of the above 4. $\\ $ A 90\\% confidence interval for the mean weight of brown M{\\&}M plain candies is [ANS] $< \\mu <$ [ANS]", "answer_v3": [ "B", "0.896582087522031", "0.924617912477969", "A", "0.894709213586554", "0.917513008635668" ], "answer_type_v3": [ "MCS", "NV", "NV", "MCS", "NV", "NV" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [], [], [ "A", "B", "C", "D", "E" ], [], [] ] }, { "id": "Statistics_0168", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - t", "level": "2", "keywords": [ "large sample", "estimates", "confidence interval" ], "problem_v1": "The scientific productivity of major world cities was the subject of a recent study. The study determined the number of scientific papers published between 1994 and 1997 by researchers from each of the 20 world cities, and is shown below.\n$\\begin{array}{cccc}\\hline City & Number of papers & City & Number of papers \\\\ \\hline City 1 & 23 & City 11 & 20 \\\\ \\hline City 2 & 18 & City 12 & 9 \\\\ \\hline City 3 & 20 & City 13 & 16 \\\\ \\hline City 4 & 22 & City 14 & 14 \\\\ \\hline City 5 & 10 & City 15 & 13 \\\\ \\hline City 6 & 11 & City 16 & 14 \\\\ \\hline City 7 & 18 & City 17 & 16 \\\\ \\hline City 8 & 18 & City 18 & 5 \\\\ \\hline City 9 & 12 & City 19 & 6 \\\\ \\hline City 10 & 16 & City 20 & 13 \\\\ \\hline \\end{array}$\nConstruct a $95$ \\% confidence interval for the average number of papers published in major world cities. [ANS] $< \\mu <$ [ANS]", "answer_v1": [ "12.3767428378906", "17.0232571621094" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The scientific productivity of major world cities was the subject of a recent study. The study determined the number of scientific papers published between 1994 and 1997 by researchers from each of the 20 world cities, and is shown below.\n$\\begin{array}{cccc}\\hline City & Number of papers & City & Number of papers \\\\ \\hline City 1 & 4 & City 11 & 20 \\\\ \\hline City 2 & 29 & City 12 & 14 \\\\ \\hline City 3 & 6 & City 13 & 25 \\\\ \\hline City 4 & 11 & City 14 & 7 \\\\ \\hline City 5 & 29 & City 15 & 7 \\\\ \\hline City 6 & 11 & City 16 & 9 \\\\ \\hline City 7 & 7 & City 17 & 18 \\\\ \\hline City 8 & 11 & City 18 & 7 \\\\ \\hline City 9 & 18 & City 19 & 11 \\\\ \\hline City 10 & 3 & City 20 & 18 \\\\ \\hline \\end{array}$\nConstruct a $80$ \\% confidence interval for the average number of papers published in major world cities. [ANS] $< \\mu <$ [ANS]", "answer_v2": [ "10.9046128759665", "15.5953871240335" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The scientific productivity of major world cities was the subject of a recent study. The study determined the number of scientific papers published between 1994 and 1997 by researchers from each of the 20 world cities, and is shown below.\n$\\begin{array}{cccc}\\hline City & Number of papers & City & Number of papers \\\\ \\hline City 1 & 11 & City 11 & 10 \\\\ \\hline City 2 & 19 & City 12 & 9 \\\\ \\hline City 3 & 10 & City 13 & 3 \\\\ \\hline City 4 & 17 & City 14 & 18 \\\\ \\hline City 5 & 7 & City 15 & 30 \\\\ \\hline City 6 & 12 & City 16 & 24 \\\\ \\hline City 7 & 25 & City 17 & 19 \\\\ \\hline City 8 & 28 & City 18 & 5 \\\\ \\hline City 9 & 27 & City 19 & 10 \\\\ \\hline City 10 & 7 & City 20 & 20 \\\\ \\hline \\end{array}$\nConstruct a $98$ \\% confidence interval for the average number of papers published in major world cities. [ANS] $< \\mu <$ [ANS]", "answer_v3": [ "10.8693333820601", "20.2306666179399" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0169", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - t", "level": "3", "keywords": [ "statistics", "inference", "hypothesis testing", "t score" ], "problem_v1": "A random sample of 16 size AA batteries for toys yield a mean of 3.37 hours with standard deviation, 1.26 hours.\n(a) Find the critical value, t*, for a 99\\% CI. t*=[ANS]\n(b) Find the margin of error for a 99\\% CI. [ANS]", "answer_v1": [ "2.947", "2.947*1.26/[sqrt(16)]" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A random sample of 9 size AA batteries for toys yield a mean of 3.9 hours with standard deviation, 0.58 hours.\n(a) Find the critical value, t*, for a 99\\% CI. t*=[ANS]\n(b) Find the margin of error for a 99\\% CI. [ANS]", "answer_v2": [ "3.355", "3.355*0.58/[sqrt(9)]" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A random sample of 11 size AA batteries for toys yield a mean of 3.41 hours with standard deviation, 0.81 hours.\n(a) Find the critical value, t*, for a 99\\% CI. t*=[ANS]\n(b) Find the margin of error for a 99\\% CI. [ANS]", "answer_v3": [ "3.169", "3.169*0.81/[sqrt(11)]" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0170", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - t", "level": "2", "keywords": [ "Statistics", "Confidence Intervals", "small sample", "confidence interval", "statistics", "estimates" ], "problem_v1": "The following random sample was selected from a normal distribution: \\begin{array}{llllllllll} 12& 13& 7& 15& 7& 12& 8& 16& 12& 10 \\cr \\end{array}\n(a) $\\ $ Construct a $90$ \\% confidence interval for the population mean $\\mu$. [ANS] $\\leq \\mu \\leq$ [ANS]\n(b) $\\ $ Construct a $95$ \\% confidence interval for the population mean $\\mu$. [ANS] $\\leq \\mu \\leq$ [ANS]", "answer_v1": [ "9.37096811406155", "13.0290318859384", "8.94287262024945", "13.4571273797505" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "The following random sample was selected from a normal distribution: \\begin{array}{llllllllll} 4& 7& 12& 19& 3& 19& 7& 2& 7& 2 \\cr \\end{array}\n(a) $\\ $ Construct a $90$ \\% confidence interval for the population mean $\\mu$. [ANS] $\\leq \\mu \\leq$ [ANS]\n(b) $\\ $ Construct a $95$ \\% confidence interval for the population mean $\\mu$. [ANS] $\\leq \\mu \\leq$ [ANS]", "answer_v2": [ "4.46517114350669", "11.9348288564933", "3.59101284374375", "12.8089871562563" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "The following random sample was selected from a normal distribution: \\begin{array}{llllllllll} 6& 11& 7& 19& 5& 17& 7& 13& 5& 18 \\cr \\end{array}\n(a) $\\ $ Construct a $90$ \\% confidence interval for the population mean $\\mu$. [ANS] $\\leq \\mu \\leq$ [ANS]\n(b) $\\ $ Construct a $95$ \\% confidence interval for the population mean $\\mu$. [ANS] $\\leq \\mu \\leq$ [ANS]", "answer_v3": [ "7.55747302962753", "14.0425269703725", "6.79854083426647", "14.8014591657335" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Statistics_0171", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - t", "level": "2", "keywords": [ "Statistics", "Confidence Intervals" ], "problem_v1": "Periodically, the county Water Department tests the drinking water of homeowners for contaminants such as lead and copper. The lead and copper levels in water specimens collected in 1998 for a sample of 10 residents of a subdevelopement of the county are shown below.\n$\\begin{array}{cc}\\hline lead (\\mug/L) & copper (mg/L) \\\\ \\hline 2 & 0.391 \\\\ \\hline 2.2 & 0.142 \\\\ \\hline 4.4 & 0.385 \\\\ \\hline 3.7 & 0.457 \\\\ \\hline 1.8 & 0.357 \\\\ \\hline 3 & 0.355 \\\\ \\hline 3.5 & 0.121 \\\\ \\hline 4.5 & 0.588 \\\\ \\hline 3.5 & 0.243 \\\\ \\hline 3.4 & 0.463 \\\\ \\hline \\end{array}$\n(a) $\\ $ Construct a $99$ \\% confidence interval for the mean lead level in water specimans of the subdevelopment. [ANS] $\\leq \\mu \\leq$ [ANS]\n(b) $\\ $ Construct a $99$ \\% confidence interval for the mean copper level in water specimans of the subdevelopment. [ANS] $\\leq \\mu \\leq$ [ANS]", "answer_v1": [ "2.22867162286499", "4.17132837713501", "0.200490371187115", "0.499909628812885" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Periodically, the county Water Department tests the drinking water of homeowners for contaminants such as lead and copper. The lead and copper levels in water specimens collected in 1998 for a sample of 10 residents of a subdevelopement of the county are shown below.\n$\\begin{array}{cc}\\hline lead (\\mug/L) & copper (mg/L) \\\\ \\hline 5.6 & 0.397 \\\\ \\hline 0.9 & 0.73 \\\\ \\hline 0.5 & 0.577 \\\\ \\hline 1.9 & 0.243 \\\\ \\hline 1.1 & 0.516 \\\\ \\hline 2 & 0.173 \\\\ \\hline 0.4 & 0.501 \\\\ \\hline 5.7 & 0.17 \\\\ \\hline 2 & 0.169 \\\\ \\hline 3.4 & 0.29 \\\\ \\hline \\end{array}$\n(a) $\\ $ Construct a $99$ \\% confidence interval for the mean lead level in water specimans of the subdevelopment. [ANS] $\\leq \\mu \\leq$ [ANS]\n(b) $\\ $ Construct a $99$ \\% confidence interval for the mean copper level in water specimans of the subdevelopment. [ANS] $\\leq \\mu \\leq$ [ANS]", "answer_v2": [ "0.345944333333333", "4.35405566666667", "0.173083411811555", "0.580116588188445" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Periodically, the county Water Department tests the drinking water of homeowners for contaminants such as lead and copper. The lead and copper levels in water specimens collected in 1998 for a sample of 10 residents of a subdevelopement of the county are shown below.\n$\\begin{array}{cc}\\hline lead (\\mug/L) & copper (mg/L) \\\\ \\hline 5.5 & 0.095 \\\\ \\hline 1.2 & 0.882 \\\\ \\hline 3.3 & 0.515 \\\\ \\hline 1.2 & 0.577 \\\\ \\hline 4.9 & 0.553 \\\\ \\hline 5.3 & 0.263 \\\\ \\hline 1.6 & 0.038 \\\\ \\hline 3.6 & 0.226 \\\\ \\hline 2.1 & 0.713 \\\\ \\hline 1.9 & 0.261 \\\\ \\hline \\end{array}$\n(a) $\\ $ Construct a $99$ \\% confidence interval for the mean lead level in water specimans of the subdevelopment. [ANS] $\\leq \\mu \\leq$ [ANS]\n(b) $\\ $ Construct a $99$ \\% confidence interval for the mean copper level in water specimans of the subdevelopment. [ANS] $\\leq \\mu \\leq$ [ANS]", "answer_v3": [ "1.31246823979528", "4.80753176020472", "0.12791689835321", "0.69668310164679" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Statistics_0172", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - t", "level": "2", "keywords": [ "statistics", "inference about a population", "inference" ], "problem_v1": "Select True or False, depending on whether the corresponding statement is true or false.\n[ANS] 1. The upper limit of the 85\\% confidence interval for the population proportion $p$, given that $n=60$ and $\\hat{p}=0.20$ is $0.274$. [ANS] 2. The lower limit of the 90\\% confidence interval for the population proportion $p$, given that $n=400$ and $\\hat{p}=0.10$ is $0.1247$. [ANS] 3. If a sample has 15 observations and a 90\\% confidence estimate for $\\mu$ is needed, the appropriate t-score is 1.341. [ANS] 4. If a sample has 18 observations and a 90\\% confidence estimate for $\\mu$ is needed, the appropriate t-score is 1.740.", "answer_v1": [ "T", "F", "F", "T" ], "answer_type_v1": [ "TF", "TF", "TF", "TF" ], "options_v1": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v2": "Select True or False, depending on whether the corresponding statement is true or false.\n[ANS] 1. The lower limit of the 90\\% confidence interval for the population proportion $p$, given that $n=400$ and $\\hat{p}=0.10$ is $0.1247$. [ANS] 2. If a sample has 18 observations and a 90\\% confidence estimate for $\\mu$ is needed, the appropriate t-score is 1.740. [ANS] 3. The upper limit of the 85\\% confidence interval for the population proportion $p$, given that $n=60$ and $\\hat{p}=0.20$ is $0.274$. [ANS] 4. If a sample of size 20 is selected, the value of $A$ for the probability $P(t \\geq A)=0.01$ is 2.528.", "answer_v2": [ "F", "T", "T", "F" ], "answer_type_v2": [ "TF", "TF", "TF", "TF" ], "options_v2": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v3": "Select True or False, depending on whether the corresponding statement is true or false.\n[ANS] 1. If a sample of size 30 is selected, the value $A$ for the probability $P(-A \\leq t \\leq A)=0.95$ is 2.045. [ANS] 2. If a sample of size 20 is selected, the value of $A$ for the probability $P(t \\geq A)=0.01$ is 2.528. [ANS] 3. The upper limit of the 85\\% confidence interval for the population proportion $p$, given that $n=60$ and $\\hat{p}=0.20$ is $0.274$. [ANS] 4. If a sample has 15 observations and a 95\\% confidence estimate for $\\mu$ is needed, the appropriate t-score is 1.753.", "answer_v3": [ "T", "F", "T", "F" ], "answer_type_v3": [ "TF", "TF", "TF", "TF" ], "options_v3": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ] }, { "id": "Statistics_0173", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - t", "level": "3", "keywords": [ "statistics", "Inference about a population" ], "problem_v1": "How much money do winners go home with from the television quiz show Jeopardy? To determine an answer, a random sample of winners was drawn and the amount of money each won was recorded and listed below. Estimate with 96\\% confidence the mean winning's for all the show's players.\n\\begin{array}{ccccc} 36686 & 37471 & 39504 & 31088 & 31673 \\\\ 36402 & 36540 & 32475 & 34941 & 38110 \\\\ 30450 & 35200 & 33608 & 32991 & 33749 \\\\ \\end{array} UCL=[ANS]\nLCL=[ANS]", "answer_v1": [ "36312.4329882857", "33139.3003450476" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "How much money do winners go home with from the television quiz show Jeopardy? To determine an answer, a random sample of winners was drawn and the amount of money each won was recorded and listed below. Estimate with 96\\% confidence the mean winning's for all the show's players.\n\\begin{array}{ccccc} 30294 & 14651 & 18342 & 30596 & 17992 \\\\ 15320 & 18225 & 22944 & 13028 & 24477 \\\\ 20481 & 27877 & 15518 & 15454 & 17076 \\\\ \\end{array} UCL=[ANS]\nLCL=[ANS]", "answer_v2": [ "23528.3996951264", "16774.933638207" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "How much money do winners go home with from the television quiz show Jeopardy? To determine an answer, a random sample of winners was drawn and the amount of money each won was recorded and listed below. Estimate with 96\\% confidence the mean winning's for all the show's players.\n\\begin{array}{ccccc} 28378 & 21841 & 27241 & 20392 & 23191 \\\\ 32442 & 34494 & 33728 & 20274 & 22066 \\\\ 21293 & 17126 & 27723 & 35860 & 32110 \\\\ \\end{array} UCL=[ANS]\nLCL=[ANS]", "answer_v3": [ "30104.3189313576", "22983.5477353091" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0174", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample mean - t", "level": "3", "keywords": [ "statistics", "Inference about a population" ], "problem_v1": "The following data were drawn from a normal population. Find a 92.4\\% confidence interval for the mean.\n\\begin{array}{ccccccccccc} 18 & 19 & 21 & 14 & 14 & 18 & 18 & 15 & 17 & 20 \\end{array} UCL=[ANS]\nLCL=[ANS]", "answer_v1": [ "18.9294957788952", "15.8705042211048" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The following data were drawn from a normal population. Find a 94\\% confidence interval for the mean.\n\\begin{array}{ccccccccccc} 20 & 7 & 10 & 21 & 10 & 8 & 10 & 14 & 6 & 15 \\end{array} UCL=[ANS]\nLCL=[ANS]", "answer_v2": [ "15.6616601960643", "8.53833980393567" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The following data were drawn from a normal population. Find a 92.2\\% confidence interval for the mean.\n\\begin{array}{ccccccccccc} 16 & 10 & 15 & 9 & 11 & 19 & 21 & 20 & 9 & 10 \\end{array} UCL=[ANS]\nLCL=[ANS]", "answer_v3": [ "17.0082348737838", "10.9917651262162" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0175", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample proportion", "level": "2", "keywords": [ "Sample Size", "Confidence", "Margin of Error", "statistics", "estimates", "population", "samples" ], "problem_v1": "College officials want to estimate the percentage of students who carry a gun, knife, or other such weapon. How many randomly selected students must be surveyed in order to be $97$ \\% confident that the sample percentage has a margin of error of $2$ percentage points?\n(a) $\\ $ Assume that there is no available information that could be used as an estimate of $\\hat{p}$. Answer: [ANS]\n(b) $\\ $ Assume that another study indicated that $6$ \\% of college students carry weapons.\nAnswer: [ANS]", "answer_v1": [ "2944", "665" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "College officials want to estimate the percentage of students who carry a gun, knife, or other such weapon. How many randomly selected students must be surveyed in order to be $90$ \\% confident that the sample percentage has a margin of error of $1.5$ percentage points?\n(a) $\\ $ Assume that there is no available information that could be used as an estimate of $\\hat{p}$. Answer: [ANS]\n(b) $\\ $ Assume that another study indicated that $7$ \\% of college students carry weapons.\nAnswer: [ANS]", "answer_v2": [ "3007", "783" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "College officials want to estimate the percentage of students who carry a gun, knife, or other such weapon. How many randomly selected students must be surveyed in order to be $93$ \\% confident that the sample percentage has a margin of error of $1.5$ percentage points?\n(a) $\\ $ Assume that there is no available information that could be used as an estimate of $\\hat{p}$. Answer: [ANS]\n(b) $\\ $ Assume that another study indicated that $6$ \\% of college students carry weapons.\nAnswer: [ANS]", "answer_v3": [ "3648", "823" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0176", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample proportion", "level": "3", "keywords": [ "Confidence Interval", "Point Estimate", "Margin of Error", "statistics", "estimates", "population", "samples" ], "problem_v1": "Use the given confidence interval limits to find the point estimate $\\hat{p}$ and the margin of error $E.$ $0.64 < p < 0.76$ $\\hat{p}=$ [ANS] $\\ \\ \\ \\ $ $E=$ [ANS]", "answer_v1": [ "0.7", "0.06" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Use the given confidence interval limits to find the point estimate $\\hat{p}$ and the margin of error $E.$ $0.07 < p < 0.25$ $\\hat{p}=$ [ANS] $\\ \\ \\ \\ $ $E=$ [ANS]", "answer_v2": [ "0.16", "0.09" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Use the given confidence interval limits to find the point estimate $\\hat{p}$ and the margin of error $E.$ $0.29 < p < 0.41$ $\\hat{p}=$ [ANS] $\\ \\ \\ \\ $ $E=$ [ANS]", "answer_v3": [ "0.35", "0.06" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0177", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample proportion", "level": "2", "keywords": [ "Confidence Interval", "statistics", "estimates", "population", "samples" ], "problem_v1": "Construct the $97$ \\% confidence interval estimate of the population proportion $p$ if the sample size is $n=600$ and the number of successes in the sample is $x=317.$ [ANS] $< p <$ [ANS]", "answer_v1": [ "0.484107727117115", "0.572558939549551" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Construct the $90$ \\% confidence interval estimate of the population proportion $p$ if the sample size is $n=1000$ and the number of successes in the sample is $x=162.$ [ANS] $< p <$ [ANS]", "answer_v2": [ "0.142835090473532", "0.181164909526468" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Construct the $93$ \\% confidence interval estimate of the population proportion $p$ if the sample size is $n=700$ and the number of successes in the sample is $x=192.$ [ANS] $< p <$ [ANS]", "answer_v3": [ "0.243731424473105", "0.304840004098323" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0178", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample proportion", "level": "1", "keywords": [ "Confidence Interval" ], "problem_v1": "A poll is taken in which $376$ out of $550$ randomly selected voters indicated their preference for a certain candidate. Find a $98$ \\% confidence interval for $p$. [ANS] $\\leq p \\leq$ [ANS]", "answer_v1": [ "0.637504694263561", "0.729768033009166" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A poll is taken in which $308$ out of $600$ randomly selected voters indicated their preference for a certain candidate. Find a $80$ \\% confidence interval for $p$. [ANS] $\\leq p \\leq$ [ANS]", "answer_v2": [ "0.487183074376246", "0.53948359229042" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A poll is taken in which $331$ out of $575$ randomly selected voters indicated their preference for a certain candidate. Find a $90$ \\% confidence interval for $p$. [ANS] $\\leq p \\leq$ [ANS]", "answer_v3": [ "0.541749465078839", "0.609554882747248" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0179", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample proportion", "level": "3", "keywords": [ "large sample", "population proportion", "estimates", "confidence intervals" ], "problem_v1": "Astronaunts often report that there are times when they become disoriented as they move around in zero-gravity. Therefore, they ususally rely on bright colors and other visual information to help them estabish a top-down orientation. A study was conducted to assses the potential of using color as body orienting. $85$ college students, reclining on their backs in the dark, found it difficult to establish orientation when positioned on under a rotating disk. This rotating disk was painted half black and half white. Out of the $85$ students, $66$ believed they were right side up when the white was on top. Use this information to estimate the true proportion of subjects who use the white color as a cue for right-side-up orientation. That is, construct a $98$ \\% confidence interval for the true proportion. [ANS] $\\leq p \\leq$ [ANS]", "answer_v1": [ "0.671348243768397", "0.881592932702191" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Astronaunts often report that there are times when they become disoriented as they move around in zero-gravity. Therefore, they ususally rely on bright colors and other visual information to help them estabish a top-down orientation. A study was conducted to assses the potential of using color as body orienting. $60$ college students, reclining on their backs in the dark, found it difficult to establish orientation when positioned on under a rotating disk. This rotating disk was painted half black and half white. Out of the $60$ students, $59$ believed they were right side up when the white was on top. Use this information to estimate the true proportion of subjects who use the white color as a cue for right-side-up orientation. That is, construct a $80$ \\% confidence interval for the true proportion. [ANS] $\\leq p \\leq$ [ANS]", "answer_v2": [ "0.962152881731348", "1.00451378493532" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Astronaunts often report that there are times when they become disoriented as they move around in zero-gravity. Therefore, they ususally rely on bright colors and other visual information to help them estabish a top-down orientation. A study was conducted to assses the potential of using color as body orienting. $70$ college students, reclining on their backs in the dark, found it difficult to establish orientation when positioned on under a rotating disk. This rotating disk was painted half black and half white. Out of the $70$ students, $58$ believed they were right side up when the white was on top. Use this information to estimate the true proportion of subjects who use the white color as a cue for right-side-up orientation. That is, construct a $90$ \\% confidence interval for the true proportion. [ANS] $\\leq p \\leq$ [ANS]", "answer_v3": [ "0.754477125645043", "0.902665731497814" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0180", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample proportion", "level": "3", "keywords": [ "statistics", "Inference", "one proportion" ], "problem_v1": "Albert wants to determine a 98 percent confidence interval for the true proportion of times he rolls a 4 (using a fair, 6-sided die). How many rolls must Albert make to get a margin of error less than or equal to.05? [To find n, use the guessed value p*=1/6 for the sample proportion and the values for z*from a z-table or t-table.] [Round to the smallest integer that works.] n=[ANS]", "answer_v1": [ "301" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Pedro wants to determine a 80 percent confidence interval for the true proportion of times he rolls a 6 (using a fair, 6-sided die). How many rolls must Pedro make to get a margin of error less than or equal to.05? [To find n, use the guessed value p*=1/6 for the sample proportion and the values for z*from a z-table or t-table.] [Round to the smallest integer that works.] n=[ANS]", "answer_v2": [ "92" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Julia wants to determine a 90 percent confidence interval for the true proportion of times she rolls a 4 (using a fair, 6-sided die). How many rolls must Julia make to get a margin of error less than or equal to.05? [To find n, use the guessed value p*=1/6 for the sample proportion and the values for z*from a z-table or t-table.] [Round to the smallest integer that works.] n=[ANS]", "answer_v3": [ "151" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0181", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample proportion", "level": "3", "keywords": [ "statistics", "Inference", "one proportion" ], "problem_v1": "A recent survey showed that among 800 randomly selected subjects who completed 4 years of college, 175 smoke and 625 do not smoke. Determine a 95\\% confidence interval for the true proportion of the given population that smokes. 95\\% CI: [ANS] to [ANS]", "answer_v1": [ "0.21875-0.0286471", "0.21875+0.0286471" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A recent survey showed that among 650 randomly selected subjects who completed 4 years of college, 142 smoke and 508 do not smoke. Determine a 95\\% confidence interval for the true proportion of the given population that smokes. 95\\% CI: [ANS] to [ANS]", "answer_v2": [ "0.218462-0.031766", "0.218462+0.031766" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A recent survey showed that among 700 randomly selected subjects who completed 4 years of college, 151 smoke and 549 do not smoke. Determine a 95\\% confidence interval for the true proportion of the given population that smokes. 95\\% CI: [ANS] to [ANS]", "answer_v3": [ "0.215714-0.0304708", "0.215714+0.0304708" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0182", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample proportion", "level": "4", "keywords": [ "statistics", "Inference", "one proportion" ], "problem_v1": "For each problem, select the best response.\n(a) A newspaper conducted a statewide survey concerning the 1998 race for state senator. The newspaper took a SRS of 1200 registered voters and found that 620 would vote for the Republican candidate. Let $p$ represent the proportion of registered voters in the state who would vote for the Republican candidate. How large a sample $n$ would you need to estimate $p$ with a margin of error 0.01 with 95 percent confidence? Use the guess $p=.5$ as the value of $p$. [ANS] A. 49 B. 1500 C. 4800 D. 9604\n(b) A newspaper conducted a statewide survey concerning the 1998 race for state senator. The newspaper took a SRS of 1200 registered voters and found that 620 would vote for the Republican candidate. Let $p$ represent the proportion of registered voters in the state who would vote for the Republican candidate. A 90 percent confidence interval for $p$ is: [ANS] A. 0.517 $\\pm$ 0.249 B. 0.517 $\\pm$ 0.014 C. 0.517 $\\pm$ 0.028 D. 0.517 $\\pm$ 0.024\n(c) A radio talk show host with a large audience is interested in the proportion $p$ of adults in his listening area who think the drinking age should be lowered to 18. He asks, 'Do you think the drinking age should be reduced to 18 in light of the fact that 18 year olds are eligible for military service?' He asks listeners to phone in and vote 'yes' if they agree the drinking age should be lowered to 18, and 'no' if not. Of the 100 people who phoned in, 70 answered 'yes.' Which of the following assumptions for inference about a proportion using a confidence interval are violated? [ANS] A. The data are an SRS from the population of interest. B. The sample size is large enough so that the count of failures $n(1-\\hat{p})$ is 15 or more. C. The sample size is large enough so that the count of successes $n\\hat{p}$ is 15 or more. D. The population is at least ten times as large as the sample.", "answer_v1": [ "D", "D", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "For each problem, select the best response.\n(a) A radio talk show host with a large audience is interested in the proportion $p$ of adults in his listening area who think the drinking age should be lowered to 18. He asks, 'Do you think the drinking age should be reduced to 18 in light of the fact that 18 year olds are eligible for military service?' He asks listeners to phone in and vote 'yes' if they agree the drinking age should be lowered to 18, and 'no' if not. Of the 100 people who phoned in, 70 answered 'yes.' Which of the following assumptions for inference about a proportion using a confidence interval are violated? [ANS] A. The sample size is large enough so that the count of successes $n\\hat{p}$ is 15 or more. B. The sample size is large enough so that the count of failures $n(1-\\hat{p})$ is 15 or more. C. The population is at least ten times as large as the sample. D. The data are an SRS from the population of interest.\n(b) A newspaper conducted a statewide survey concerning the 1998 race for state senator. The newspaper took a SRS of 1200 registered voters and found that 620 would vote for the Republican candidate. Let $p$ represent the proportion of registered voters in the state who would vote for the Republican candidate. How large a sample $n$ would you need to estimate $p$ with a margin of error 0.01 with 95 percent confidence? Use the guess $p=.5$ as the value of $p$. [ANS] A. 49 B. 9604 C. 4800 D. 1500\n(c) A newspaper conducted a statewide survey concerning the 1998 race for state senator. The newspaper took a SRS of 1200 registered voters and found that 620 would vote for the Republican candidate. Let $p$ represent the proportion of registered voters in the state who would vote for the Republican candidate. A 90 percent confidence interval for $p$ is: [ANS] A. 0.517 $\\pm$ 0.024 B. 0.517 $\\pm$ 0.028 C. 0.517 $\\pm$ 0.249 D. 0.517 $\\pm$ 0.014", "answer_v2": [ "D", "B", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "For each problem, select the best response.\n(a) A radio talk show host with a large audience is interested in the proportion $p$ of adults in his listening area who think the drinking age should be lowered to 18. He asks, 'Do you think the drinking age should be reduced to 18 in light of the fact that 18 year olds are eligible for military service?' He asks listeners to phone in and vote 'yes' if they agree the drinking age should be lowered to 18, and 'no' if not. Of the 100 people who phoned in, 70 answered 'yes.' Which of the following assumptions for inference about a proportion using a confidence interval are violated? [ANS] A. The data are an SRS from the population of interest. B. The population is at least ten times as large as the sample. C. The sample size is large enough so that the count of successes $n\\hat{p}$ is 15 or more. D. The sample size is large enough so that the count of failures $n(1-\\hat{p})$ is 15 or more.\n(b) A newspaper conducted a statewide survey concerning the 1998 race for state senator. The newspaper took a SRS of 1200 registered voters and found that 620 would vote for the Republican candidate. Let $p$ represent the proportion of registered voters in the state who would vote for the Republican candidate. How large a sample $n$ would you need to estimate $p$ with a margin of error 0.01 with 95 percent confidence? Use the guess $p=.5$ as the value of $p$. [ANS] A. 1500 B. 9604 C. 4800 D. 49\n(c) A newspaper conducted a statewide survey concerning the 1998 race for state senator. The newspaper took a SRS of 1200 registered voters and found that 620 would vote for the Republican candidate. Let $p$ represent the proportion of registered voters in the state who would vote for the Republican candidate. A 90 percent confidence interval for $p$ is: [ANS] A. 0.517 $\\pm$ 0.024 B. 0.517 $\\pm$ 0.028 C. 0.517 $\\pm$ 0.014 D. 0.517 $\\pm$ 0.249", "answer_v3": [ "A", "B", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0183", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample proportion", "level": "1", "keywords": [ "statistics", "estimates", "population", "samples" ], "problem_v1": "Construct the $99$ \\% confidence interval estimate of the population proportion $p$ if the sample size is $n=800$ and the number of successes in the sample is $x=393.$ [ANS] $< p <$ [ANS]\nWhich of the following is the correct interpretation for your answer in part (a)? [ANS] A. We can be 99\\% confident that the percentage of successes in the population lies in the interval B. We can be 99\\% confident that the percentage of successes in the sample lies in the interval C. There is a 99\\% chance that the percentage of successes in the population lies in the interval D. None of the above", "answer_v1": [ "0.445722313755966", "0.536777686244034", "A" ], "answer_type_v1": [ "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "A", "B", "C", "D" ] ], "problem_v2": "Construct the $99$ \\% confidence interval estimate of the population proportion $p$ if the sample size is $n=100$ and the number of successes in the sample is $x=78.$ [ANS] $< p <$ [ANS]\nWhich of the following is the correct interpretation for your answer in part (a)? [ANS] A. There is a 99\\% chance that the percentage of successes in the population lies in the interval B. We can be 99\\% confident that the percentage of successes in the population lies in the interval C. We can be 99\\% confident that the percentage of successes in the sample lies in the interval D. None of the above", "answer_v2": [ "0.67329722305768", "0.88670277694232", "B" ], "answer_type_v2": [ "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "A", "B", "C", "D" ] ], "problem_v3": "Construct the $99$ \\% confidence interval estimate of the population proportion $p$ if the sample size is $n=400$ and the number of successes in the sample is $x=214.$ [ANS] $< p <$ [ANS]\nWhich of the following is the correct interpretation for your answer in part (a)? [ANS] A. We can be 99\\% confident that the percentage of successes in the population lies in the interval B. There is a 99\\% chance that the percentage of successes in the population lies in the interval C. We can be 99\\% confident that the percentage of successes in the sample lies in the interval D. None of the above", "answer_v3": [ "0.470762230586354", "0.599237769413646", "A" ], "answer_type_v3": [ "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0184", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample proportion", "level": "2", "keywords": [ "statistics", "estimation", "samples" ], "problem_v1": "A random sample of elementary school children in New York state is to be selected to estimate the proportion $p$ who have received a medical examination during the past year. An interval estimate of the proportion $p$ with a margin of error of $0.08$ and $98$ \\% confidence is required.\n(a) $\\ $ Assuming no prior information about $\\hat{p}$ is available, approximately how large of a sample size is needed? $n=$ [ANS]\n(b) $\\ $ If a planning study indicates that $\\hat{p}$ is around $0.8$, approximately how large of a sample size is needed? $n=$ [ANS]", "answer_v1": [ "212", "136" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A random sample of elementary school children in New York state is to be selected to estimate the proportion $p$ who have received a medical examination during the past year. An interval estimate of the proportion $p$ with a margin of error of $0.045$ and $95$ \\% confidence is required.\n(a) $\\ $ Assuming no prior information about $\\hat{p}$ is available, approximately how large of a sample size is needed? $n=$ [ANS]\n(b) $\\ $ If a planning study indicates that $\\hat{p}$ is around $0.9$, approximately how large of a sample size is needed? $n=$ [ANS]", "answer_v2": [ "475", "171" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A random sample of elementary school children in New York state is to be selected to estimate the proportion $p$ who have received a medical examination during the past year. An interval estimate of the proportion $p$ with a margin of error of $0.06$ and $90$ \\% confidence is required.\n(a) $\\ $ Assuming no prior information about $\\hat{p}$ is available, approximately how large of a sample size is needed? $n=$ [ANS]\n(b) $\\ $ If a planning study indicates that $\\hat{p}$ is around $0.8$, approximately how large of a sample size is needed? $n=$ [ANS]", "answer_v3": [ "188", "121" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0185", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample proportion", "level": "3", "keywords": [ "margin of error", "confidence interval", "sample" ], "problem_v1": "A poll was taken of 2400 NAU students to determine the proportion of students who eat at the union for most meals. It was found that 60 \\% of those surveyed ate most of their meals at the union. Recall that this sample statistic is meant to approximate the population parameter. For a 95\\% confidence level, what is the margin of error for this poll? (HINT: The margin of error is two times the standard deviation of the sampling distribution)\nMOE: [ANS] \\%", "answer_v1": [ "2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A poll was taken of 3750 NAU students to determine the proportion of students who eat at the union for most meals. It was found that 33 \\% of those surveyed ate most of their meals at the union. Recall that this sample statistic is meant to approximate the population parameter. For a 95\\% confidence level, what is the margin of error for this poll? (HINT: The margin of error is two times the standard deviation of the sampling distribution)\nMOE: [ANS] \\%", "answer_v2": [ "1.5357083056362" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A poll was taken of 2500 NAU students to determine the proportion of students who eat at the union for most meals. It was found that 42 \\% of those surveyed ate most of their meals at the union. Recall that this sample statistic is meant to approximate the population parameter. For a 95\\% confidence level, what is the margin of error for this poll? (HINT: The margin of error is two times the standard deviation of the sampling distribution)\nMOE: [ANS] \\%", "answer_v3": [ "1.97423402868049" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0186", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample proportion", "level": "3", "keywords": [ "Confidence Intervals" ], "problem_v1": "A random sample of 1700 registered voters in Flagstaff found 1173 registered voters who support immigration reform. Find a 95\\% confidence interval for the true percent of registered voters in Flagstaff who support immigration reform. Express your results to the nearest hundredth of a percent.. Answer: [ANS] to [ANS] \\%", "answer_v1": [ "66.7565782437378", "71.2434217562622" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A random sample of 600 registered voters in Flagstaff found 324 registered voters who support immigration reform. Find a 95\\% confidence interval for the true percent of registered voters in Flagstaff who support immigration reform. Express your results to the nearest hundredth of a percent.. Answer: [ANS] to [ANS] \\%", "answer_v2": [ "49.9306020101248", "58.0693979898752" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A random sample of 1000 registered voters in Flagstaff found 580 registered voters who support immigration reform. Find a 95\\% confidence interval for the true percent of registered voters in Flagstaff who support immigration reform. Express your results to the nearest hundredth of a percent.. Answer: [ANS] to [ANS] \\%", "answer_v3": [ "54.8784619175797", "61.1215380824203" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0187", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample proportion", "level": "3", "keywords": [ "Confidence Intervals" ], "problem_v1": "A random sample of 1700 home owners in a particular city found 595 home owners who had a swimming pool in their backyard. Find a 95\\% confidence interval for the true percent of home owners in this city who have a swimming pool in their backyard. Express your results to the nearest hundredth of a percent.. Answer: [ANS] to [ANS] \\%", "answer_v1": [ "32.6863575953768", "37.3136424046232" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A random sample of 600 car owners in a particular city found 96 car owners who received a speeding ticket this year. Find a 95\\% confidence interval for the true percent of car owners in this city who received a speeding ticket this year. Express your results to the nearest hundredth of a percent.. Answer: [ANS] to [ANS] \\%", "answer_v2": [ "13.0066740905808", "18.9933259094192" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A random sample of 1000 home owners in a particular city found 210 home owners who had a swimming pool in their backyard. Find a 95\\% confidence interval for the true percent of home owners in this city who have a swimming pool in their backyard. Express your results to the nearest hundredth of a percent.. Answer: [ANS] to [ANS] \\%", "answer_v3": [ "18.423956522106", "23.576043477894" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0188", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample proportion", "level": "3", "keywords": [ "statistics", "sampling" ], "problem_v1": "Refer to the following scenario. An epidemiologist is worried about the prevalence of the flu in East Vancouver and the potential shortage of vaccines for the area. She will need to provide a recommendation for how to allocate the vaccines appropriately across the city. She takes a simple random sample of 336 people living in East Vancouver and finds that 38 have recently had the flu. Suppose that the epidemiologist wants to re-estimate the population proportion and wishes for her 95\\% confidence interval to have a margin of error no larger than 0.04. How large a sample should she take to achieve this? Please carry answers to at least six decimal places in intermediate steps. Sample size=[ANS] (using a critical value of 1.96)", "answer_v1": [ "241" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Refer to the following scenario. An epidemiologist is worried about the prevalence of the flu in East Vancouver and the potential shortage of vaccines for the area. She will need to provide a recommendation for how to allocate the vaccines appropriately across the city. She takes a simple random sample of 340 people living in East Vancouver and finds that 30 have recently had the flu. Suppose that the epidemiologist wants to re-estimate the population proportion and wishes for her 95\\% confidence interval to have a margin of error no larger than 0.03. How large a sample should she take to achieve this? Please carry answers to at least six decimal places in intermediate steps. Sample size=[ANS] (using a critical value of 1.96)", "answer_v2": [ "344" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Refer to the following scenario. An epidemiologist is worried about the prevalence of the flu in East Vancouver and the potential shortage of vaccines for the area. She will need to provide a recommendation for how to allocate the vaccines appropriately across the city. She takes a simple random sample of 336 people living in East Vancouver and finds that 33 have recently had the flu. Suppose that the epidemiologist wants to re-estimate the population proportion and wishes for her 95\\% confidence interval to have a margin of error no larger than 0.03. How large a sample should she take to achieve this? Please carry answers to at least six decimal places in intermediate steps. Sample size=[ANS] (using a critical value of 1.96)", "answer_v3": [ "379" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0189", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample proportion", "level": "3", "keywords": [ "statistics", "sampling" ], "problem_v1": "Refer to the following scenario. A government official is in charge of allocating social programs throughout the city of Vancouver. He will decide where these social outreach programs should be located based on the percentage of residents living below the poverty line in each region of the city. He takes a simple random sample of 126 people living in Gastown and finds that 24 have an annual income that is below the poverty line. Suppose that the government official wants to re-estimate the population proportion and wishes for his 95\\% confidence interval to have a margin of error no larger than 0.04. How large a sample should he take to achieve this? Please carry answers to at least six decimal places in intermediate steps. Sample size=[ANS] (using a critical value of 1.96)", "answer_v1": [ "371" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Refer to the following scenario. A government official is in charge of allocating social programs throughout the city of Vancouver. He will decide where these social outreach programs should be located based on the percentage of residents living below the poverty line in each region of the city. He takes a simple random sample of 130 people living in Gastown and finds that 20 have an annual income that is below the poverty line. Suppose that the government official wants to re-estimate the population proportion and wishes for his 95\\% confidence interval to have a margin of error no larger than 0.03. How large a sample should he take to achieve this? Please carry answers to at least six decimal places in intermediate steps. Sample size=[ANS] (using a critical value of 1.96)", "answer_v2": [ "556" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Refer to the following scenario. A government official is in charge of allocating social programs throughout the city of Vancouver. He will decide where these social outreach programs should be located based on the percentage of residents living below the poverty line in each region of the city. He takes a simple random sample of 126 people living in Gastown and finds that 21 have an annual income that is below the poverty line. Suppose that the government official wants to re-estimate the population proportion and wishes for his 95\\% confidence interval to have a margin of error no larger than 0.03. How large a sample should he take to achieve this? Please carry answers to at least six decimal places in intermediate steps. Sample size=[ANS] (using a critical value of 1.96)", "answer_v3": [ "593" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0190", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample proportion", "level": "3", "keywords": [], "problem_v1": "McBeans magazine recently published a news article about caffeine consumption in universities that claims that 80\\% of people at universities drink coffee regularly. Moonbucks, a popular coffee chain, is interested in opening a new store on UBC campus. After reading McBeans' article, they will consider opening a store in UBC if more than 80\\% of the people in UBC drink coffee regularly. A random sample of people from UBC was taken, and it was found that 680 out of 810 survey participants considered themselves as regular coffee drinkers. Does Moonbucks' survey result provide sufficient evidence to support opening a store at UBC?\nPart i) What is the parameter of interest? [ANS] A. The proportion of all people at UBC that drink coffee regularly. B. The proportion of people at UBC that drink coffee regularly out of the 810 surveyed. C. All people at UBC that drinks coffee regularly. D. Whether a person at UBC drinks coffee regularly.\nPart ii) Let $p$ be the population proportion of people at UBC that drink coffee regularly. What are the null and alternative hypotheses? [ANS] A. Null: $p=0.80$. Alternative: $p=0.84$. B. Null: $p=0.84$. Alternative: $p > 0.80$. C. Null: $p=0.80$. Alternative: $p > 0.80$. D. Null: $p=0.84$. Alternative: $p \\ne 0.84$. E. Null: $p=0.80$. Alternative: $p \\ne 0.80$. F. Null: $p=0.84$. Alternative: $p > 0.84$.\nPart iii) The $P$-value is found to be about 0.0025. Using all the information available to you, which of the following is/are correct? (check all that apply) [ANS] A. The observed proportion of people at UBC that drink coffee regularly is unusually low if the reported value 80\\% is incorrect. B. The observed proportion of people at UBC that drink coffee regularly is unusually low if the reported value 80\\% is correct. C. Assuming the reported value 80\\% is incorrect, there is a 0.0025 probability that in a random sample of 810, at least 680 of the people at UBC regularly drink coffee D. Assuming the reported value 80\\% is correct, there is a 0.0025 probability that in a random sample of 810, at least 680 of the people at UBC regularly drink coffee. E. The reported value 80\\% must be false. F. The observed proportion of people at UBC that drink coffee regularly is unusually high if the reported value 80\\% is incorrect. G. The observed proportion of people at UBC that drink coffee regularly is unusually high if the reported value 80\\% is correct.\nPart iv) What is an appropriate conclusion for the hypothesis test at the 5\\% significance level? [ANS] A. There is sufficient evidence to support opening a store at UBC. B. There is insufficient evidence to support opening a store at UBC. C. There is a 5\\% probability that the reported value 80\\% is true. D. Both A. and C. E. Both B. and C.\nPart v) Which of the following scenarios describe the Type II error of the test? [ANS] A. The data do not provide sufficient evidence to support opening a store at UBC when in fact the true proportion of UBC people who drink coffee regularly is equal to the reported value 80\\%. B. The data provide sufficient evidence to support opening a store at UBC when in fact the true proportion of UBC people who drink coffee regularly is equal to the reported value 80\\%. C. The data provide sufficient evidence to support opening a store at UBC when in fact the true proportion of UBC people who drink coffee regularly exceeds the reported value 80\\%. D. The data do not provide sufficient evidence to support opening a store at UBC when in fact the true proportion of UBC people who drink coffee regularly exceeds the reported value 80\\%.\nPart vi) Based on the result of the hypothesis test, which of the following types of errors are we in a position of committing? [ANS] A. Type II error only. B. Both Type I and Type II errors. C. Neither Type I nor Type II errors. D. Type I error only.", "answer_v1": [ "A", "C", "DG", "A", "D", "D" ], "answer_type_v1": [ "MCS", "MCS", "MCM", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "McBeans magazine recently published a news article about caffeine consumption in universities that claims that 80\\% of people at universities drink coffee regularly. Moonbucks, a popular coffee chain, is interested in opening a new store on UBC campus. After reading McBeans' article, they will consider opening a store in UBC if more than 80\\% of the people in UBC drink coffee regularly. A random sample of people from UBC was taken, and it was found that 680 out of 810 survey participants considered themselves as regular coffee drinkers. Does Moonbucks' survey result provide sufficient evidence to support opening a store at UBC?\nPart i) What is the parameter of interest? [ANS] A. The proportion of all people at UBC that drink coffee regularly. B. The proportion of people at UBC that drink coffee regularly out of the 810 surveyed. C. All people at UBC that drinks coffee regularly. D. Whether a person at UBC drinks coffee regularly.\nPart ii) Let $p$ be the population proportion of people at UBC that drink coffee regularly. What are the null and alternative hypotheses? [ANS] A. Null: $p=0.80$. Alternative: $p > 0.80$. B. Null: $p=0.84$. Alternative: $p > 0.80$. C. Null: $p=0.84$. Alternative: $p > 0.84$. D. Null: $p=0.80$. Alternative: $p \\ne 0.80$. E. Null: $p=0.80$. Alternative: $p=0.84$. F. Null: $p=0.84$. Alternative: $p \\ne 0.84$.\nPart iii) The $P$-value is found to be about 0.0025. Using all the information available to you, which of the following is/are correct? (check all that apply) [ANS] A. The observed proportion of people at UBC that drink coffee regularly is unusually low if the reported value 80\\% is correct. B. Assuming the reported value 80\\% is correct, there is a 0.0025 probability that in a random sample of 810, at least 680 of the people at UBC regularly drink coffee. C. Assuming the reported value 80\\% is incorrect, there is a 0.0025 probability that in a random sample of 810, at least 680 of the people at UBC regularly drink coffee D. The observed proportion of people at UBC that drink coffee regularly is unusually high if the reported value 80\\% is correct. E. The observed proportion of people at UBC that drink coffee regularly is unusually low if the reported value 80\\% is incorrect. F. The observed proportion of people at UBC that drink coffee regularly is unusually high if the reported value 80\\% is incorrect. G. The reported value 80\\% must be false.\nPart iv) What is an appropriate conclusion for the hypothesis test at the 5\\% significance level? [ANS] A. There is sufficient evidence to support opening a store at UBC. B. There is insufficient evidence to support opening a store at UBC. C. There is a 5\\% probability that the reported value 80\\% is true. D. Both A. and C. E. Both B. and C.\nPart v) Which of the following scenarios describe the Type II error of the test? [ANS] A. The data do not provide sufficient evidence to support opening a store at UBC when in fact the true proportion of UBC people who drink coffee regularly exceeds the reported value 80\\%. B. The data provide sufficient evidence to support opening a store at UBC when in fact the true proportion of UBC people who drink coffee regularly exceeds the reported value 80\\%. C. The data do not provide sufficient evidence to support opening a store at UBC when in fact the true proportion of UBC people who drink coffee regularly is equal to the reported value 80\\%. D. The data provide sufficient evidence to support opening a store at UBC when in fact the true proportion of UBC people who drink coffee regularly is equal to the reported value 80\\%.\nPart vi) Based on the result of the hypothesis test, which of the following types of errors are we in a position of committing? [ANS] A. Type II error only. B. Type I error only. C. Neither Type I nor Type II errors. D. Both Type I and Type II errors.", "answer_v2": [ "A", "A", "BD", "A", "A", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCM", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "McBeans magazine recently published a news article about caffeine consumption in universities that claims that 80\\% of people at universities drink coffee regularly. Moonbucks, a popular coffee chain, is interested in opening a new store on UBC campus. After reading McBeans' article, they will consider opening a store in UBC if more than 80\\% of the people in UBC drink coffee regularly. A random sample of people from UBC was taken, and it was found that 680 out of 810 survey participants considered themselves as regular coffee drinkers. Does Moonbucks' survey result provide sufficient evidence to support opening a store at UBC?\nPart i) What is the parameter of interest? [ANS] A. The proportion of all people at UBC that drink coffee regularly. B. All people at UBC that drinks coffee regularly. C. The proportion of people at UBC that drink coffee regularly out of the 810 surveyed. D. Whether a person at UBC drinks coffee regularly.\nPart ii) Let $p$ be the population proportion of people at UBC that drink coffee regularly. What are the null and alternative hypotheses? [ANS] A. Null: $p=0.84$. Alternative: $p > 0.84$. B. Null: $p=0.84$. Alternative: $p > 0.80$. C. Null: $p=0.80$. Alternative: $p \\ne 0.80$. D. Null: $p=0.80$. Alternative: $p > 0.80$. E. Null: $p=0.80$. Alternative: $p=0.84$. F. Null: $p=0.84$. Alternative: $p \\ne 0.84$.\nPart iii) The $P$-value is found to be about 0.0025. Using all the information available to you, which of the following is/are correct? (check all that apply) [ANS] A. Assuming the reported value 80\\% is incorrect, there is a 0.0025 probability that in a random sample of 810, at least 680 of the people at UBC regularly drink coffee B. The observed proportion of people at UBC that drink coffee regularly is unusually low if the reported value 80\\% is incorrect. C. Assuming the reported value 80\\% is correct, there is a 0.0025 probability that in a random sample of 810, at least 680 of the people at UBC regularly drink coffee. D. The observed proportion of people at UBC that drink coffee regularly is unusually low if the reported value 80\\% is correct. E. The observed proportion of people at UBC that drink coffee regularly is unusually high if the reported value 80\\% is incorrect. F. The observed proportion of people at UBC that drink coffee regularly is unusually high if the reported value 80\\% is correct. G. The reported value 80\\% must be false.\nPart iv) What is an appropriate conclusion for the hypothesis test at the 5\\% significance level? [ANS] A. There is sufficient evidence to support opening a store at UBC. B. There is insufficient evidence to support opening a store at UBC. C. There is a 5\\% probability that the reported value 80\\% is true. D. Both A. and C. E. Both B. and C.\nPart v) Which of the following scenarios describe the Type II error of the test? [ANS] A. The data do not provide sufficient evidence to support opening a store at UBC when in fact the true proportion of UBC people who drink coffee regularly exceeds the reported value 80\\%. B. The data provide sufficient evidence to support opening a store at UBC when in fact the true proportion of UBC people who drink coffee regularly exceeds the reported value 80\\%. C. The data provide sufficient evidence to support opening a store at UBC when in fact the true proportion of UBC people who drink coffee regularly is equal to the reported value 80\\%. D. The data do not provide sufficient evidence to support opening a store at UBC when in fact the true proportion of UBC people who drink coffee regularly is equal to the reported value 80\\%.\nPart vi) Based on the result of the hypothesis test, which of the following types of errors are we in a position of committing? [ANS] A. Both Type I and Type II errors. B. Type I error only. C. Neither Type I nor Type II errors. D. Type II error only.", "answer_v3": [ "A", "D", "CF", "A", "A", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCM", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0191", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample proportion", "level": "3", "keywords": [], "problem_v1": "A report says that 82\\% of British Columbians over the age of 25 are high school graduates. A survey of randomly selected British Columbians included 1290 who were over the age of 25, and 1135 of them were high school graduates. Does the city\u2019s survey result provide sufficient evidence to contradict the reported value, 82\\%?\nPart i) What is the parameter of interest? [ANS] A. The proportion of all British Columbians (aged above 25) who are high school graduates. B. The proportion of 1290 British Columbians (aged above 25) who are high school graduates. C. All British Columbians aged above 25. D. Whether a British Columbian is a high school graduate.\nPart ii) Let $p$ be the population proportion of British Columbians aged above 25 who are high school graduates. What are the null and alternative hypotheses? [ANS] A. Null: $p=0.82$. Alternative: $p > 0.82$. B. Null: $p=0.88$. Alternative: $p \\ne 0.82$. C. Null: $p=0.82$. Alternative: $p \\ne 0.82$. D. Null: $p=0.88$. Alternative: $p \\ne 0.88$. E. Null: $p=0.82$. Alternative: $p=0.88$. F. Null: $p=0.88$. Alternative: $p > 0.88$.\nPart iii) The $P$-value is less than 0.0001. Using all the information available to you, which of the following is/are correct? (check all that apply) [ANS] A. The observed proportion of British Columbians who are high school graduates is unusually low if the reported value 82\\% is incorrect. B. The observed proportion of British Columbians who are high school graduates is unusually low if the reported value 82\\% is correct. C. Assuming the reported value 82\\% is incorrect, it is nearly impossible that in a random sample of 1290 British Columbians aged above 25, 1135 or more are high school graduates. D. Assuming the reported value 82\\% is correct, it is nearly impossible that in a random sample of 1290 British Columbians aged above 25, 1135 or more are high school graduates. E. The reported value 82\\% must be false. F. The observed proportion of British Columbians who are high school graduates is unusually high if the reported value 82\\% is incorrect. G. The observed proportion of British Columbians who are high school graduates is unusually high if the reported value 82\\% is correct.\nPart iv) What is an appropriate conclusion for the hypothesis test at the 5\\% significance level? [ANS] A. There is sufficient evidence to contradict the reported value 82\\%. B. There is insufficient evidence to contradict the reported value 82\\%. C. There is a 5\\% probability that the reported value 82\\% is true. D. Both A. and C. E. Both B. and C.\nPart v) Which of the following scenarios describe the Type II error of the test? [ANS] A. The data suggest that reported value is incorrect when in fact the value is correct. B. The data suggest that reported value is correct when in fact the value is correct. C. The data suggest that reported value is incorrect when in fact the value is incorrect. D. The data suggest that reported value is correct when in fact the value is incorrect.\nPart vi) Based on the result of the hypothesis test, which of the following types of errors are we in a position of committing? [ANS] A. Type II error only. B. Both Type I and Type II errors. C. Neither Type I nor Type II errors. D. Type I error only.", "answer_v1": [ "A", "C", "DG", "A", "D", "D" ], "answer_type_v1": [ "MCS", "MCS", "MCM", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "A report says that 82\\% of British Columbians over the age of 25 are high school graduates. A survey of randomly selected British Columbians included 1290 who were over the age of 25, and 1135 of them were high school graduates. Does the city\u2019s survey result provide sufficient evidence to contradict the reported value, 82\\%?\nPart i) What is the parameter of interest? [ANS] A. The proportion of all British Columbians (aged above 25) who are high school graduates. B. The proportion of 1290 British Columbians (aged above 25) who are high school graduates. C. All British Columbians aged above 25. D. Whether a British Columbian is a high school graduate.\nPart ii) Let $p$ be the population proportion of British Columbians aged above 25 who are high school graduates. What are the null and alternative hypotheses? [ANS] A. Null: $p=0.82$. Alternative: $p \\ne 0.82$. B. Null: $p=0.88$. Alternative: $p \\ne 0.82$. C. Null: $p=0.88$. Alternative: $p > 0.88$. D. Null: $p=0.82$. Alternative: $p=0.88$. E. Null: $p=0.82$. Alternative: $p > 0.82$. F. Null: $p=0.88$. Alternative: $p \\ne 0.88$.\nPart iii) The $P$-value is less than 0.0001. Using all the information available to you, which of the following is/are correct? (check all that apply) [ANS] A. The observed proportion of British Columbians who are high school graduates is unusually low if the reported value 82\\% is correct. B. Assuming the reported value 82\\% is correct, it is nearly impossible that in a random sample of 1290 British Columbians aged above 25, 1135 or more are high school graduates. C. Assuming the reported value 82\\% is incorrect, it is nearly impossible that in a random sample of 1290 British Columbians aged above 25, 1135 or more are high school graduates. D. The observed proportion of British Columbians who are high school graduates is unusually high if the reported value 82\\% is correct. E. The observed proportion of British Columbians who are high school graduates is unusually low if the reported value 82\\% is incorrect. F. The observed proportion of British Columbians who are high school graduates is unusually high if the reported value 82\\% is incorrect. G. The reported value 82\\% must be false.\nPart iv) What is an appropriate conclusion for the hypothesis test at the 5\\% significance level? [ANS] A. There is sufficient evidence to contradict the reported value 82\\%. B. There is insufficient evidence to contradict the reported value 82\\%. C. There is a 5\\% probability that the reported value 82\\% is true. D. Both A. and C. E. Both B. and C.\nPart v) Which of the following scenarios describe the Type II error of the test? [ANS] A. The data suggest that reported value is correct when in fact the value is incorrect. B. The data suggest that reported value is incorrect when in fact the value is incorrect. C. The data suggest that reported value is incorrect when in fact the value is correct. D. The data suggest that reported value is correct when in fact the value is correct.\nPart vi) Based on the result of the hypothesis test, which of the following types of errors are we in a position of committing? [ANS] A. Type II error only. B. Type I error only. C. Neither Type I nor Type II errors. D. Both Type I and Type II errors.", "answer_v2": [ "A", "A", "BD", "A", "A", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCM", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "A report says that 82\\% of British Columbians over the age of 25 are high school graduates. A survey of randomly selected British Columbians included 1290 who were over the age of 25, and 1135 of them were high school graduates. Does the city\u2019s survey result provide sufficient evidence to contradict the reported value, 82\\%?\nPart i) What is the parameter of interest? [ANS] A. The proportion of all British Columbians (aged above 25) who are high school graduates. B. All British Columbians aged above 25. C. The proportion of 1290 British Columbians (aged above 25) who are high school graduates. D. Whether a British Columbian is a high school graduate.\nPart ii) Let $p$ be the population proportion of British Columbians aged above 25 who are high school graduates. What are the null and alternative hypotheses? [ANS] A. Null: $p=0.88$. Alternative: $p > 0.88$. B. Null: $p=0.88$. Alternative: $p \\ne 0.82$. C. Null: $p=0.82$. Alternative: $p=0.88$. D. Null: $p=0.82$. Alternative: $p \\ne 0.82$. E. Null: $p=0.82$. Alternative: $p > 0.82$. F. Null: $p=0.88$. Alternative: $p \\ne 0.88$.\nPart iii) The $P$-value is less than 0.0001. Using all the information available to you, which of the following is/are correct? (check all that apply) [ANS] A. Assuming the reported value 82\\% is incorrect, it is nearly impossible that in a random sample of 1290 British Columbians aged above 25, 1135 or more are high school graduates. B. The observed proportion of British Columbians who are high school graduates is unusually low if the reported value 82\\% is incorrect. C. Assuming the reported value 82\\% is correct, it is nearly impossible that in a random sample of 1290 British Columbians aged above 25, 1135 or more are high school graduates. D. The observed proportion of British Columbians who are high school graduates is unusually low if the reported value 82\\% is correct. E. The observed proportion of British Columbians who are high school graduates is unusually high if the reported value 82\\% is incorrect. F. The observed proportion of British Columbians who are high school graduates is unusually high if the reported value 82\\% is correct. G. The reported value 82\\% must be false.\nPart iv) What is an appropriate conclusion for the hypothesis test at the 5\\% significance level? [ANS] A. There is sufficient evidence to contradict the reported value 82\\%. B. There is insufficient evidence to contradict the reported value 82\\%. C. There is a 5\\% probability that the reported value 82\\% is true. D. Both A. and C. E. Both B. and C.\nPart v) Which of the following scenarios describe the Type II error of the test? [ANS] A. The data suggest that reported value is correct when in fact the value is incorrect. B. The data suggest that reported value is incorrect when in fact the value is incorrect. C. The data suggest that reported value is correct when in fact the value is correct. D. The data suggest that reported value is incorrect when in fact the value is correct.\nPart vi) Based on the result of the hypothesis test, which of the following types of errors are we in a position of committing? [ANS] A. Both Type I and Type II errors. B. Type I error only. C. Neither Type I nor Type II errors. D. Type II error only.", "answer_v3": [ "A", "D", "CF", "A", "A", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCM", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0192", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample proportion", "level": "3", "keywords": [], "problem_v1": "BlueSky Air claims that at least 80\\% of its flights arrive on time. A random sample of 160 BlueSky Air flights revealed that 115 arrive on time. Do the data provide sufficient evidence to contradict the claim by BlueSky Air (i.e., you would like to see whether the percentage of the airline's flight is below what the airline claims)?\nPart i) What is the parameter of interest? [ANS] A. The proportion of all BlueSky Air flights that arrive on time. B. All BlueSky Air flights. C. The proportion of the 160 BlueSky Air flights that arrive on time. D. Whether a BlueSky Air flight arrives on time.\nPart ii) Let $p$ be the population proportion of flights that arrive on time. What are the null and alternative hypotheses? [ANS] A. Null: $p > 0.80$. Alternative: $p \\le 0.80$. B. Null: $p=115/160$. Alternative: $p \\ne 115/160$. C. Null: $p=0.80$. Alternative: $p < 0.80$. D. Null: $p=0.80$. Alternative: $p > 0.80$. E. Null: $p < 0.80$. Alternative: $p \\ge 0.80$. F. Null: $p=0.80$. Alternative: $p \\ne 0.80$.\nPart iii) Compute the P-value (please round to four decimal places): [ANS]\nPart iv) The $P$-value is computed to be [P-value] (your answer in\nPart iii). Using all of the information available to you, which of the following is/are correct? (check all that apply) [ANS] A. The observed proportion of flights that arrive on time is unusually low if BlueSky Air's claim is false. B. The observed proportion of flights that arrive on time is unusually high if BlueSky Air's claim is true. C. Assuming BlueSky's claim is false, there is a [P-value] probability that in a random sample of 160 flights, 115 or fewer flights arrive on time. D. Assuming BlueSky's claim is true, there is a [P-value] probability that in a random sample of 160 flights, 115 or fewer flights arrive on time. E. There is a [P-value] probability that BlueSky Air's claim is true. F. The observed proportion of flights that arrive on time is unusually high if BlueSky Air's claim is false. G. The observed proportion of flights that arrive on time is unusually low if BlueSky Air's claim is true.\nPart v) What is an appropriate conclusion for the hypothesis test at the 1\\% significance level? [ANS] A. There is sufficient evidence to contradict BlueSky Air's claim. B. BlueSky Air's claim is false. C. BlueSky Air's claim is true. D. There is sufficient evidence to support BlueSky Air's claim.\nPart vi) Which of the following scenarios describe the Type I error of the test? [ANS] A. The data suggest that BlueSky Air's claim is true when in fact the claim is false. B. The data suggest that BlueSky Air's claim is false when in fact the claim is false. C. The data suggest that BlueSky Air's claim is true when in fact the claim is true. D. The data suggest that BlueSky Air's claim is false when in fact the claim is true.\nPart vii) Based on the result of the hypothesis test, which of the following types of errors are we in a position of committing? [ANS] A. Type II error only. B. Both Type I and Type II errors. C. Neither Type I nor Type II errors. D. Type I error only.", "answer_v1": [ "A", "C", "0.0051", "DG", "A", "D", "D" ], "answer_type_v1": [ "MCS", "MCS", "NV", "MCM", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E", "F" ], [], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "BlueSky Air claims that at least 80\\% of its flights arrive on time. A random sample of 160 BlueSky Air flights revealed that 115 arrive on time. Do the data provide sufficient evidence to contradict the claim by BlueSky Air (i.e., you would like to see whether the percentage of the airline's flight is below what the airline claims)?\nPart i) What is the parameter of interest? [ANS] A. The proportion of all BlueSky Air flights that arrive on time. B. All BlueSky Air flights. C. The proportion of the 160 BlueSky Air flights that arrive on time. D. Whether a BlueSky Air flight arrives on time.\nPart ii) Let $p$ be the population proportion of flights that arrive on time. What are the null and alternative hypotheses? [ANS] A. Null: $p=0.80$. Alternative: $p < 0.80$. B. Null: $p=115/160$. Alternative: $p \\ne 115/160$. C. Null: $p=0.80$. Alternative: $p \\ne 0.80$. D. Null: $p < 0.80$. Alternative: $p \\ge 0.80$. E. Null: $p > 0.80$. Alternative: $p \\le 0.80$. F. Null: $p=0.80$. Alternative: $p > 0.80$.\nPart iii) Compute the P-value (please round to four decimal places): [ANS]\nPart iv) The $P$-value is computed to be [P-value] (your answer in\nPart iii). Using all of the information available to you, which of the following is/are correct? (check all that apply) [ANS] A. The observed proportion of flights that arrive on time is unusually high if BlueSky Air's claim is true. B. Assuming BlueSky's claim is true, there is a [P-value] probability that in a random sample of 160 flights, 115 or fewer flights arrive on time. C. Assuming BlueSky's claim is false, there is a [P-value] probability that in a random sample of 160 flights, 115 or fewer flights arrive on time. D. The observed proportion of flights that arrive on time is unusually low if BlueSky Air's claim is true. E. The observed proportion of flights that arrive on time is unusually low if BlueSky Air's claim is false. F. The observed proportion of flights that arrive on time is unusually high if BlueSky Air's claim is false. G. There is a [P-value] probability that BlueSky Air's claim is true.\nPart v) What is an appropriate conclusion for the hypothesis test at the 1\\% significance level? [ANS] A. BlueSky Air's claim is true. B. BlueSky Air's claim is false. C. There is sufficient evidence to support BlueSky Air's claim. D. There is sufficient evidence to contradict BlueSky Air's claim.\nPart vi) Which of the following scenarios describe the Type I error of the test? [ANS] A. The data suggest that BlueSky Air's claim is false when in fact the claim is true. B. The data suggest that BlueSky Air's claim is true when in fact the claim is true. C. The data suggest that BlueSky Air's claim is true when in fact the claim is false. D. The data suggest that BlueSky Air's claim is false when in fact the claim is false.\nPart vii) Based on the result of the hypothesis test, which of the following types of errors are we in a position of committing? [ANS] A. Type II error only. B. Type I error only. C. Neither Type I nor Type II errors. D. Both Type I and Type II errors.", "answer_v2": [ "A", "A", "0.0051", "BD", "D", "A", "B" ], "answer_type_v2": [ "MCS", "MCS", "NV", "MCM", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E", "F" ], [], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "BlueSky Air claims that at least 80\\% of its flights arrive on time. A random sample of 160 BlueSky Air flights revealed that 115 arrive on time. Do the data provide sufficient evidence to contradict the claim by BlueSky Air (i.e., you would like to see whether the percentage of the airline's flight is below what the airline claims)?\nPart i) What is the parameter of interest? [ANS] A. The proportion of all BlueSky Air flights that arrive on time. B. The proportion of the 160 BlueSky Air flights that arrive on time. C. All BlueSky Air flights. D. Whether a BlueSky Air flight arrives on time.\nPart ii) Let $p$ be the population proportion of flights that arrive on time. What are the null and alternative hypotheses? [ANS] A. Null: $p=0.80$. Alternative: $p \\ne 0.80$. B. Null: $p=115/160$. Alternative: $p \\ne 115/160$. C. Null: $p < 0.80$. Alternative: $p \\ge 0.80$. D. Null: $p=0.80$. Alternative: $p < 0.80$. E. Null: $p > 0.80$. Alternative: $p \\le 0.80$. F. Null: $p=0.80$. Alternative: $p > 0.80$.\nPart iii) Compute the P-value (please round to four decimal places): [ANS]\nPart iv) The $P$-value is computed to be [P-value] (your answer in\nPart iii). Using all of the information available to you, which of the following is/are correct? (check all that apply) [ANS] A. Assuming BlueSky's claim is false, there is a [P-value] probability that in a random sample of 160 flights, 115 or fewer flights arrive on time. B. The observed proportion of flights that arrive on time is unusually low if BlueSky Air's claim is false. C. Assuming BlueSky's claim is true, there is a [P-value] probability that in a random sample of 160 flights, 115 or fewer flights arrive on time. D. The observed proportion of flights that arrive on time is unusually high if BlueSky Air's claim is true. E. The observed proportion of flights that arrive on time is unusually high if BlueSky Air's claim is false. F. The observed proportion of flights that arrive on time is unusually low if BlueSky Air's claim is true. G. There is a [P-value] probability that BlueSky Air's claim is true.\nPart v) What is an appropriate conclusion for the hypothesis test at the 1\\% significance level? [ANS] A. There is sufficient evidence to contradict BlueSky Air's claim. B. There is sufficient evidence to support BlueSky Air's claim. C. BlueSky Air's claim is true. D. BlueSky Air's claim is false.\nPart vi) Which of the following scenarios describe the Type I error of the test? [ANS] A. The data suggest that BlueSky Air's claim is false when in fact the claim is true. B. The data suggest that BlueSky Air's claim is true when in fact the claim is true. C. The data suggest that BlueSky Air's claim is false when in fact the claim is false. D. The data suggest that BlueSky Air's claim is true when in fact the claim is false.\nPart vii) Based on the result of the hypothesis test, which of the following types of errors are we in a position of committing? [ANS] A. Both Type I and Type II errors. B. Type I error only. C. Neither Type I nor Type II errors. D. Type II error only.", "answer_v3": [ "A", "D", "0.0051", "CF", "A", "A", "B" ], "answer_type_v3": [ "MCS", "MCS", "NV", "MCM", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E", "F" ], [], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0193", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample proportion", "level": "3", "keywords": [ "statistics", "Inference about a population" ], "problem_v1": "A. Determine the sample size required to estimate a population proportion to within 0.038 with 95.5\\% confidence, assuming that you have no knowledge of the approximate value of the sample proportion. Sample Size=[ANS]\nB. Repeat part the previous problem, but now with the knowledge that the population proportion is approximately 0.31. Sample Size=[ANS]", "answer_v1": [ "696", "596" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A. Determine the sample size required to estimate a population proportion to within 0.03 with 98.9\\% confidence, assuming that you have no knowledge of the approximate value of the sample proportion. Sample Size=[ANS]\nB. Repeat part the previous problem, but now with the knowledge that the population proportion is approximately 0.26. Sample Size=[ANS]", "answer_v2": [ "1796", "1383" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A. Determine the sample size required to estimate a population proportion to within 0.032 with 95.8\\% confidence, assuming that you have no knowledge of the approximate value of the sample proportion. Sample Size=[ANS]\nB. Repeat part the previous problem, but now with the knowledge that the population proportion is approximately 0.27. Sample Size=[ANS]", "answer_v3": [ "1010", "796" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0194", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample proportion", "level": "3", "keywords": [ "statistics", "Inference", "inference about a population" ], "problem_v1": "Under which of the following circumstances is it impossible to construct a confidence interval for the population mean? [ANS] A. A non-normal population with a small sample and an unknown population variance. B. A normal population with a small sample and an unknown population variance. C. A normal population with a large sample and a known population variance. D. A non-normal population with a large sample and an unknown population variance.\nThe use of the standard normal distribution for constructing a confidence interval estimate for the population proportion $p$ requires: [ANS] A. that the sample size is greater than 30 B. $np$ and $n(1-p)$ are both greater than 5 C. $n\\hat{p}$ and $n(1-\\hat{p}$) are both greater than 5, where $\\hat{p}$ denotes the sample proportion D. $n(p+\\hat{p})$ and $n(p-\\hat{p})$ are both greater than 5", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Under which of the following circumstances is it impossible to construct a confidence interval for the population mean? [ANS] A. A normal population with a large sample and a known population variance. B. A normal population with a small sample and an unknown population variance. C. A non-normal population with a large sample and an unknown population variance. D. A non-normal population with a small sample and an unknown population variance.\nIn selecting the sample size to estimate the population proportion $p$, if we have no knowledge of even the approximate value of the sample proportion $\\hat{p}$, we: [ANS] A. let $\\hat{p}=0.50$ B. let $\\hat{p}=0.95$ C. take another sample and estimate $\\hat{p}$ D. take two more samples and find the average of their $\\hat{p}$", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Under which of the following circumstances is it impossible to construct a confidence interval for the population mean? [ANS] A. A normal population with a large sample and a known population variance. B. A non-normal population with a small sample and an unknown population variance. C. A normal population with a small sample and an unknown population variance. D. A non-normal population with a large sample and an unknown population variance.\nAs its degrees of freedom increase, the chi-squared distribution approaches the shape of the [ANS] A. normal distribution B. exponential distribution C. Student t distribution D. Poisson distribution", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0195", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample proportion", "level": "3", "keywords": [ "statistics", "Inference about a population" ], "problem_v1": "A polling organization has been hired to conduct a survey to determine the proportion of people who are likely to support an upcoming ballot initiative. To help determine the size of the sample required for the main poll, a preliminary poll of 20 people is taken that yields the results below. (Yes indicates that the person will support the initiative, No indicates that they won't.)\n\\begin{array}{lllll} \\mbox{No} & \\mbox{Yes} & \\mbox{Yes} & \\mbox{No} & \\mbox{No} \\\\ \\mbox{No} & \\mbox{Yes} & \\mbox{Yes} & \\mbox{Yes} & \\mbox{Yes} \\\\ \\mbox{Yes} & \\mbox{No} & \\mbox{Yes} & \\mbox{Yes} & \\mbox{Yes} \\\\ \\mbox{Yes} & \\mbox{Yes} & \\mbox{No} & \\mbox{No} & \\mbox{Yes} \\\\ \\end{array} Based on the results of the peliminary poll, how large a sample is required for the main poll, assuming that the main poll should have a margin of error of 3.1\\% and a confidence level of 98.2\\%. Sample Size=[ANS]", "answer_v1": [ "1325" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A polling organization has been hired to conduct a survey to determine the proportion of people who are likely to support an upcoming ballot initiative. To help determine the size of the sample required for the main poll, a preliminary poll of 20 people is taken that yields the results below. (Yes indicates that the person will support the initiative, No indicates that they won't.)\n\\begin{array}{lllll} \\mbox{No} & \\mbox{No} & \\mbox{No} & \\mbox{Yes} & \\mbox{No} \\\\ \\mbox{No} & \\mbox{No} & \\mbox{No} & \\mbox{Yes} & \\mbox{No} \\\\ \\mbox{Yes} & \\mbox{Yes} & \\mbox{No} & \\mbox{No} & \\mbox{No} \\\\ \\mbox{No} & \\mbox{Yes} & \\mbox{No} & \\mbox{No} & \\mbox{Yes} \\\\ \\end{array} Based on the results of the peliminary poll, how large a sample is required for the main poll, assuming that the main poll should have a margin of error of 2.7\\% and a confidence level of 95.5\\%. Sample Size=[ANS]", "answer_v2": [ "1158" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A polling organization has been hired to conduct a survey to determine the proportion of people who are likely to support an upcoming ballot initiative. To help determine the size of the sample required for the main poll, a preliminary poll of 20 people is taken that yields the results below. (Yes indicates that the person will support the initiative, No indicates that they won't.)\n\\begin{array}{lllll} \\mbox{No} & \\mbox{Yes} & \\mbox{No} & \\mbox{Yes} & \\mbox{No} \\\\ \\mbox{Yes} & \\mbox{No} & \\mbox{No} & \\mbox{No} & \\mbox{No} \\\\ \\mbox{No} & \\mbox{No} & \\mbox{No} & \\mbox{Yes} & \\mbox{No} \\\\ \\mbox{No} & \\mbox{Yes} & \\mbox{No} & \\mbox{No} & \\mbox{Yes} \\\\ \\end{array} Based on the results of the peliminary poll, how large a sample is required for the main poll, assuming that the main poll should have a margin of error of 3.2\\% and a confidence level of 97.1\\%. Sample Size=[ANS]", "answer_v3": [ "978" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0196", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "One sample proportion", "level": "3", "keywords": [ "statistics", "Inference about a population" ], "problem_v1": "In a random sample of 740 Americans, 38.6\\% indicated that they have a cat for a pet. Estimate with 96\\% confidence the proportion of all Americans that have cats as pets. (Give the confidence interval in percentages.) Confidence Interval: [ANS]", "answer_v1": [ "(34.9245581634078,42.2754418365922)" ], "answer_type_v1": [ "INT" ], "options_v1": [ [] ], "problem_v2": "In a random sample of 830 Americans, 29.1\\% indicated that they have a cat for a pet. Estimate with 93\\% confidence the proportion of all Americans that have cats as pets. (Give the confidence interval in percentages.) Confidence Interval: [ANS]", "answer_v2": [ "(26.2432832068666,31.9567167931334)" ], "answer_type_v2": [ "INT" ], "options_v2": [ [] ], "problem_v3": "In a random sample of 750 Americans, 32.4\\% indicated that they have a cat for a pet. Estimate with 94\\% confidence the proportion of all Americans that have cats as pets. (Give the confidence interval in percentages.) Confidence Interval: [ANS]", "answer_v3": [ "(29.185928540138,35.614071459862)" ], "answer_type_v3": [ "INT" ], "options_v3": [ [] ] }, { "id": "Statistics_0197", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "Two sample proportion", "level": "2", "keywords": [ "statistics", "inferences", "two samples", "2 samples" ], "problem_v1": "Find the size of each sample needed to estimate the difference between the proportions of boys and girls under 10 years old who are afraid of spiders. Assume that we want $98$ \\% confidence that the error is smaller than $0.06.$ $n=$ [ANS]", "answer_v1": [ "376" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the size of each sample needed to estimate the difference between the proportions of boys and girls under 10 years old who are afraid of spiders. Assume that we want $95$ \\% confidence that the error is smaller than $0.1.$ $n=$ [ANS]", "answer_v2": [ "97" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the size of each sample needed to estimate the difference between the proportions of boys and girls under 10 years old who are afraid of spiders. Assume that we want $96$ \\% confidence that the error is smaller than $0.07.$ $n=$ [ANS]", "answer_v3": [ "216" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0198", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "Two sample proportion", "level": "3", "keywords": [ "statistics", "Inference about a population" ], "problem_v1": "Two random samples are taken, one from among UVA students and the other from among UNC students. Both groups are asked if academics are their top priority. A summary of the sample sizes and proportions of each group answering ``yes'' are given below:\n\\begin{array}{lll} \\mbox{UVA (Pop. 1)}: & n_1=95, & \\hat{p}_1=0.793 \\\\ \\mbox{UNC (Pop. 2)}: & n_2=92, & \\hat{p}_2=0.658 \\\\ \\end{array} Find a 93.8\\% confidence interval for the difference $p_1-p_2$ of the population proportions. Confidence interval=[ANS]", "answer_v1": [ "(0.0144256271318742,0.255574372868126)" ], "answer_type_v1": [ "INT" ], "options_v1": [ [] ], "problem_v2": "Two random samples are taken, one from among UVA students and the other from among UNC students. Both groups are asked if academics are their top priority. A summary of the sample sizes and proportions of each group answering ``yes'' are given below:\n\\begin{array}{lll} \\mbox{UVA (Pop. 1)}: & n_1=81, & \\hat{p}_1=0.722 \\\\ \\mbox{UNC (Pop. 2)}: & n_2=99, & \\hat{p}_2=0.6 \\\\ \\end{array} Find a 97.7\\% confidence interval for the difference $p_1-p_2$ of the population proportions. Confidence interval=[ANS]", "answer_v2": [ "(-0.0371761092151127,0.281176109215112)" ], "answer_type_v2": [ "INT" ], "options_v2": [ [] ], "problem_v3": "Two random samples are taken, one from among UVA students and the other from among UNC students. Both groups are asked if academics are their top priority. A summary of the sample sizes and proportions of each group answering ``yes'' are given below:\n\\begin{array}{lll} \\mbox{UVA (Pop. 1)}: & n_1=86, & \\hat{p}_1=0.742 \\\\ \\mbox{UNC (Pop. 2)}: & n_2=92, & \\hat{p}_2=0.632 \\\\ \\end{array} Find a 93.2\\% confidence interval for the difference $p_1-p_2$ of the population proportions. Confidence interval=[ANS]", "answer_v3": [ "(-0.0158330829769159,0.235833082976916)" ], "answer_type_v3": [ "INT" ], "options_v3": [ [] ] }, { "id": "Statistics_0199", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "Paired samples", "level": "4", "keywords": [ "statistics", "Inference about a population" ], "problem_v1": "To test the efficacy of a new cholesterol-lowering medication, 10 people are selected at random. Each has their LDL levels measured (shown below as Before), then take the medicine for 10 weeks, and then has their LDL levels measured again (After).\n\\begin{array}{ccc} \\mbox{Subject} & \\mbox{Before} & \\mbox{After} \\\\ 1 & 178 & 174 \\\\ 2 & 162 & 137 \\\\ 3 & 166 & 154 \\\\ 4 & 175 & 158 \\\\ 5 & 137 & 119 \\\\ 6 & 140 & 124 \\\\ 7 & 161 & 149 \\\\ 8 & 162 & 129 \\\\ 9 & 143 & 111 \\\\ 10 & 154 & 136 \\\\ \\end{array}\nGive a 94.6\\% confidence interval for $\\mu_B-\\mu_A$, the difference between LDL levels before and after taking the medication. Confidence Interval=[ANS]", "answer_v1": [ "(12.356689642138,25.043310357862)" ], "answer_type_v1": [ "INT" ], "options_v1": [ [] ], "problem_v2": "To test the efficacy of a new cholesterol-lowering medication, 10 people are selected at random. Each has their LDL levels measured (shown below as Before), then take the medicine for 10 weeks, and then has their LDL levels measured again (After).\n\\begin{array}{ccc} \\mbox{Subject} & \\mbox{Before} & \\mbox{After} \\\\ 1 & 117 & 112 \\\\ 2 & 194 & 178 \\\\ 3 & 123 & 128 \\\\ 4 & 140 & 110 \\\\ 5 & 196 & 166 \\\\ 6 & 138 & 113 \\\\ 7 & 126 & 118 \\\\ 8 & 139 & 109 \\\\ 9 & 161 & 139 \\\\ 10 & 116 & 107 \\\\ \\end{array}\nGive a 91.8\\% confidence interval for $\\mu_B-\\mu_A$, the difference between LDL levels before and after taking the medication. Confidence Interval=[ANS]", "answer_v2": [ "(9.36346362158283,24.6365363784172)" ], "answer_type_v2": [ "INT" ], "options_v2": [ [] ], "problem_v3": "To test the efficacy of a new cholesterol-lowering medication, 10 people are selected at random. Each has their LDL levels measured (shown below as Before), then take the medicine for 10 weeks, and then has their LDL levels measured again (After).\n\\begin{array}{ccc} \\mbox{Subject} & \\mbox{Before} & \\mbox{After} \\\\ 1 & 138 & 114 \\\\ 2 & 165 & 139 \\\\ 3 & 135 & 97 \\\\ 4 & 159 & 151 \\\\ 5 & 128 & 142 \\\\ 6 & 141 & 145 \\\\ 7 & 183 & 177 \\\\ 8 & 192 & 157 \\\\ 9 & 189 & 165 \\\\ 10 & 128 & 123 \\\\ \\end{array}\nGive a 95.9\\% confidence interval for $\\mu_B-\\mu_A$, the difference between LDL levels before and after taking the medication. Confidence Interval=[ANS]", "answer_v3": [ "(1.88275468254257,27.7172453174574)" ], "answer_type_v3": [ "INT" ], "options_v3": [ [] ] }, { "id": "Statistics_0200", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "Independent samples - z", "level": "2", "keywords": [ "statistics", "inferences", "sample size" ], "problem_v1": "Suppose you wanted to estimate the difference between two population means correct to within $3.3$ with probability $0.96$. If prior information suggests that the popluation variances are approximately equal to $\\sigma^2_1=\\sigma^2_2=16$ and you want to select independent random samples of equal size from the poulations, how large should the sample sizes, $n_1$ and $n_2$ be? answer: $n_1=n_2=$ [ANS]", "answer_v1": [ "12.3941512253475" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose you wanted to estimate the difference between two population means correct to within $4.8$ with probability $0.9$. If prior information suggests that the popluation variances are approximately equal to $\\sigma^2_1=\\sigma^2_2=11$ and you want to select independent random samples of equal size from the poulations, how large should the sample sizes, $n_1$ and $n_2$ be? answer: $n_1=n_2=$ [ANS]", "answer_v2": [ "2.58341822876165" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose you wanted to estimate the difference between two population means correct to within $3.4$ with probability $0.92$. If prior information suggests that the popluation variances are approximately equal to $\\sigma^2_1=\\sigma^2_2=13$ and you want to select independent random samples of equal size from the poulations, how large should the sample sizes, $n_1$ and $n_2$ be? answer: $n_1=n_2=$ [ANS]", "answer_v3": [ "6.89337757126912" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0201", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "Independent samples - z", "level": "3", "keywords": [ "statistics", "Inference about a population" ], "problem_v1": "Random samples of female and male UVA undergraduates are asked to estimate the number of alcoholic drinks that each consumes on a typical weekend. The data is below: Females (Population 1): 5, 5, 3, 3, 4, 4, 3, 4, 5, 3 Males (Population 2): 6, 6, 5, 6, 6, 4, 4, 5, 6, 7 Give a 94.2\\% confidence interval for the difference between mean female and male drink consumption. (Assume that the population variances are equal.) Confidence Interval=[ANS]", "answer_v1": [ "(-2.43750405502063,-0.762495944979369)" ], "answer_type_v1": [ "INT" ], "options_v1": [ [] ], "problem_v2": "Random samples of female and male UVA undergraduates are asked to estimate the number of alcoholic drinks that each consumes on a typical weekend. The data is below: Females (Population 1): 0, 1, 4, 1, 0, 1, 2, 0, 3, 2 Males (Population 2): 9, 5, 5, 6, 7, 5, 6, 7, 5, 5 Give a 91.6\\% confidence interval for the difference between mean female and male drink consumption. (Assume that the population variances are equal.) Confidence Interval=[ANS]", "answer_v2": [ "(-5.697466,-3.502534)" ], "answer_type_v2": [ "INT" ], "options_v2": [ [] ], "problem_v3": "Random samples of female and male UVA undergraduates are asked to estimate the number of alcoholic drinks that each consumes on a typical weekend. The data is below: Females (Population 1): 1, 2, 1, 1, 4, 4, 4, 1, 1, 1 Males (Population 2): 4, 6, 8, 7, 7, 4, 5, 7, 7, 6 Give a 93.9\\% confidence interval for the difference between mean female and male drink consumption. (Assume that the population variances are equal.) Confidence Interval=[ANS]", "answer_v3": [ "(-5.34448958137222,-2.85551041862778)" ], "answer_type_v3": [ "INT" ], "options_v3": [ [] ] }, { "id": "Statistics_0202", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "Independent samples - z", "level": "4", "keywords": [ "statistics", "Inference about a population" ], "problem_v1": "Two random samples are selected from two independent populations. A summary of the samples sizes and sample means is given below:\n\\begin{array}{ll} n_1=47, & \\bar{x}_1=56.2 \\\\ n_2=46, & \\bar{x}_2=77.2 \\\\ \\end{array} If the 95.5\\% confidence interval for the difference $\\mu_1-\\mu_2$ of the means is (-24.4421,-17.5579), what is the value of the pooled variance estimator? (You may assume equal population variances.) Pooled Variance Estimator=[ANS]", "answer_v1": [ "66.6614285714286" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Two random samples are selected from two independent populations. A summary of the samples sizes and sample means is given below:\n\\begin{array}{ll} n_1=35, & \\bar{x}_1=51.5 \\\\ n_2=52, & \\bar{x}_2=73.3 \\\\ \\end{array} If the 93\\% confidence interval for the difference $\\mu_1-\\mu_2$ of the means is (-25.3446,-18.2554), what is the value of the pooled variance estimator? (You may assume equal population variances.) Pooled Variance Estimator=[ANS]", "answer_v2": [ "78.054" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Two random samples are selected from two independent populations. A summary of the samples sizes and sample means is given below:\n\\begin{array}{ll} n_1=39, & \\bar{x}_1=52.8 \\\\ n_2=46, & \\bar{x}_2=75.5 \\\\ \\end{array} If the 97\\% confidence interval for the difference $\\mu_1-\\mu_2$ of the means is (-26.7183,-18.6817), what is the value of the pooled variance estimator? (You may assume equal population variances.) Pooled Variance Estimator=[ANS]", "answer_v3": [ "69.8984337349398" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0203", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "Independent samples - z", "level": "4", "keywords": [ "statistics", "Inference about a population" ], "problem_v1": "Samples are collected from two independent populations to conduct a hypothesis test for the difference of the means $\\mu_1-\\mu_2$. If the sample size $n_2$ from population 2 is 5 times larger than the sample size $n_1$ from population 1, and the degrees of freedom for the test statistic is 112, then what are the sample sizes? $n_1$=[ANS]\n$n_2$=[ANS]", "answer_v1": [ "19", "95" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Samples are collected from two independent populations to conduct a hypothesis test for the difference of the means $\\mu_1-\\mu_2$. If the sample size $n_2$ from population 2 is 7 times larger than the sample size $n_1$ from population 1, and the degrees of freedom for the test statistic is 102, then what are the sample sizes? $n_1$=[ANS]\n$n_2$=[ANS]", "answer_v2": [ "13", "91" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Samples are collected from two independent populations to conduct a hypothesis test for the difference of the means $\\mu_1-\\mu_2$. If the sample size $n_2$ from population 2 is 6 times larger than the sample size $n_1$ from population 1, and the degrees of freedom for the test statistic is 103, then what are the sample sizes? $n_1$=[ANS]\n$n_2$=[ANS]", "answer_v3": [ "15", "90" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0204", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "Independent samples - t", "level": "4", "keywords": [ "statistics", "variance", "ANOVA", "2-sample t" ], "problem_v1": "Some car tires can develop what is known as \"heel and toe\" wear if not rotated after a certain mileage. To assess this issue, a consumer group investigated the tire wear on two brands of tire, A and B, say. Fifteen cars were fitted with new brand A tires and thirteen with brand B tires, the cars assigned to brand at random. (Two cars initially assigned to brand B suffered serious tire faults other than heel and toe wear, and were excluded from the study.) The cars were driven in regular driving conditions, and the mileage at which heal and toe wear could be observed was recorded on each car. For the cars with brand A tires, the mean mileage observed was $25.84$ (in $10^3$ miles) and the variance was $3.76$ (in $10^6$ miles $^2$). For the cars with brand B, the corresponding statistics were $24.64$ (in $10^3$ miles) and $8.80$ (in $10^6$ miles $^2$) respectively. The mileage before heal and toe wear is detectable is assumed to be Normally distributed for both brands. Part a) Calculate the pooled variance $s^2$ to 3 decimal places. During intermediate steps to arrive at the answer, make sure you keep as many decimal places as possible so that you can achieve the precision required in this question. [ANS] $\\times 10^6$ miles $^2$ Part b) Determine a 95\\% confidence interval for $\\mu_A-\\mu_B$, the difference in the mean $10^3$ mileages before heal and toe wear for the two brands of tire. Leave your answer to 2 decimal places. ([ANS]) Part c) Based on the 95\\% confidence interval constructed in the previous part, which of the following conclusions can be drawn when we test $H_0: \\mu_A=\\mu_B$ vs. $H_a: \\mu_A \\ne \\mu_B$ with $\\alpha=0.05$. [ANS] A. Do not reject $H_0$ since 0 is within the interval found in part (b). B. Reject $H_0$ since 0 is in the interval found in part (b). C. Reject $H_0$ since 0 is not within the interval found in part (b). D. Do not reject $H_0$ since 0 is not in the interval found in part (b). E. Do not reject $H_0$ since $1.20$ is within the interval found in part (b).", "answer_v1": [ "6.08615", "(-0.722013, 3.12201)", "A" ], "answer_type_v1": [ "NV", "INT", "MCS" ], "options_v1": [ [], [], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Some car tires can develop what is known as \"heel and toe\" wear if not rotated after a certain mileage. To assess this issue, a consumer group investigated the tire wear on two brands of tire, A and B, say. Fifteen cars were fitted with new brand A tires and thirteen with brand B tires, the cars assigned to brand at random. (Two cars initially assigned to brand B suffered serious tire faults other than heel and toe wear, and were excluded from the study.) The cars were driven in regular driving conditions, and the mileage at which heal and toe wear could be observed was recorded on each car. For the cars with brand A tires, the mean mileage observed was $24.99$ (in $10^3$ miles) and the variance was $7.75$ (in $10^6$ miles $^2$). For the cars with brand B, the corresponding statistics were $32.92$ (in $10^3$ miles) and $6.47$ (in $10^6$ miles $^2$) respectively. The mileage before heal and toe wear is detectable is assumed to be Normally distributed for both brands. Part a) Calculate the pooled variance $s^2$ to 3 decimal places. During intermediate steps to arrive at the answer, make sure you keep as many decimal places as possible so that you can achieve the precision required in this question. [ANS] $\\times 10^6$ miles $^2$ Part b) Determine a 95\\% confidence interval for $\\mu_A-\\mu_B$, the difference in the mean $10^3$ mileages before heal and toe wear for the two brands of tire. Leave your answer to 2 decimal places. ([ANS]) Part c) Based on the 95\\% confidence interval constructed in the previous part, which of the following conclusions can be drawn when we test $H_0: \\mu_A=\\mu_B$ vs. $H_a: \\mu_A \\ne \\mu_B$ with $\\alpha=0.05$. [ANS] A. Reject $H_0$ since 0 is in the interval found in part (b). B. Do not reject $H_0$ since 0 is within the interval found in part (b). C. Do not reject $H_0$ since 0 is not in the interval found in part (b). D. Reject $H_0$ since 0 is not within the interval found in part (b). E. Do not reject $H_0$ since $-7.93$ is within the interval found in part (b).", "answer_v2": [ "7.15923", "(-10.0146, -5.84542)", "D" ], "answer_type_v2": [ "NV", "INT", "MCS" ], "options_v2": [ [], [], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Some car tires can develop what is known as \"heel and toe\" wear if not rotated after a certain mileage. To assess this issue, a consumer group investigated the tire wear on two brands of tire, A and B, say. Fifteen cars were fitted with new brand A tires and thirteen with brand B tires, the cars assigned to brand at random. (Two cars initially assigned to brand B suffered serious tire faults other than heel and toe wear, and were excluded from the study.) The cars were driven in regular driving conditions, and the mileage at which heal and toe wear could be observed was recorded on each car. For the cars with brand A tires, the mean mileage observed was $23.69$ (in $10^3$ miles) and the variance was $6.40$ (in $10^6$ miles $^2$). For the cars with brand B, the corresponding statistics were $23.92$ (in $10^3$ miles) and $5.53$ (in $10^6$ miles $^2$) respectively. The mileage before heal and toe wear is detectable is assumed to be Normally distributed for both brands. Part a) Calculate the pooled variance $s^2$ to 3 decimal places. During intermediate steps to arrive at the answer, make sure you keep as many decimal places as possible so that you can achieve the precision required in this question. [ANS] $\\times 10^6$ miles $^2$ Part b) Determine a 95\\% confidence interval for $\\mu_A-\\mu_B$, the difference in the mean $10^3$ mileages before heal and toe wear for the two brands of tire. Leave your answer to 2 decimal places. ([ANS]) Part c) Based on the 95\\% confidence interval constructed in the previous part, which of the following conclusions can be drawn when we test $H_0: \\mu_A=\\mu_B$ vs. $H_a: \\mu_A \\ne \\mu_B$ with $\\alpha=0.05$. [ANS] A. Do not reject $H_0$ since $-0.23$ is within the interval found in part (b). B. Reject $H_0$ since 0 is in the interval found in part (b). C. Do not reject $H_0$ since 0 is not in the interval found in part (b). D. Reject $H_0$ since 0 is not within the interval found in part (b). E. Do not reject $H_0$ since 0 is within the interval found in part (b).", "answer_v3": [ "5.99846", "(-2.13812, 1.67812)", "E" ], "answer_type_v3": [ "NV", "INT", "MCS" ], "options_v3": [ [], [], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Statistics_0205", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "Variance", "level": "2", "keywords": [ "Sample Size", "Confidence", "Variance", "statistics", "estimates", "population", "samples" ], "problem_v1": "Find the minimum sample size needed to be $99$ \\% confident that the sample variance is within $30$ \\% of the population variance. [ANS]", "answer_v1": [ "171" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the minimum sample size needed to be $95$ \\% confident that the sample variance is within $50$ \\% of the population variance. [ANS]", "answer_v2": [ "37" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the minimum sample size needed to be $95$ \\% confident that the sample variance is within $30$ \\% of the population variance. [ANS]", "answer_v3": [ "97" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0206", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "Variance", "level": "2", "keywords": [ "Confidence Interval", "Chi Squared", "statistics", "estimates", "population", "samples" ], "problem_v1": "Find the critical values $\\chi_L^2=\\chi_{1-\\alpha/2}^2$ and $\\chi_R^2=\\chi_{\\alpha/2}^2$ that correspond to $98$ \\% degree of confidence and the sample size $n=20.$ $\\chi_L^2=$ [ANS] $\\ \\ \\ \\ $ $\\chi_R^2=$ [ANS]", "answer_v1": [ "7.63273", "36.1909" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the critical values $\\chi_L^2=\\chi_{1-\\alpha/2}^2$ and $\\chi_R^2=\\chi_{\\alpha/2}^2$ that correspond to $80$ \\% degree of confidence and the sample size $n=29.$ $\\chi_L^2=$ [ANS] $\\ \\ \\ \\ $ $\\chi_R^2=$ [ANS]", "answer_v2": [ "18.9392", "37.9159" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the critical values $\\chi_L^2=\\chi_{1-\\alpha/2}^2$ and $\\chi_R^2=\\chi_{\\alpha/2}^2$ that correspond to $90$ \\% degree of confidence and the sample size $n=20.$ $\\chi_L^2=$ [ANS] $\\ \\ \\ \\ $ $\\chi_R^2=$ [ANS]", "answer_v3": [ "10.1170", "30.1435" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0207", "subject": "Statistics", "topic": "Confidence intervals", "subtopic": "Variance", "level": "3", "keywords": [ "statistics", "Inference about a population" ], "problem_v1": "The weights of a random sample of cereal boxes that are supposed to weigh 1 pound are given below. Estimate the standard deviation of the entire population with 93.5\\% confidence. Assume the population is normally distributed. $ \\qquad \\begin{array}{cccc} 1.04 & 1.01 & 1.02 & 1.03 \\\\ 0.98 & 0.98 & 1.01 & 1.01 \\\\ \\end{array}$ LCL=[ANS]\nUCL=[ANS]", "answer_v1": [ "0.0144685", "0.0415504" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The weights of a random sample of cereal boxes that are supposed to weigh 1 pound are given below. Estimate the standard deviation of the entire population with 95.4\\% confidence. Assume the population is normally distributed. $ \\qquad \\begin{array}{cccc} 0.95 & 1.06 & 0.96 & 0.99 \\\\ 1.06 & 0.98 & 0.97 & 0.98 \\\\ \\end{array}$ LCL=[ANS]\nUCL=[ANS]", "answer_v2": [ "0.0280596", "0.0882598" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The weights of a random sample of cereal boxes that are supposed to weigh 1 pound are given below. Estimate the standard deviation of the entire population with 98.3\\% confidence. Assume the population is normally distributed. $ \\qquad \\begin{array}{cccc} 0.98 & 1.02 & 0.98 & 1.01 \\\\ 0.97 & 0.99 & 1.04 & 1.05 \\\\ \\end{array}$ LCL=[ANS]\nUCL=[ANS]", "answer_v3": [ "0.018111", "0.0726525" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0208", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Concepts", "level": "1", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "Find the critical $z$ value using a significance level of $\\alpha=0.08$ if the alternative hypothesis $H_0$ is $\\mu < 59$. [ANS]", "answer_v1": [ "-1.40507" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the critical $z$ value using a significance level of $\\alpha=0.01$ if the null hypothesis $H_0$ is $\\mu \\ge 94$. [ANS]", "answer_v2": [ "-2.32635" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the critical $z$ value using a significance level of $\\alpha=0.04$ if the null hypothesis $H_0$ is $\\mu \\le 61$. [ANS]", "answer_v3": [ "1.75069" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0209", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Concepts", "level": "1", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "For each statement, express the null hypothesis $H_0$ and alternative hypothesis $H_1$ in symbolic form. 1. $\\ $ At least one-half of all Internet users make on-line purchases. [ANS] A. $H_0: p \\ge 0.5,$ $H_1: p < 0.5$ B. $H_0: \\mu \\le 0.5,$ $H_1: \\mu > 0.5$ C. $H_0: \\mu \\ge 0.5,$ $H_1: \\mu < 0.5$ D. $H_0: p \\le 0.5,$ $H_1: p > 0.5$\n2. $\\ $ IQ scores of statistics students have a standard deviation at most 15. [ANS] A. $H_0: \\sigma \\ge 15,$ $H_1: \\sigma < 15$ B. $H_0: \\mu \\le 15,$ $H_1: \\mu > 15$ C. $H_0: \\mu < 15,$ $H_1: \\mu \\ge 15$ D. $H_0: \\sigma \\le 15,$ $H_1: \\sigma > 15$\n3. $\\ $ The mean salary of statistics professors is less than 70,000 dollars. [ANS] A. $H_0: \\mu \\ge 70,000,$ $H_1: \\mu < 70,000$ B. $H_0: \\mu > 70,000,$ $H_1: \\mu \\le 70,000$ C. $H_0: \\mu < 70,000,$ $H_1: \\mu \\ge 70,000$ D. $H_0: \\mu \\le 70,000,$ $H_1: \\mu > 70,000$", "answer_v1": [ "A", "D", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "For each statement, express the null hypothesis $H_0$ and alternative hypothesis $H_1$ in symbolic form. 1. $\\ $ The mean salary of statistics professors is at least 70,000 dollars. [ANS] A. $H_0: \\mu < 70,000,$ $H_1: \\mu \\ge 70,000$ B. $H_0: \\mu > 70,000,$ $H_1: \\mu \\le 70,000$ C. $H_0: \\mu \\le 70,000,$ $H_1: \\mu > 70,000$ D. $H_0: \\mu \\ge 70,000,$ $H_1: \\mu < 70,000$\n2. $\\ $ Fewer than one-half of all Internet users make on-line purchases. [ANS] A. $H_0: \\mu \\ge 0.5,$ $H_1: \\mu < 0.5$ B. $H_0: \\mu \\le 0.5,$ $H_1: \\mu > 0.5$ C. $H_0: p \\le 0.5,$ $H_1: p > 0.5$ D. $H_0: p \\ge 0.5,$ $H_1: p < 0.5$\n3. $\\ $ IQ scores of statistics students have a standard deviation more than 15. [ANS] A. $H_0: \\sigma \\ge 15,$ $H_1: \\sigma < 15$ B. $H_0: \\sigma \\le 15,$ $H_1: \\sigma > 15$ C. $H_0: \\mu < 15,$ $H_1: \\mu \\ge 15$ D. $H_0: \\mu \\le 15,$ $H_1: \\mu > 15$", "answer_v2": [ "D", "D", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "For each statement, express the null hypothesis $H_0$ and alternative hypothesis $H_1$ in symbolic form. 1. $\\ $ The mean salary of statistics professors is at least 70,000 dollars. [ANS] A. $H_0: \\mu < 70,000,$ $H_1: \\mu \\ge 70,000$ B. $H_0: \\mu \\ge 70,000,$ $H_1: \\mu < 70,000$ C. $H_0: \\mu > 70,000,$ $H_1: \\mu \\le 70,000$ D. $H_0: \\mu \\le 70,000,$ $H_1: \\mu > 70,000$\n2. $\\ $ IQ scores of statistics students have a standard deviation at most 15. [ANS] A. $H_0: \\mu \\le 15,$ $H_1: \\mu > 15$ B. $H_0: \\sigma \\le 15,$ $H_1: \\sigma > 15$ C. $H_0: \\mu < 15,$ $H_1: \\mu \\ge 15$ D. $H_0: \\sigma \\ge 15,$ $H_1: \\sigma < 15$\n3. $\\ $ More than one-half of all Internet users make on-line purchases. [ANS] A. $H_0: p \\le 0.5,$ $H_1: p > 0.5$ B. $H_0: p \\ge 0.5,$ $H_1: p < 0.5$ C. $H_0: \\mu \\ge 0.5,$ $H_1: \\mu < 0.5$ D. $H_0: \\mu \\le 0.5,$ $H_1: \\mu > 0.5$", "answer_v3": [ "B", "B", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0210", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Concepts", "level": "1", "keywords": [], "problem_v1": "Given the significance level $\\alpha=0.08$ find the following:\n(a) $\\ $ lower-tailed $z$ value $z=$ [ANS]\n(b) $\\ $ right-tailed $z$ value $z=$ [ANS]\n(c) $\\ $ two-tailed $z$ value $|z|=$ [ANS]", "answer_v1": [ "-1.40507", "1.40507", "1.75069" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Given the significance level $\\alpha=0.01$ find the following:\n(a) $\\ $ lower-tailed $z$ value $z=$ [ANS]\n(b) $\\ $ right-tailed $z$ value $z=$ [ANS]\n(c) $\\ $ two-tailed $z$ value $|z|=$ [ANS]", "answer_v2": [ "-2.32635", "2.32635", "2.57583" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Given the significance level $\\alpha=0.035$ find the following:\n(a) $\\ $ lower-tailed $z$ value $z=$ [ANS]\n(b) $\\ $ right-tailed $z$ value $z=$ [ANS]\n(c) $\\ $ two-tailed $z$ value $|z|=$ [ANS]", "answer_v3": [ "-1.81191", "1.81191", "2.10836" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Statistics_0211", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Concepts", "level": "1", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "Find the critical $z$ value for a left-tailed test using a significance level of $\\alpha=0.06.$ [ANS]", "answer_v1": [ "-1.55477" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the critical $z$ value for a right-tailed test using a significance level of $\\alpha=0.1.$ [ANS]", "answer_v2": [ "1.28155" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the critical $z$ value for a right-tailed test using a significance level of $\\alpha=0.07.$ [ANS]", "answer_v3": [ "1.47579" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0212", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Concepts", "level": "2", "keywords": [ "statistics", "inference" ], "problem_v1": "For each problem, select the best response.\n(a) Does 30 minutes of aerobic exercise each day provide significant improvement in mental performance? To investigate this issue, a researcher conducted a study with 150 adult subjects who performed aerobic exercise each day for a period of six months. At the end of the study, 200 variables related to the mental performance of the subjects were measured on each subject and the means compared to known means for these variables in the population of all adults. Nine of these variables were significantly better (in the sense of statistical significance) at the $\\alpha=0.05$ level for the group that performed 30 minutes of aerobic exercise each day as compared to the population as a whole, and one variable was significantly better at the $\\alpha=0.01$ level for the group that performed 30 minutes of aerobic exercise each day as compared to the population as a whole. It would be correct to conclude [ANS] A. that these results would have provided very good statistical evidence that 30 minutes of aerobic exercise each day provides some improvement in mental performance if the number of subjects had been larger. It is premature to draw statistical conclusions from studies in which the number of subjects is less than the number of variables measured. B. that there is very good statistical evidence that 30 minutes of aerobic exercise each day provides improvement for the variable that was significant at the $\\alpha=0.01$ level. We should be somewhat cautious about making claims for the variables that were significant at the $\\alpha=0.05$ level. C. that there is very good statistical evidence that 30 minutes of aerobic exercise each day provides some improvement in mental performance. D. none of the above.\n(b) The nicotine content in cigarettes of a certain brand is normally distributed with mean (in milligrams) $\\mu$ and standard deviation $\\sigma=0.1$. The brand advertises that the mean nicotine content of their cigarettes is 1.5, but you believe that the mean nicotine content is actually higher than advertised. To explore this, you test the hypotheses $H_0: \\mu=1.5$, $H_a: \\mu > 1.5$ and you obtain a P-value of 0.052. Which of the following is true? [ANS] A. At the $\\alpha=0.05$ significance level, you have proven that $H_0$ is true. B. This should be viewed as a pilot study and the data suggests that further investigation of the hypotheses will not be fruitful at the $\\alpha=0.05$ significance level. C. You have failed to obtain any evidence for $H_a.$ D. There is some evidence against $H_0$, and a study using a larger sample size may be worthwhile.\n(c) An engineer designs an improved light bulb. The previous design had an average lifetime of 1200 hours. The new bulb had a lifetime of 1200.2 hours, using a sample of 40,000 bulbs. Although the difference is quite small, the effect was statistically significant. The explanation is [ANS] A. that the sample size is very large. B. that new designs typically have more variability than standard designs. C. that the mean of 1200 is large. D. all of the above.", "answer_v1": [ "D", "D", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "For each problem, select the best response.\n(a) An engineer designs an improved light bulb. The previous design had an average lifetime of 1200 hours. The new bulb had a lifetime of 1200.2 hours, using a sample of 40,000 bulbs. Although the difference is quite small, the effect was statistically significant. The explanation is [ANS] A. that the mean of 1200 is large. B. that new designs typically have more variability than standard designs. C. that the sample size is very large. D. all of the above.\n(b) Does 30 minutes of aerobic exercise each day provide significant improvement in mental performance? To investigate this issue, a researcher conducted a study with 150 adult subjects who performed aerobic exercise each day for a period of six months. At the end of the study, 200 variables related to the mental performance of the subjects were measured on each subject and the means compared to known means for these variables in the population of all adults. Nine of these variables were significantly better (in the sense of statistical significance) at the $\\alpha=0.05$ level for the group that performed 30 minutes of aerobic exercise each day as compared to the population as a whole, and one variable was significantly better at the $\\alpha=0.01$ level for the group that performed 30 minutes of aerobic exercise each day as compared to the population as a whole. It would be correct to conclude [ANS] A. that these results would have provided very good statistical evidence that 30 minutes of aerobic exercise each day provides some improvement in mental performance if the number of subjects had been larger. It is premature to draw statistical conclusions from studies in which the number of subjects is less than the number of variables measured. B. that there is very good statistical evidence that 30 minutes of aerobic exercise each day provides some improvement in mental performance. C. that there is very good statistical evidence that 30 minutes of aerobic exercise each day provides improvement for the variable that was significant at the $\\alpha=0.01$ level. We should be somewhat cautious about making claims for the variables that were significant at the $\\alpha=0.05$ level. D. none of the above.\n(c) The nicotine content in cigarettes of a certain brand is normally distributed with mean (in milligrams) $\\mu$ and standard deviation $\\sigma=0.1$. The brand advertises that the mean nicotine content of their cigarettes is 1.5, but you believe that the mean nicotine content is actually higher than advertised. To explore this, you test the hypotheses $H_0: \\mu=1.5$, $H_a: \\mu > 1.5$ and you obtain a P-value of 0.052. Which of the following is true? [ANS] A. There is some evidence against $H_0$, and a study using a larger sample size may be worthwhile. B. You have failed to obtain any evidence for $H_a.$ C. At the $\\alpha=0.05$ significance level, you have proven that $H_0$ is true. D. This should be viewed as a pilot study and the data suggests that further investigation of the hypotheses will not be fruitful at the $\\alpha=0.05$ significance level.", "answer_v2": [ "C", "D", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "For each problem, select the best response.\n(a) An engineer designs an improved light bulb. The previous design had an average lifetime of 1200 hours. The new bulb had a lifetime of 1200.2 hours, using a sample of 40,000 bulbs. Although the difference is quite small, the effect was statistically significant. The explanation is [ANS] A. that the sample size is very large. B. that new designs typically have more variability than standard designs. C. that the mean of 1200 is large. D. all of the above.\n(b) Does 30 minutes of aerobic exercise each day provide significant improvement in mental performance? To investigate this issue, a researcher conducted a study with 150 adult subjects who performed aerobic exercise each day for a period of six months. At the end of the study, 200 variables related to the mental performance of the subjects were measured on each subject and the means compared to known means for these variables in the population of all adults. Nine of these variables were significantly better (in the sense of statistical significance) at the $\\alpha=0.05$ level for the group that performed 30 minutes of aerobic exercise each day as compared to the population as a whole, and one variable was significantly better at the $\\alpha=0.01$ level for the group that performed 30 minutes of aerobic exercise each day as compared to the population as a whole. It would be correct to conclude [ANS] A. that there is very good statistical evidence that 30 minutes of aerobic exercise each day provides improvement for the variable that was significant at the $\\alpha=0.01$ level. We should be somewhat cautious about making claims for the variables that were significant at the $\\alpha=0.05$ level. B. that there is very good statistical evidence that 30 minutes of aerobic exercise each day provides some improvement in mental performance. C. that these results would have provided very good statistical evidence that 30 minutes of aerobic exercise each day provides some improvement in mental performance if the number of subjects had been larger. It is premature to draw statistical conclusions from studies in which the number of subjects is less than the number of variables measured. D. none of the above.\n(c) The nicotine content in cigarettes of a certain brand is normally distributed with mean (in milligrams) $\\mu$ and standard deviation $\\sigma=0.1$. The brand advertises that the mean nicotine content of their cigarettes is 1.5, but you believe that the mean nicotine content is actually higher than advertised. To explore this, you test the hypotheses $H_0: \\mu=1.5$, $H_a: \\mu > 1.5$ and you obtain a P-value of 0.052. Which of the following is true? [ANS] A. There is some evidence against $H_0$, and a study using a larger sample size may be worthwhile. B. You have failed to obtain any evidence for $H_a.$ C. This should be viewed as a pilot study and the data suggests that further investigation of the hypotheses will not be fruitful at the $\\alpha=0.05$ significance level. D. At the $\\alpha=0.05$ significance level, you have proven that $H_0$ is true.", "answer_v3": [ "A", "D", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0213", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Concepts", "level": "2", "keywords": [ "statistics", "inference" ], "problem_v1": "For each problem, select the best response.\n(a) How much do you plan to spend for gifts this holiday season? How much do you plan to spend for gifts this holiday season? An interviewer asks this question of 250 customers at a large shopping mall. The distribution of individual responses is skewed, but the sample mean and standard deviation of the responses are $\\bar x$=437 dollars and s=65 dollars. Which of the following are true? [ANS] A. The Central Limit Theorem informs us that we can act as if $\\bar x$ is approximately Normally distributed. B. If we calculate a confidence interval, it cannot be trusted since the sample responses may be badly biased. C. The margin of error for a 95\\% confidence interval will be less than the margin of error for a 99\\% confidence interval. D. All of the above.\n(b) A Gallup Poll asked the question How would you rate the overall quality of the environment in this country today-as excellent, good, only fair, or poor? How would you rate the overall quality of the environment in this country today-as excellent, good, only fair, or poor? In all, 46\\% of the sample rated the environment as good or excellent. Gallup announced the poll's margin of error for 95\\% confidence as $\\pm 3$ percentage points. Which of the following sources of error are included in the margin of error? [ANS] A. There is chance variation in the random selection of telephone numbers. B. Nonresponse-some people whose numbers were chosen never answered the phone in several calls or answered but refused to participate in the poll. C. The poll dialed telephone numbers at random and so missed all people without phones. D. All of the above.\n(c) What is significance good for? Which of the following questions does a test of significance answer? [ANS] A. Is the observed effect due to chance? B. Is the sample or experiment properly designed? C. Is the observed effect important? D. All of the above.", "answer_v1": [ "D", "A", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "For each problem, select the best response.\n(a) What is significance good for? Which of the following questions does a test of significance answer? [ANS] A. Is the observed effect important? B. Is the sample or experiment properly designed? C. Is the observed effect due to chance? D. All of the above.\n(b) How much do you plan to spend for gifts this holiday season? How much do you plan to spend for gifts this holiday season? An interviewer asks this question of 250 customers at a large shopping mall. The distribution of individual responses is skewed, but the sample mean and standard deviation of the responses are $\\bar x$=437 dollars and s=65 dollars. Which of the following are true? [ANS] A. The Central Limit Theorem informs us that we can act as if $\\bar x$ is approximately Normally distributed. B. The margin of error for a 95\\% confidence interval will be less than the margin of error for a 99\\% confidence interval. C. If we calculate a confidence interval, it cannot be trusted since the sample responses may be badly biased. D. All of the above.\n(c) A Gallup Poll asked the question How would you rate the overall quality of the environment in this country today-as excellent, good, only fair, or poor? How would you rate the overall quality of the environment in this country today-as excellent, good, only fair, or poor? In all, 46\\% of the sample rated the environment as good or excellent. Gallup announced the poll's margin of error for 95\\% confidence as $\\pm 3$ percentage points. Which of the following sources of error are included in the margin of error? [ANS] A. The poll dialed telephone numbers at random and so missed all people without phones. B. Nonresponse-some people whose numbers were chosen never answered the phone in several calls or answered but refused to participate in the poll. C. There is chance variation in the random selection of telephone numbers. D. All of the above.", "answer_v2": [ "C", "D", "C" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "For each problem, select the best response.\n(a) What is significance good for? Which of the following questions does a test of significance answer? [ANS] A. Is the observed effect due to chance? B. Is the sample or experiment properly designed? C. Is the observed effect important? D. All of the above.\n(b) How much do you plan to spend for gifts this holiday season? How much do you plan to spend for gifts this holiday season? An interviewer asks this question of 250 customers at a large shopping mall. The distribution of individual responses is skewed, but the sample mean and standard deviation of the responses are $\\bar x$=437 dollars and s=65 dollars. Which of the following are true? [ANS] A. If we calculate a confidence interval, it cannot be trusted since the sample responses may be badly biased. B. The margin of error for a 95\\% confidence interval will be less than the margin of error for a 99\\% confidence interval. C. The Central Limit Theorem informs us that we can act as if $\\bar x$ is approximately Normally distributed. D. All of the above.\n(c) A Gallup Poll asked the question How would you rate the overall quality of the environment in this country today-as excellent, good, only fair, or poor? How would you rate the overall quality of the environment in this country today-as excellent, good, only fair, or poor? In all, 46\\% of the sample rated the environment as good or excellent. Gallup announced the poll's margin of error for 95\\% confidence as $\\pm 3$ percentage points. Which of the following sources of error are included in the margin of error? [ANS] A. The poll dialed telephone numbers at random and so missed all people without phones. B. There is chance variation in the random selection of telephone numbers. C. Nonresponse-some people whose numbers were chosen never answered the phone in several calls or answered but refused to participate in the poll. D. All of the above.", "answer_v3": [ "A", "D", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0214", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Concepts", "level": "2", "keywords": [ "statistics", "inference", "confidence interval" ], "problem_v1": "For each problem, select the best response.\n(a) In testing hypotheses, which of the following would be strong evidence against the null hypothesis? [ANS] A. Obtaining data with a large P-value. B. Using a small level of significance. C. Using a large level of significance. D. Obtaining data with a small P-value.\n(b) The P-value of a test of a null hypothesis is [ANS] A. the probability the null hypothesis is false. B. the probability, assuming the null hypothesis is false, that the test statistic will take a value at least as extreme as that actually observed. C. the probability the null hypothesis is true. D. the probability, assuming the null hypothesis is true, that the test statistic will take a value at least as extreme as that actually observed.\n(c) In formulating hypotheses for a statistical test of significance, the null hypothesis is often [ANS] A. a statement of ''no effect'' or ''no difference''. B. 0.05 C. a statement that the data are all 0. D. the probability of observing the data you actually obtained", "answer_v1": [ "D", "D", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "For each problem, select the best response.\n(a) In formulating hypotheses for a statistical test of significance, the null hypothesis is often [ANS] A. a statement that the data are all 0. B. 0.05 C. the probability of observing the data you actually obtained D. a statement of ''no effect'' or ''no difference''.\n(b) In testing hypotheses, which of the following would be strong evidence against the null hypothesis? [ANS] A. Obtaining data with a large P-value. B. Obtaining data with a small P-value. C. Using a large level of significance. D. Using a small level of significance.\n(c) The P-value of a test of a null hypothesis is [ANS] A. the probability, assuming the null hypothesis is true, that the test statistic will take a value at least as extreme as that actually observed. B. the probability the null hypothesis is true. C. the probability the null hypothesis is false. D. the probability, assuming the null hypothesis is false, that the test statistic will take a value at least as extreme as that actually observed.", "answer_v2": [ "D", "B", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "For each problem, select the best response.\n(a) In formulating hypotheses for a statistical test of significance, the null hypothesis is often [ANS] A. a statement of ''no effect'' or ''no difference''. B. the probability of observing the data you actually obtained C. a statement that the data are all 0. D. 0.05\n(b) In testing hypotheses, which of the following would be strong evidence against the null hypothesis? [ANS] A. Using a small level of significance. B. Obtaining data with a small P-value. C. Using a large level of significance. D. Obtaining data with a large P-value.\n(c) The P-value of a test of a null hypothesis is [ANS] A. the probability, assuming the null hypothesis is true, that the test statistic will take a value at least as extreme as that actually observed. B. the probability the null hypothesis is true. C. the probability, assuming the null hypothesis is false, that the test statistic will take a value at least as extreme as that actually observed. D. the probability the null hypothesis is false.", "answer_v3": [ "A", "B", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0215", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Concepts", "level": "2", "keywords": [ "statistics", "inference", "confidence interval" ], "problem_v1": "For each statement, select the correct null hypothesis, $H_0$, and alternative hypothesis, $H_a$, in symbolic form.\n(a) According to the Merck Veterinary Manual, the average resting heart rate for a certain type of sheep dog is 115 beats per minute (bpm). A Montana farmer notices his aging sheep dog has been acting more lethargic than usual and wonders if her heart rate is slowing. He measures her heart rate on 15 occasions and finds a sample mean heart rate of 118.2 bpm. [ANS] A. $H_0: \\mu=115$, $\\ \\ H_a: \\mu \\neq 115$ B. $H_0: {\\bar x}=118.2$, $\\ \\ H_a: {\\bar x} \\neq 118.2$ C. $H_0: \\mu=115$, $\\ \\ H_a: \\mu > 115$ D. $H_0: {\\bar x}=115$, $\\ \\ H_a: {\\bar x} > 115$ E. $H_0: \\mu=118.2$, $\\ \\ H_a: \\mu < 118.2$ F. $H_0: \\mu=115$, $\\ \\ H_a: \\mu < 115$\n(b) The mean height of all adult American males is 69 inches (5 ft 9 in). A researcher wonders if young American males between the ages of 18 and 21 tend to be taller than 69 inches. A random sample of 100 young American males ages 18 to 21 yielded a sample mean of 71 inches. [ANS] A. $H_0: \\mu > 69$, $\\ \\ H_a: \\mu < 69$ B. $H_0: {\\bar x}=69$, $\\ \\ H_a: {\\bar x} > 69$ C. $H_0: \\mu=71$, $\\ \\ H_a: \\mu < 71$ D. $H_0: {\\bar x}=71$, $\\ \\ H_a: {\\bar x} < 71$ E. $H_0: \\mu=69$, $\\ \\ H_a: \\mu \\neq 69$ F. $H_0: \\mu=69$, $\\ \\ H_a: \\mu > 69$\n(c) A certain type of hummingbird is known to have an average weight of 4.55 grams. A researcher wonders if hummingbirds (of this same type) living in the Grand Canyon differ in weight from the population as a whole. The researcher finds a sample of 30 such hummingbirds from the Grand Canyon and calculates their average weight to be 3.75 grams. [ANS] A. $H_0: {\\bar x}=4.55$, $\\ \\ H_a: {\\bar x} < 4.55$ B. $H_0: \\mu=4.55$, $\\ \\ H_a: \\mu < 4.55$ C. $H_0: \\mu=4.55$, $\\ \\ H_a: \\mu \\neq 4.55$ D. $H_0: \\mu < 4.55$, $\\ \\ H_a: \\mu=4.55$ E. $H_0: \\mu=3.75$, $\\ \\ H_a: \\mu \\neq 3.75$ F. $H_0: {\\bar x}=3.75$, $\\ \\ H_a: {\\bar x} > 3.75$", "answer_v1": [ "F", "F", "C" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "For each statement, select the correct null hypothesis, $H_0$, and alternative hypothesis, $H_a$, in symbolic form.\n(a) A certain type of hummingbird is known to have an average weight of 4.55 grams. A researcher wonders if hummingbirds (of this same type) living in the Grand Canyon differ in weight from the population as a whole. The researcher finds a sample of 30 such hummingbirds from the Grand Canyon and calculates their average weight to be 3.75 grams. [ANS] A. $H_0: {\\bar x}=3.75$, $\\ \\ H_a: {\\bar x} > 3.75$ B. $H_0: \\mu=4.55$, $\\ \\ H_a: \\mu < 4.55$ C. $H_0: \\mu=4.55$, $\\ \\ H_a: \\mu \\neq 4.55$ D. $H_0: \\mu=3.75$, $\\ \\ H_a: \\mu \\neq 3.75$ E. $H_0: {\\bar x}=4.55$, $\\ \\ H_a: {\\bar x} < 4.55$ F. $H_0: \\mu < 4.55$, $\\ \\ H_a: \\mu=4.55$\n(b) According to the Merck Veterinary Manual, the average resting heart rate for a certain type of sheep dog is 115 beats per minute (bpm). A Montana farmer notices his aging sheep dog has been acting more lethargic than usual and wonders if her heart rate is slowing. He measures her heart rate on 15 occasions and finds a sample mean heart rate of 118.2 bpm. [ANS] A. $H_0: \\mu=115$, $\\ \\ H_a: \\mu < 115$ B. $H_0: {\\bar x}=118.2$, $\\ \\ H_a: {\\bar x} \\neq 118.2$ C. $H_0: \\mu=115$, $\\ \\ H_a: \\mu > 115$ D. $H_0: {\\bar x}=115$, $\\ \\ H_a: {\\bar x} > 115$ E. $H_0: \\mu=115$, $\\ \\ H_a: \\mu \\neq 115$ F. $H_0: \\mu=118.2$, $\\ \\ H_a: \\mu < 118.2$\n(c) The mean height of all adult American males is 69 inches (5 ft 9 in). A researcher wonders if young American males between the ages of 18 and 21 tend to be taller than 69 inches. A random sample of 100 young American males ages 18 to 21 yielded a sample mean of 71 inches. [ANS] A. $H_0: {\\bar x}=71$, $\\ \\ H_a: {\\bar x} < 71$ B. $H_0: {\\bar x}=69$, $\\ \\ H_a: {\\bar x} > 69$ C. $H_0: \\mu=71$, $\\ \\ H_a: \\mu < 71$ D. $H_0: \\mu > 69$, $\\ \\ H_a: \\mu < 69$ E. $H_0: \\mu=69$, $\\ \\ H_a: \\mu > 69$ F. $H_0: \\mu=69$, $\\ \\ H_a: \\mu \\neq 69$", "answer_v2": [ "C", "A", "E" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "For each statement, select the correct null hypothesis, $H_0$, and alternative hypothesis, $H_a$, in symbolic form.\n(a) A certain type of hummingbird is known to have an average weight of 4.55 grams. A researcher wonders if hummingbirds (of this same type) living in the Grand Canyon differ in weight from the population as a whole. The researcher finds a sample of 30 such hummingbirds from the Grand Canyon and calculates their average weight to be 3.75 grams. [ANS] A. $H_0: {\\bar x}=4.55$, $\\ \\ H_a: {\\bar x} < 4.55$ B. $H_0: \\mu=3.75$, $\\ \\ H_a: \\mu \\neq 3.75$ C. $H_0: \\mu=4.55$, $\\ \\ H_a: \\mu \\neq 4.55$ D. $H_0: \\mu < 4.55$, $\\ \\ H_a: \\mu=4.55$ E. $H_0: {\\bar x}=3.75$, $\\ \\ H_a: {\\bar x} > 3.75$ F. $H_0: \\mu=4.55$, $\\ \\ H_a: \\mu < 4.55$\n(b) According to the Merck Veterinary Manual, the average resting heart rate for a certain type of sheep dog is 115 beats per minute (bpm). A Montana farmer notices his aging sheep dog has been acting more lethargic than usual and wonders if her heart rate is slowing. He measures her heart rate on 15 occasions and finds a sample mean heart rate of 118.2 bpm. [ANS] A. $H_0: \\mu=115$, $\\ \\ H_a: \\mu > 115$ B. $H_0: {\\bar x}=115$, $\\ \\ H_a: {\\bar x} > 115$ C. $H_0: {\\bar x}=118.2$, $\\ \\ H_a: {\\bar x} \\neq 118.2$ D. $H_0: \\mu=118.2$, $\\ \\ H_a: \\mu < 118.2$ E. $H_0: \\mu=115$, $\\ \\ H_a: \\mu \\neq 115$ F. $H_0: \\mu=115$, $\\ \\ H_a: \\mu < 115$\n(c) The mean height of all adult American males is 69 inches (5 ft 9 in). A researcher wonders if young American males between the ages of 18 and 21 tend to be taller than 69 inches. A random sample of 100 young American males ages 18 to 21 yielded a sample mean of 71 inches. [ANS] A. $H_0: {\\bar x}=71$, $\\ \\ H_a: {\\bar x} < 71$ B. $H_0: \\mu > 69$, $\\ \\ H_a: \\mu < 69$ C. $H_0: {\\bar x}=69$, $\\ \\ H_a: {\\bar x} > 69$ D. $H_0: \\mu=69$, $\\ \\ H_a: \\mu \\neq 69$ E. $H_0: \\mu=69$, $\\ \\ H_a: \\mu > 69$ F. $H_0: \\mu=71$, $\\ \\ H_a: \\mu < 71$", "answer_v3": [ "C", "F", "E" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Statistics_0216", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Concepts", "level": "1", "keywords": [ "statistics", "inference", "confidence interval" ], "problem_v1": "Lindsey thinks a certain potato chip maker is putting fewer chips in their regular bags of chips. From a random sample of 17 bags of potato chips she calculated a P value of 0.095 for the sample.\n(a) At a 5\\% level of significance, is there evidence that Lindsey is correct? (Type: Yes or No): [ANS]\n(b) At a 10\\% level of significance, is there evidence that she is correct? (Type: Yes or No): [ANS]\n(c) In a statistical test of hypotheses, we say that the data are statistically significant at level $\\alpha$ if [ANS] A. the P-value is less than $\\alpha$. B. the P-value is larger than $\\alpha$. C. $\\alpha$ is small. D. $\\alpha=0.05$.", "answer_v1": [ "No", "Yes", "A" ], "answer_type_v1": [ "TF", "TF", "MCS" ], "options_v1": [ [], [], [ "A", "B", "C", "D" ] ], "problem_v2": "Calvin thinks a certain potato chip maker is putting fewer chips in their regular bags of chips. From a random sample of 22 bags of potato chips he calculated a P value of 0.026 for the sample.\n(a) At a 5\\% level of significance, is there evidence that Calvin is correct? (Type: Yes or No): [ANS]\n(b) At a 10\\% level of significance, is there evidence that he is correct? (Type: Yes or No): [ANS]\n(c) In a statistical test of hypotheses, we say that the data are statistically significant at level $\\alpha$ if [ANS] A. $\\alpha$ is small. B. the P-value is larger than $\\alpha$. C. $\\alpha=0.05$. D. the P-value is less than $\\alpha$.", "answer_v2": [ "Yes", "Yes", "D" ], "answer_type_v2": [ "TF", "TF", "MCS" ], "options_v2": [ [], [], [ "A", "B", "C", "D" ] ], "problem_v3": "Dan thinks a certain potato chip maker is putting fewer chips in their regular bags of chips. From a random sample of 17 bags of potato chips he calculated a P value of 0.045 for the sample.\n(a) At a 5\\% level of significance, is there evidence that Dan is correct? (Type: Yes or No): [ANS]\n(b) At a 10\\% level of significance, is there evidence that he is correct? (Type: Yes or No): [ANS]\n(c) In a statistical test of hypotheses, we say that the data are statistically significant at level $\\alpha$ if [ANS] A. the P-value is less than $\\alpha$. B. $\\alpha=0.05$. C. $\\alpha$ is small. D. the P-value is larger than $\\alpha$.", "answer_v3": [ "Yes", "Yes", "A" ], "answer_type_v3": [ "TF", "TF", "MCS" ], "options_v3": [ [], [], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0218", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Concepts", "level": "3", "keywords": [ "statistics", "inference", "hypothesis testing", "t score" ], "problem_v1": "(a) Suppose we were not sure if the distribution of a population was normal. In which of the following circumstances would we NOT be safe using a t procedure? [ANS] A. The mean and median of the data are nearly equal. B. A histogram of the data shows moderate skewness. C. The sample standard deviation is large. D. A stemplot of the data has a large outlier.\n(b) Which of the following is an example of a matched pairs design? [ANS] A. A teacher calculates the average scores of students on a pair of tests and wishes to see if this average is larger than 80\\%. B. A teacher compares the scores of students using a computer based method of instruction with scores of other students using a traditional method of instruction. C. A teacher compares the scores of students in her class on a standardized test with the national average score. D. A teacher compares the pre-test and post-test scores of students.\n(c) You are thinking of using a t procedure to test hypotheses about the mean of a population using a significance level of 0.05. You suspect the distribution of the population is not normal and may be moderately skewed. Which of the following statements is correct? [ANS] A. You may use the t procedure, provided your sample size is large, say at least fifty. B. You may not use the t procedure. t procedures are robust to nonnormality for confidence intervals but not for tests of hypotheses. C. You should not use the t procedure because the population does not have a normal distribution. D. You may use the t procedure, but you should probably only claim the significance level is 0.10.", "answer_v1": [ "D", "D", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "(a) You are thinking of using a t procedure to test hypotheses about the mean of a population using a significance level of 0.05. You suspect the distribution of the population is not normal and may be moderately skewed. Which of the following statements is correct? [ANS] A. You should not use the t procedure because the population does not have a normal distribution. B. You may not use the t procedure. t procedures are robust to nonnormality for confidence intervals but not for tests of hypotheses. C. You may use the t procedure, but you should probably only claim the significance level is 0.10. D. You may use the t procedure, provided your sample size is large, say at least fifty.\n(b) Suppose we were not sure if the distribution of a population was normal. In which of the following circumstances would we NOT be safe using a t procedure? [ANS] A. The mean and median of the data are nearly equal. B. A stemplot of the data has a large outlier. C. The sample standard deviation is large. D. A histogram of the data shows moderate skewness.\n(c) Which of the following is an example of a matched pairs design? [ANS] A. A teacher compares the pre-test and post-test scores of students. B. A teacher compares the scores of students in her class on a standardized test with the national average score. C. A teacher calculates the average scores of students on a pair of tests and wishes to see if this average is larger than 80\\%. D. A teacher compares the scores of students using a computer based method of instruction with scores of other students using a traditional method of instruction.", "answer_v2": [ "D", "B", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "(a) You are thinking of using a t procedure to test hypotheses about the mean of a population using a significance level of 0.05. You suspect the distribution of the population is not normal and may be moderately skewed. Which of the following statements is correct? [ANS] A. You may use the t procedure, provided your sample size is large, say at least fifty. B. You may use the t procedure, but you should probably only claim the significance level is 0.10. C. You should not use the t procedure because the population does not have a normal distribution. D. You may not use the t procedure. t procedures are robust to nonnormality for confidence intervals but not for tests of hypotheses.\n(b) Suppose we were not sure if the distribution of a population was normal. In which of the following circumstances would we NOT be safe using a t procedure? [ANS] A. A histogram of the data shows moderate skewness. B. A stemplot of the data has a large outlier. C. The sample standard deviation is large. D. The mean and median of the data are nearly equal.\n(c) Which of the following is an example of a matched pairs design? [ANS] A. A teacher compares the pre-test and post-test scores of students. B. A teacher compares the scores of students in her class on a standardized test with the national average score. C. A teacher compares the scores of students using a computer based method of instruction with scores of other students using a traditional method of instruction. D. A teacher calculates the average scores of students on a pair of tests and wishes to see if this average is larger than 80\\%.", "answer_v3": [ "A", "B", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0219", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Concepts", "level": "1", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "In each part, we have given the significance level and the P-value for a hypothesis test. For each case determine if the null hypothesis should be rejected. Write \"reject\" or \"do not reject\" (without quotations).\n(a) $\\alpha=0.01, P=0.06$ answer: [ANS]\n(b) $\\alpha=0.06, P=0.06$ answer: [ANS]\n(c) $\\alpha=0.07, P=0.06$ answer: [ANS]", "answer_v1": [ "DO NOT REJECT", "REJECT", "REJECT" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "REJECT", "DO NOT REJECT" ], [ "REJECT", "DO NOT REJECT" ], [ "REJECT", "DO NOT REJECT" ] ], "problem_v2": "In each part, we have given the significance level and the P-value for a hypothesis test. For each case determine if the null hypothesis should be rejected. Write \"reject\" or \"do not reject\" (without quotations).\n(a) $\\alpha=0.07, P=0.06$ answer: [ANS]\n(b) $\\alpha=0.01, P=0.06$ answer: [ANS]\n(c) $\\alpha=0.06, P=0.06$ answer: [ANS]", "answer_v2": [ "REJECT", "DO NOT REJECT", "REJECT" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "REJECT", "DO NOT REJECT" ], [ "REJECT", "DO NOT REJECT" ], [ "REJECT", "DO NOT REJECT" ] ], "problem_v3": "In each part, we have given the significance level and the P-value for a hypothesis test. For each case determine if the null hypothesis should be rejected. Write \"reject\" or \"do not reject\" (without quotations).\n(a) $\\alpha=0.07, P=0.06$ answer: [ANS]\n(b) $\\alpha=0.06, P=0.06$ answer: [ANS]\n(c) $\\alpha=0.01, P=0.06$ answer: [ANS]", "answer_v3": [ "REJECT", "REJECT", "DO NOT REJECT" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "REJECT", "DO NOT REJECT" ], [ "REJECT", "DO NOT REJECT" ], [ "REJECT", "DO NOT REJECT" ] ] }, { "id": "Statistics_0220", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Concepts", "level": "1", "keywords": [ "Statistics" ], "problem_v1": "Suppose that you have obtained data by taking a random sample from a population. Before performing a statistical inference, what should you do? [ANS] A. Make sure the sample size is greater than 30, otherwise throw the data out and start over B. Throw out any potential outliers because it's easier to examine data with no outliers C. Graph the data to make sure that none of the conditions required for the contemplated statistical procedure have been violated D. None of the above", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "Suppose that you have obtained data by taking a random sample from a population. Before performing a statistical inference, what should you do? [ANS] A. Graph the data to make sure that none of the conditions required for the contemplated statistical procedure have been violated B. Throw out any potential outliers because it's easier to examine data with no outliers C. Make sure the sample size is greater than 30, otherwise throw the data out and start over D. None of the above", "answer_v2": [ "A" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "Suppose that you have obtained data by taking a random sample from a population. Before performing a statistical inference, what should you do? [ANS] A. Make sure the sample size is greater than 30, otherwise throw the data out and start over B. Graph the data to make sure that none of the conditions required for the contemplated statistical procedure have been violated C. Throw out any potential outliers because it's easier to examine data with no outliers D. None of the above", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0221", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Concepts", "level": "3", "keywords": [ "Statistics" ], "problem_v1": "Suppose we were comparing the mean age of buyers of new domestic cars to the mean age of buyers of new imported cars.\nWhat is the variable under consideration? [ANS] A. buyers of imported cars B. buyers of domestic cars C. age of buyers D. None of the above\nWhat are the two populations under consideration? [ANS] A. buyers of new domestic cars and buyers of new imported cars B. cars and buyers C. age of buyers of new domestic cars and age of buyers of new imported cars D. None of the above", "answer_v1": [ "C", "A" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Suppose we were comparing the mean age of buyers of new domestic cars to the mean age of buyers of new imported cars.\nWhat is the variable under consideration? [ANS] A. age of buyers B. buyers of domestic cars C. buyers of imported cars D. None of the above\nWhat are the two populations under consideration? [ANS] A. buyers of new domestic cars and buyers of new imported cars B. age of buyers of new domestic cars and age of buyers of new imported cars C. cars and buyers D. None of the above", "answer_v2": [ "A", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Suppose we were comparing the mean age of buyers of new domestic cars to the mean age of buyers of new imported cars.\nWhat is the variable under consideration? [ANS] A. buyers of imported cars B. age of buyers C. buyers of domestic cars D. None of the above\nWhat are the two populations under consideration? [ANS] A. cars and buyers B. buyers of new domestic cars and buyers of new imported cars C. age of buyers of new domestic cars and age of buyers of new imported cars D. None of the above", "answer_v3": [ "B", "B" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0222", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Concepts", "level": "2", "keywords": [ "Statistics", "Hypothesis Testing" ], "problem_v1": "Why is $s_p$ called the pooled sample standard deviation? [ANS] A. Because it is obtained by combining information on variation from two distinct population standard deviations to get a single new population standard deviation B. Because it is obtained by combining information on variation from the individual samples into one estimate of the common sample standard deviation C. Because it is obtained by combining information on variation from the individual samples into one estimate of the common population standard deviation D. None of the above", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "Why is $s_p$ called the pooled sample standard deviation? [ANS] A. Because it is obtained by combining information on variation from the individual samples into one estimate of the common population standard deviation B. Because it is obtained by combining information on variation from the individual samples into one estimate of the common sample standard deviation C. Because it is obtained by combining information on variation from two distinct population standard deviations to get a single new population standard deviation D. None of the above", "answer_v2": [ "A" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "Why is $s_p$ called the pooled sample standard deviation? [ANS] A. Because it is obtained by combining information on variation from two distinct population standard deviations to get a single new population standard deviation B. Because it is obtained by combining information on variation from the individual samples into one estimate of the common population standard deviation C. Because it is obtained by combining information on variation from the individual samples into one estimate of the common sample standard deviation D. None of the above", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0224", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Concepts", "level": "3", "keywords": [ "hypothesis test", "p-value", "statistics", "statistical inference", "hypothesis test", "interpretation" ], "problem_v1": "Prof. Johnson conducts a hypothesis test on whether the proportion of all UBC students who bike to school (denoted as $p)$ equals 30\\%. Specifically, Prof. Johnson has $H_0: p=0.3$ versus $H_A: p \\ne 0.3$. He obtains a $P$-value of 0.01. On the other hand, Prof. Smith would like to test if there is sufficient evidence to support that $p$ is greater than 0.3 at the 10\\% significance level. Based on Prof. Johnson's result, will the null hypothesis of Prof. Smith's test be rejected? [ANS] A. No. B. Yes. C. There is insufficient information to tell.", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C" ] ], "problem_v2": "Prof. Johnson conducts a hypothesis test on whether the proportion of all UBC students who bike to school (denoted as $p)$ equals 30\\%. Specifically, Prof. Johnson has $H_0: p=0.3$ versus $H_A: p \\ne 0.3$. He obtains a $P$-value of 0.01. On the other hand, Prof. Smith would like to test if there is sufficient evidence to support that $p$ is greater than 0.3 at the 10\\% significance level. Based on Prof. Johnson's result, will the null hypothesis of Prof. Smith's test be rejected? [ANS] A. There is insufficient information to tell. B. Yes. C. No.", "answer_v2": [ "A" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C" ] ], "problem_v3": "Prof. Johnson conducts a hypothesis test on whether the proportion of all UBC students who bike to school (denoted as $p)$ equals 30\\%. Specifically, Prof. Johnson has $H_0: p=0.3$ versus $H_A: p \\ne 0.3$. He obtains a $P$-value of 0.01. On the other hand, Prof. Smith would like to test if there is sufficient evidence to support that $p$ is greater than 0.3 at the 10\\% significance level. Based on Prof. Johnson's result, will the null hypothesis of Prof. Smith's test be rejected? [ANS] A. No. B. There is insufficient information to tell. C. Yes.", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C" ] ] }, { "id": "Statistics_0226", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Concepts", "level": "3", "keywords": [ "Probability", "Independence" ], "problem_v1": "Air Canada is preparing to launch a new airline specializing in servicing international flights, and is conducting a survey to collect data on the reasons for travel and the types of flights taken by its current customers. The survey was randomly given to individuals who had booked a flight for travel within the next month.\n$\\begin{array}{cccc}\\hline & International & Domestic & Total \\\\ \\hline Leisure Purposes & 250 & 450 & 700 \\\\ \\hline Business Purposes & 550 & 300 & 850 \\\\ \\hline Total & 800 & 750 & 1550 \\\\ \\hline \\end{array}$\nPart i A traveller is selected at random. What is the probability that he is on an international flight given that he is travelling for leisure purposes? [ANS] A. 250/700 B. 250/800 C. 700/1550 D. 250/1550\nPart ii Which of the following statements is true? [ANS] A. Reasons for travel and destination are dependent, but they would be independent if we changed the second row of counts from 550, 300, 850 to 550, 1110, 1660. B. Reasons for travel and destination are dependent, but they would be independent if we changed the second row of counts from 550, 300, 850 to 250, 600, 850. C. Reasons for travel and destination are independent. D. Reasons for travel and destination are dependent, but they would be independent if we changed the second row of counts from 550, 300, 850 to 550, 1540, 2090. E. Reasons for travel and destination are dependent, but they would be independent if we changed the second row of counts from 550, 300, 850 to 550, 990, 1540. F. Reasons for travel and destination are dependent, but they would be independent if we changed the second row of counts from 550, 300, 850 to 550, 550, 1100.\nPart iii In order to attract potential customers to its new airline, Air Canada is giving discounts to individuals that are flying to international destinations during its first month of service. The 800 individuals who were travelling outside of Canada were invited to participate in the following experiment. Half of the individuals travelling for business related reasons are randomized to receive the discounted air fare and half do not receive a discount. The individuals travelling for leisure purposes are randomized in a similar fashion. After six months, the proportion of business travellers who continue to fly with Air Canada's new airline is compared between the discounted airfare group and the regular airfare group. The same comparison is also done among the leisure travellers.\nIn the experiment described above, reasons for travel (leisure versus business) [CHECK ALL THAT APPLY] [ANS] A. is the blocking variable. B. defines the experimental units. C. is the response variable. D. defines the treatments. E. none of the above.\nPart iv What is the purpose of performing the treatment randomization and comparison for the leisure and business travellers separately? Choose the most appropriate answer. [ANS] A. To control for the effect of reason for travel on the travellers' willingness to switch to the new airline. B. To evaluate the effect of reasons for travel on the travellers' willingness to taking international flights. C. To ensure that both leisure and business travellers can participate in the study.\nPart v To display the data for the two variables: reasons for travel and whether travellers switch to taking the new airline after six months, what is the most appropriate display to use? [ANS] A. A contingency table. B. A stem-and-leaf display. C. A scatterplot. D. Side-by-side boxplots. E. A histogram.", "answer_v1": [ "A", "E", "A", "A", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Air Canada is preparing to launch a new airline specializing in servicing international flights, and is conducting a survey to collect data on the reasons for travel and the types of flights taken by its current customers. The survey was randomly given to individuals who had booked a flight for travel within the next month.\n$\\begin{array}{cccc}\\hline & International & Domestic & Total \\\\ \\hline Leisure Purposes & 250 & 450 & 700 \\\\ \\hline Business Purposes & 550 & 300 & 850 \\\\ \\hline Total & 800 & 750 & 1550 \\\\ \\hline \\end{array}$\nPart i A traveller is selected at random. What is the probability that he is travelling for business reasons given that his destination is outside of Canada? [ANS] A. 850/1550 B. 550/850 C. 550/1550 D. 550/800\nPart ii Which of the following statements is true? [ANS] A. Reasons for travel and destination are dependent, but they would be independent if we changed the second row of counts from 550, 300, 850 to 550, 550, 1100. B. Reasons for travel and destination are dependent, but they would be independent if we changed the second row of counts from 550, 300, 850 to 550, 1540, 2090. C. Reasons for travel and destination are dependent, but they would be independent if we changed the second row of counts from 550, 300, 850 to 550, 1110, 1660. D. Reasons for travel and destination are dependent, but they would be independent if we changed the second row of counts from 550, 300, 850 to 250, 600, 850. E. Reasons for travel and destination are dependent, but they would be independent if we changed the second row of counts from 550, 300, 850 to 550, 990, 1540. F. Reasons for travel and destination are independent.\nPart iii In order to attract potential customers to its new airline, Air Canada is giving discounts to individuals that are flying to international destinations during its first month of service. The 800 individuals who were travelling outside of Canada were invited to participate in the following experiment. Half of the individuals travelling for business related reasons are randomized to receive the discounted air fare and half do not receive a discount. The individuals travelling for leisure purposes are randomized in a similar fashion. After six months, the proportion of business travellers who continue to fly with Air Canada's new airline is compared between the discounted airfare group and the regular airfare group. The same comparison is also done among the leisure travellers.\nIn the experiment described above, reasons for travel (leisure versus business) [CHECK ALL THAT APPLY] [ANS] A. is the response variable. B. defines the experimental units. C. defines the treatments. D. is the blocking variable. E. none of the above.\nPart iv What is the purpose of performing the treatment randomization and comparison for the leisure and business travellers separately? Choose the most appropriate answer. [ANS] A. To ensure that both leisure and business travellers can participate in the study. B. To evaluate the effect of reasons for travel on the travellers' willingness to taking international flights. C. To control for the effect of reason for travel on the travellers' willingness to switch to the new airline.\nPart v To display the data for the two variables: reasons for travel and whether travellers switch to taking the new airline after six months, what is the most appropriate display to use? [ANS] A. A contingency table. B. A histogram. C. Side-by-side boxplots. D. A stem-and-leaf display. E. A scatterplot.", "answer_v2": [ "D", "E", "D", "C", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Air Canada is preparing to launch a new airline specializing in servicing international flights, and is conducting a survey to collect data on the reasons for travel and the types of flights taken by its current customers. The survey was randomly given to individuals who had booked a flight for travel within the next month.\n$\\begin{array}{cccc}\\hline & International & Domestic & Total \\\\ \\hline Leisure Purposes & 250 & 450 & 700 \\\\ \\hline Business Purposes & 550 & 300 & 850 \\\\ \\hline Total & 800 & 750 & 1550 \\\\ \\hline \\end{array}$\nPart i A traveller is selected at random. What is the probability that he is travelling for business reasons given that his destination is outside of Canada? [ANS] A. 850/1550 B. 550/800 C. 550/850 D. 550/1550\nPart ii Which of the following statements is true? [ANS] A. Reasons for travel and destination are dependent, but they would be independent if we changed the second row of counts from 550, 300, 850 to 550, 1540, 2090. B. Reasons for travel and destination are dependent, but they would be independent if we changed the second row of counts from 550, 300, 850 to 550, 550, 1100. C. Reasons for travel and destination are dependent, but they would be independent if we changed the second row of counts from 550, 300, 850 to 550, 990, 1540. D. Reasons for travel and destination are dependent, but they would be independent if we changed the second row of counts from 550, 300, 850 to 250, 600, 850. E. Reasons for travel and destination are dependent, but they would be independent if we changed the second row of counts from 550, 300, 850 to 550, 1110, 1660. F. Reasons for travel and destination are independent.\nPart iii In order to attract potential customers to its new airline, Air Canada is giving discounts to individuals that are flying to international destinations during its first month of service. The 800 individuals who were travelling outside of Canada were invited to participate in the following experiment. Half of the individuals travelling for business related reasons are randomized to receive the discounted air fare and half do not receive a discount. The individuals travelling for leisure purposes are randomized in a similar fashion. After six months, the proportion of business travellers who continue to fly with Air Canada's new airline is compared between the discounted airfare group and the regular airfare group. The same comparison is also done among the leisure travellers.\nIn the experiment described above, reasons for travel (leisure versus business) [CHECK ALL THAT APPLY] [ANS] A. is the blocking variable. B. defines the treatments. C. is the response variable. D. defines the experimental units. E. none of the above.\nPart iv What is the purpose of performing the treatment randomization and comparison for the leisure and business travellers separately? Choose the most appropriate answer. [ANS] A. To ensure that both leisure and business travellers can participate in the study. B. To control for the effect of reason for travel on the travellers' willingness to switch to the new airline. C. To evaluate the effect of reasons for travel on the travellers' willingness to taking international flights.\nPart v To display the data for the two variables: reasons for travel and whether travellers switch to taking the new airline after six months, what is the most appropriate display to use? [ANS] A. A contingency table. B. A scatterplot. C. A histogram. D. Side-by-side boxplots. E. A stem-and-leaf display.", "answer_v3": [ "B", "C", "A", "B", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Statistics_0227", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Concepts", "level": "3", "keywords": [ "statistics", "statistical inference", "hypothesis test", "p-value", "interpretatin" ], "problem_v1": "When one changes the significance level of a hypothesis test from 0.01 to 0.05, which of the following will happen? Check all that apply. [ANS] A. The chance of committing a Type II error changes from 0.01 to 0.05. B. The test becomes more stringent to reject the null hypothesis (i.e. it becomes harder to reject the null hypothesis). C. The test becomes less stringent to reject the null hypothesis (i.e., it becomes easier to reject the null hypothesis). D. It becomes harder to prove that the null hypothesis is true. E. The chance that the null hypothesis is true changes from 0.01 to 0.05. F. The chance of committing a Type I error changes from 0.01 to 0.05. G. It becomes easier to prove that the null hypothesis is true.", "answer_v1": [ "CF" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v2": "When one changes the significance level of a hypothesis test from 0.01 to 0.05, which of the following will happen? Check all that apply. [ANS] A. It becomes easier to prove that the null hypothesis is true. B. The chance of committing a Type I error changes from 0.01 to 0.05. C. The test becomes less stringent to reject the null hypothesis (i.e., it becomes easier to reject the null hypothesis). D. The chance that the null hypothesis is true changes from 0.01 to 0.05. E. The test becomes more stringent to reject the null hypothesis (i.e. it becomes harder to reject the null hypothesis). F. The chance of committing a Type II error changes from 0.01 to 0.05. G. It becomes harder to prove that the null hypothesis is true.", "answer_v2": [ "BC" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v3": "When one changes the significance level of a hypothesis test from 0.01 to 0.05, which of the following will happen? Check all that apply. [ANS] A. The test becomes more stringent to reject the null hypothesis (i.e. it becomes harder to reject the null hypothesis). B. The chance of committing a Type I error changes from 0.01 to 0.05. C. It becomes harder to prove that the null hypothesis is true. D. The chance that the null hypothesis is true changes from 0.01 to 0.05. E. The test becomes less stringent to reject the null hypothesis (i.e., it becomes easier to reject the null hypothesis). F. The chance of committing a Type II error changes from 0.01 to 0.05. G. It becomes easier to prove that the null hypothesis is true.", "answer_v3": [ "BE" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G" ] ] }, { "id": "Statistics_0228", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Concepts", "level": "2", "keywords": [ "statistics", "hypothesis testing", "hypothesis tests" ], "problem_v1": "A professor of statistics refutes the claim that the average student spends 3 hours studying for a midterm exam. Which hypothesis is used to test the claim? [ANS] A. $H_0: \\mu=3, H_1: \\mu \\not=3$ B. $H_0: \\mu=3, H_1: \\mu < 3$ C. $H_0: \\mu \\not=3, H_1: \\mu=3$ D. $H_0: \\mu \\not=3, H_1: \\mu > 3$\nSuppose that we reject a null hypothesis at the 0.05 level of significance. Then for which of the following $\\alpha-$ values do we also reject the null hypothesis? [ANS] A. 0.02 B. 0.04 C. 0.06 D. 0.03", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "A professor of statistics refutes the claim that the average student spends 3 hours studying for a midterm exam. Which hypothesis is used to test the claim? [ANS] A. $H_0: \\mu \\not=3, H_1: \\mu=3$ B. $H_0: \\mu=3, H_1: \\mu < 3$ C. $H_0: \\mu \\not=3, H_1: \\mu > 3$ D. $H_0: \\mu=3, H_1: \\mu \\not=3$\nWhich of the following $p-$ values will lead us to reject the null hypothesis if the level of significance equals 0.05? [ANS] A. 0.025 B. 0.055 C. 0.15 D. 0.10", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "A professor of statistics refutes the claim that the average student spends 3 hours studying for a midterm exam. Which hypothesis is used to test the claim? [ANS] A. $H_0: \\mu \\not=3, H_1: \\mu=3$ B. $H_0: \\mu=3, H_1: \\mu \\not=3$ C. $H_0: \\mu=3, H_1: \\mu < 3$ D. $H_0: \\mu \\not=3, H_1: \\mu > 3$\nUsing the confidence interval when conducting a two-tail test for the population mean $\\mu$, we do not reject the null hypothesis if the hypothesized value for $\\mu$: [ANS] A. falls between the lower confidence limit (LCL) and the upper confidence limit (UCL) B. is the the right of the upper confidence limit (UCL) C. is the to the left of the lower confidence limit (LCL) D. falls in the rejection region", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0229", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - z", "level": "3", "keywords": [ "hypothesis testing", "statistics", "large sample" ], "problem_v1": "A random sample of $100$ observations from a population with standard deviation $20.3575072960673$ yielded a sample mean of $93.5$.\n(a) $\\ $ Given that the null hypothesis is $\\mu=90$ and the alternative hypothesis is $\\mu > 90$ using $\\alpha=.05$, find the following: (i) $\\ $ critical $z$ score $\\ $ [ANS]\n(ii) $\\ $ test statistic $=$ [ANS]\n(b) $\\ $ Given that the null hypothesis is $\\mu=90$ and the alternative hypothesis is $\\mu \\ne 90$ using $\\alpha=.05$, find the following: (i) $\\ $ the positive critical $z$ score $\\ $ [ANS]\n(ii) $\\ $ the negative critical $z$ score $\\ $ [ANS]\n(iii) $\\ $ test statistic $=$ [ANS]\nThe conclusion from part (a) is: [ANS] A. There is insufficient evidence to reject the null hypothesis B. Reject the null hypothesis C. None of the above\nThe conclusion from part (b) is: [ANS] A. There is insufficient evidence to reject the null hypothesis B. Reject the null hypothesis C. None of the above", "answer_v1": [ "1.64485", "1.71926746683576", "1.95996", "-1.95996", "1.71926746683576", "B", "A" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "MCS", "MCS" ], "options_v1": [ [], [], [], [], [], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v2": "A random sample of $100$ observations from a population with standard deviation $8.66343190677358$ yielded a sample mean of $91.6$.\n(a) $\\ $ Given that the null hypothesis is $\\mu=90$ and the alternative hypothesis is $\\mu > 90$ using $\\alpha=.05$, find the following: (i) $\\ $ critical $z$ score $\\ $ [ANS]\n(ii) $\\ $ test statistic $=$ [ANS]\n(b) $\\ $ Given that the null hypothesis is $\\mu=90$ and the alternative hypothesis is $\\mu \\ne 90$ using $\\alpha=.05$, find the following: (i) $\\ $ the positive critical $z$ score $\\ $ [ANS]\n(ii) $\\ $ the negative critical $z$ score $\\ $ [ANS]\n(iii) $\\ $ test statistic $=$ [ANS]\nThe conclusion from part (a) is: [ANS] A. There is insufficient evidence to reject the null hypothesis B. Reject the null hypothesis C. None of the above\nThe conclusion from part (b) is: [ANS] A. Reject the null hypothesis B. There is insufficient evidence to reject the null hypothesis C. None of the above", "answer_v2": [ "1.64485", "1.84684316471515", "1.95996", "-1.95996", "1.84684316471515", "B", "B" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "MCS", "MCS" ], "options_v2": [ [], [], [], [], [], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v3": "A random sample of $100$ observations from a population with standard deviation $11.8145043776403$ yielded a sample mean of $92.1$.\n(a) $\\ $ Given that the null hypothesis is $\\mu=90$ and the alternative hypothesis is $\\mu > 90$ using $\\alpha=.05$, find the following: (i) $\\ $ critical $z$ score $\\ $ [ANS]\n(ii) $\\ $ test statistic $=$ [ANS]\n(b) $\\ $ Given that the null hypothesis is $\\mu=90$ and the alternative hypothesis is $\\mu \\ne 90$ using $\\alpha=.05$, find the following: (i) $\\ $ the positive critical $z$ score $\\ $ [ANS]\n(ii) $\\ $ the negative critical $z$ score $\\ $ [ANS]\n(iii) $\\ $ test statistic $=$ [ANS]\nThe conclusion from part (a) is: [ANS] A. There is insufficient evidence to reject the null hypothesis B. Reject the null hypothesis C. None of the above\nThe conclusion from part (b) is: [ANS] A. Reject the null hypothesis B. There is insufficient evidence to reject the null hypothesis C. None of the above", "answer_v3": [ "1.64485", "1.7774761707096", "1.95996", "-1.95996", "1.7774761707096", "B", "B" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "MCS", "MCS" ], "options_v3": [ [], [], [], [], [], [ "A", "B", "C" ], [ "A", "B", "C" ] ] }, { "id": "Statistics_0230", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - z", "level": "2", "keywords": [ "large sample", "p value", "hypothesis testing", "statistics" ], "problem_v1": "Golf-course designers have become concerned that old courses are becoming obsolete since new technology has given golfers the ability to hit the ball so far. Designers, therefore, have proposed that new golf courses need to be built expecting that the average golfer can hit the ball more than $255$ yards on average. Suppose a random sample of $162$ golfers be chosen so that their mean driving distance is $258.5$ yards, with a standard deviation of $47.2$. Conduct a hypothesis test where $H_0: \\mu=255$ and $H_1:\\mu > 255$ by computing the following:\n(a) $\\ $ test statistic $\\ $ [ANS]\n(b) $\\ $ p-value $p=$ [ANS]\n(c) $\\ $ If this was a two-tailed test, then the p-value is [ANS]", "answer_v1": [ "0.94380777997357", "0.172634", "0.345268" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Golf-course designers have become concerned that old courses are becoming obsolete since new technology has given golfers the ability to hit the ball so far. Designers, therefore, have proposed that new golf courses need to be built expecting that the average golfer can hit the ball more than $230$ yards on average. Suppose a random sample of $115$ golfers be chosen so that their mean driving distance is $235.6$ yards, with a standard deviation of $43.3$. Conduct a hypothesis test where $H_0: \\mu=230$ and $H_1:\\mu > 230$ by computing the following:\n(a) $\\ $ test statistic $\\ $ [ANS]\n(b) $\\ $ p-value $p=$ [ANS]\n(c) $\\ $ If this was a two-tailed test, then the p-value is [ANS]", "answer_v2": [ "1.38691246306411", "0.0827341", "0.1654682" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Golf-course designers have become concerned that old courses are becoming obsolete since new technology has given golfers the ability to hit the ball so far. Designers, therefore, have proposed that new golf courses need to be built expecting that the average golfer can hit the ball more than $240$ yards on average. Suppose a random sample of $128$ golfers be chosen so that their mean driving distance is $243.7$ yards, with a standard deviation of $45.5$. Conduct a hypothesis test where $H_0: \\mu=240$ and $H_1:\\mu > 240$ by computing the following:\n(a) $\\ $ test statistic $\\ $ [ANS]\n(b) $\\ $ p-value $p=$ [ANS]\n(c) $\\ $ If this was a two-tailed test, then the p-value is [ANS]", "answer_v3": [ "0.920015855961395", "0.178782", "0.357564" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Statistics_0231", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - z", "level": "2", "keywords": [ "statistics", "hypothesis testing", "inference", "confidence interval" ], "problem_v1": "$50$ people are randomly selected and the accuracy of their wristwatches is checked, with positive errors representing watches that are ahead of the correct time and negative errors representing watches that are behind the correct time. The $50$ values have a mean of $108$ sec and a standard deviation of $213$ sec. Use a $0.02$ significance level to test the claim that the population of all watches has a mean of $0$ sec. The test statistic is [ANS]\nThe P-Value is [ANS]\nThe final conclustion is [ANS] A. There is not sufficient evidence to warrant rejection of the claim that the mean is equal to 0 B. There is sufficient evidence to warrant rejection of the claim that the mean is equal to 0", "answer_v1": [ "3.58533015812897", "0.000336652", "B" ], "answer_type_v1": [ "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "A", "B" ] ], "problem_v2": "$35$ people are randomly selected and the accuracy of their wristwatches is checked, with positive errors representing watches that are ahead of the correct time and negative errors representing watches that are behind the correct time. The $35$ values have a mean of $118$ sec and a standard deviation of $165$ sec. Use a $0.01$ significance level to test the claim that the population of all watches has a mean of $0$ sec. The test statistic is [ANS]\nThe P-Value is [ANS]\nThe final conclustion is [ANS] A. There is sufficient evidence to warrant rejection of the claim that the mean is equal to 0 B. There is not sufficient evidence to warrant rejection of the claim that the mean is equal to 0", "answer_v2": [ "4.23089342064094", "2.32764E-05", "A" ], "answer_type_v2": [ "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "A", "B" ] ], "problem_v3": "$40$ people are randomly selected and the accuracy of their wristwatches is checked, with positive errors representing watches that are ahead of the correct time and negative errors representing watches that are behind the correct time. The $40$ values have a mean of $109$ sec and a standard deviation of $178$ sec. Use a $0.02$ significance level to test the claim that the population of all watches has a mean of $0$ sec. The test statistic is [ANS]\nThe P-Value is [ANS]\nThe final conclustion is [ANS] A. There is not sufficient evidence to warrant rejection of the claim that the mean is equal to 0 B. There is sufficient evidence to warrant rejection of the claim that the mean is equal to 0", "answer_v3": [ "3.87290185346464", "0.0001075472", "B" ], "answer_type_v3": [ "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "A", "B" ] ] }, { "id": "Statistics_0232", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - z", "level": "2", "keywords": [ "large sample", "hypothesis testing", "statistics" ], "problem_v1": "It is necessary for an automobile producer to estimate the number of miles per gallon achieved by its cars. Suppose that the sample mean for a random sample of $140$ cars is $29.5$ miles and assume the standard deviation is $3.2$ miles. Now suppose the car producer wants to test the hypothesis that $\\mu$, the mean number of miles per gallon, is $30.9$ against the alternative hypothesis that it is not $30.9$. Conduct a test using $\\alpha=.05$ by giving the following:\n(a) $\\ $ positive critical $z$ score $\\ $ [ANS]\n(b) $\\ $ negative critical $z$ score $\\ $ [ANS]\n(c) $\\ $ test statistic $\\ $ [ANS]\nThe final conclustion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that $\\mu=30.9$. B. We can reject the null hypothesis that $\\mu=30.9$ and accept that $\\mu \\ne 30.9$.", "answer_v1": [ "1.95996398465615", "-1.95996398465615", "-5.17656981021216", "B" ], "answer_type_v1": [ "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B" ] ], "problem_v2": "It is necessary for an automobile producer to estimate the number of miles per gallon achieved by its cars. Suppose that the sample mean for a random sample of $100$ cars is $27.6$ miles and assume the standard deviation is $3.9$ miles. Now suppose the car producer wants to test the hypothesis that $\\mu$, the mean number of miles per gallon, is $26.6$ against the alternative hypothesis that it is not $26.6$. Conduct a test using $\\alpha=.05$ by giving the following:\n(a) $\\ $ positive critical $z$ score $\\ $ [ANS]\n(b) $\\ $ negative critical $z$ score $\\ $ [ANS]\n(c) $\\ $ test statistic $\\ $ [ANS]\nThe final conclustion is [ANS] A. We can reject the null hypothesis that $\\mu=26.6$ and accept that $\\mu \\ne 26.6$. B. There is not sufficient evidence to reject the null hypothesis that $\\mu=26.6$.", "answer_v2": [ "1.95996398465615", "-1.95996398465615", "2.56410256410256", "A" ], "answer_type_v2": [ "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B" ] ], "problem_v3": "It is necessary for an automobile producer to estimate the number of miles per gallon achieved by its cars. Suppose that the sample mean for a random sample of $110$ cars is $28.1$ miles and assume the standard deviation is $3.2$ miles. Now suppose the car producer wants to test the hypothesis that $\\mu$, the mean number of miles per gallon, is $28.4$ against the alternative hypothesis that it is not $28.4$. Conduct a test using $\\alpha=.05$ by giving the following:\n(a) $\\ $ positive critical $z$ score $\\ $ [ANS]\n(b) $\\ $ negative critical $z$ score $\\ $ [ANS]\n(c) $\\ $ test statistic $\\ $ [ANS]\nThe final conclustion is [ANS] A. We can reject the null hypothesis that $\\mu=28.4$ and accept that $\\mu \\ne 28.4$. B. There is not sufficient evidence to reject the null hypothesis that $\\mu=28.4$.", "answer_v3": [ "1.95996398465615", "-1.95996398465615", "-0.983258295159519", "B" ], "answer_type_v3": [ "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B" ] ] }, { "id": "Statistics_0233", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - z", "level": "2", "keywords": [ "statistics", "inference" ], "problem_v1": "The time needed for college students to complete a certain paper-and-pencil maze follows a normal distribution with a mean of 30 seconds and a standard deviation of 3.2 seconds. You wish to see if the mean time $\\mu$ is changed by vigorous exercise, so you have a group of 25 college students exercise vigorously for 30 minutes and then complete the maze. It takes them an average of $\\bar{x}=29.5$ seconds to complete the maze. Use this information to test the hypotheses $H_0: \\mu=30$ $H_a: \\mu \\neq 30$ Conduct a test using a significance level of $\\alpha=0.05$.\n(a) The test statistic [ANS]\n(b) The positive critical value, z*=[ANS]\n(c) The final conclusion is [ANS] A. We can reject the null hypothesis that $\\mu=30$ and accept that $\\mu \\ne 30$. B. There is not sufficient evidence to reject the null hypothesis that $\\mu=30$.", "answer_v1": [ "-0.77", "1.96", "B" ], "answer_type_v1": [ "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "A", "B" ] ], "problem_v2": "The time needed for college students to complete a certain paper-and-pencil maze follows a normal distribution with a mean of 30 seconds and a standard deviation of 3.9 seconds. You wish to see if the mean time $\\mu$ is changed by vigorous exercise, so you have a group of 15 college students exercise vigorously for 30 minutes and then complete the maze. It takes them an average of $\\bar{x}=27.6$ seconds to complete the maze. Use this information to test the hypotheses $H_0: \\mu=30$ $H_a: \\mu \\neq 30$ Conduct a test using a significance level of $\\alpha=0.01$.\n(a) The test statistic [ANS]\n(b) The positive critical value, z*=[ANS]\n(c) The final conclusion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that $\\mu=30$. B. We can reject the null hypothesis that $\\mu=30$ and accept that $\\mu \\ne 30$.", "answer_v2": [ "-2.37", "2.576", "A" ], "answer_type_v2": [ "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "A", "B" ] ], "problem_v3": "The time needed for college students to complete a certain paper-and-pencil maze follows a normal distribution with a mean of 30 seconds and a standard deviation of 3.2 seconds. You wish to see if the mean time $\\mu$ is changed by vigorous exercise, so you have a group of 15 college students exercise vigorously for 30 minutes and then complete the maze. It takes them an average of $\\bar{x}=28.1$ seconds to complete the maze. Use this information to test the hypotheses $H_0: \\mu=30$ $H_a: \\mu \\neq 30$ Conduct a test using a significance level of $\\alpha=0.05$.\n(a) The test statistic [ANS]\n(b) The positive critical value, z*=[ANS]\n(c) The final conclusion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that $\\mu=30$. B. We can reject the null hypothesis that $\\mu=30$ and accept that $\\mu \\ne 30$.", "answer_v3": [ "-2.29", "1.96", "B" ], "answer_type_v3": [ "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "A", "B" ] ] }, { "id": "Statistics_0234", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - z", "level": "2", "keywords": [ "statistics", "inference" ], "problem_v1": "The nicotine content in cigarettes of a certain brand is normally distributed with mean (in milligrams) $\\mu$ and standard deviation $\\sigma=0.1$. The brand advertises that the mean nicotine content of their cigarettes is 1.5 mg. Now, suppose a reporter wants to test whether the mean nicotine content is actually higher than advertised. He takes measurements from a SRS of 20 cigarettes of this brand. The sample yields an average of 1.55 mg of nicotine. Conduct a test using a significance level of $\\alpha=0.05$.\n(a) The test statistic [ANS]\n(b) The critical value, z*=[ANS]\n(c) The final conclusion is [ANS] A. The nicotine content is probably higher than advertised. B. There is not sufficient evidence to show that the ad is misleading.", "answer_v1": [ "2.24", "1.645", "A" ], "answer_type_v1": [ "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "A", "B" ] ], "problem_v2": "The nicotine content in cigarettes of a certain brand is normally distributed with mean (in milligrams) $\\mu$ and standard deviation $\\sigma=0.1$. The brand advertises that the mean nicotine content of their cigarettes is 1.5 mg. Now, suppose a reporter wants to test whether the mean nicotine content is actually higher than advertised. He takes measurements from a SRS of 10 cigarettes of this brand. The sample yields an average of 1.6 mg of nicotine. Conduct a test using a significance level of $\\alpha=0.01$.\n(a) The test statistic [ANS]\n(b) The critical value, z*=[ANS]\n(c) The final conclusion is [ANS] A. The nicotine content is probably higher than advertised. B. There is not sufficient evidence to show that the ad is misleading.", "answer_v2": [ "3.16", "2.33", "A" ], "answer_type_v2": [ "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "A", "B" ] ], "problem_v3": "The nicotine content in cigarettes of a certain brand is normally distributed with mean (in milligrams) $\\mu$ and standard deviation $\\sigma=0.1$. The brand advertises that the mean nicotine content of their cigarettes is 1.5 mg. Now, suppose a reporter wants to test whether the mean nicotine content is actually higher than advertised. He takes measurements from a SRS of 10 cigarettes of this brand. The sample yields an average of 1.55 mg of nicotine. Conduct a test using a significance level of $\\alpha=0.01$.\n(a) The test statistic [ANS]\n(b) The critical value, z*=[ANS]\n(c) The final conclusion is [ANS] A. The nicotine content is probably higher than advertised. B. There is not sufficient evidence to show that the ad is misleading.", "answer_v3": [ "1.58", "2.33", "B" ], "answer_type_v3": [ "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "A", "B" ] ] }, { "id": "Statistics_0235", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - z", "level": "3", "keywords": [ "statistics", "inference", "confidence interval" ], "problem_v1": "A study is conducted to determine if a newly designed text book is more helpful to learning the material than the old edition. The mean score on the final exam for a course using the old edition is 75. Ten randomly selected people who used the new text take the final exam. Their scores are shown in the table below.\n$\\begin{array}{ccccccccccc}\\hline Person & A & B & C & D & E & F & G & H & I & J \\\\ \\hline Test Score & 88 & 74 & 84 & 78 & 81 & 91 & 94 & 69 & 71 & 96 \\\\ \\hline \\end{array}$\nUse a $0.05$ significance level to test the claim that people do better with the new edition. Assume the standard deviation is 10.3. (Note: You may wish to use statistical software.)\n(a) What kind of test should be used? [ANS] A. Two-Tailed B. One-Tailed C. It does not matter.\n(b) The test statistic is [ANS] (rounded to 2 decimals). (c) The P-value is [ANS]\n(d) Is there sufficient evidence to support the claim that people do better than 75 on this exam? [ANS] A. Yes B. No\n(e) Construct a $95$ \\% confidence interval for the mean score for students using the new text. [ANS] $< \\mu <$ [ANS]", "answer_v1": [ "B", "2.33", "0.0099", "A", "76.2161", "88.9839" ], "answer_type_v1": [ "MCS", "NV", "NV", "MCS", "NV", "NV" ], "options_v1": [ [ "A", "B", "C" ], [], [], [ "A", "B" ], [], [] ], "problem_v2": "A study is conducted to determine if a newly designed text book is more helpful to learning the material than the old edition. The mean score on the final exam for a course using the old edition is 75. Ten randomly selected people who used the new text take the final exam. Their scores are shown in the table below.\n$\\begin{array}{ccccccccccc}\\hline Person & A & B & C & D & E & F & G & H & I & J \\\\ \\hline Test Score & 87 & 76 & 94 & 72 & 73 & 83 & 90 & 67 & 84 & 96 \\\\ \\hline \\end{array}$\nUse a $0.01$ significance level to test the claim that people do better with the new edition. Assume the standard deviation is 10.2. (Note: You may wish to use statistical software.)\n(a) What kind of test should be used? [ANS] A. One-Tailed B. Two-Tailed C. It does not matter.\n(b) The test statistic is [ANS] (rounded to 2 decimals). (c) The P-value is [ANS]\n(d) Is there sufficient evidence to support the claim that people do better than 75 on this exam? [ANS] A. No B. Yes\n(e) Construct a $99$ \\% confidence interval for the mean score for students using the new text. [ANS] $< \\mu <$ [ANS]", "answer_v2": [ "A", "2.23", "0.0129", "A", "73.8916", "90.5084" ], "answer_type_v2": [ "MCS", "NV", "NV", "MCS", "NV", "NV" ], "options_v2": [ [ "A", "B", "C" ], [], [], [ "A", "B" ], [], [] ], "problem_v3": "A study is conducted to determine if a newly designed text book is more helpful to learning the material than the old edition. The mean score on the final exam for a course using the old edition is 75. Ten randomly selected people who used the new text take the final exam. Their scores are shown in the table below.\n$\\begin{array}{ccccccccccc}\\hline Person & A & B & C & D & E & F & G & H & I & J \\\\ \\hline Test Score & 73 & 77 & 67 & 92 & 96 & 89 & 85 & 71 & 80 & 95 \\\\ \\hline \\end{array}$\nUse a $0.05$ significance level to test the claim that people do better with the new edition. Assume the standard deviation is 11.1. (Note: You may wish to use statistical software.)\n(a) What kind of test should be used? [ANS] A. Two-Tailed B. One-Tailed C. It does not matter.\n(b) The test statistic is [ANS] (rounded to 2 decimals). (c) The P-value is [ANS]\n(d) Is there sufficient evidence to support the claim that people do better than 75 on this exam? [ANS] A. No B. Yes\n(e) Construct a $95$ \\% confidence interval for the mean score for students using the new text. [ANS] $< \\mu <$ [ANS]", "answer_v3": [ "B", "2.14", "0.0162", "B", "75.6203", "89.3797" ], "answer_type_v3": [ "MCS", "NV", "NV", "MCS", "NV", "NV" ], "options_v3": [ [ "A", "B", "C" ], [], [], [ "A", "B" ], [], [] ] }, { "id": "Statistics_0236", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - z", "level": "1", "keywords": [ "statistics", "inference", "confidence interval" ], "problem_v1": "The contents of $38$ cans of Coke have a mean of $\\overline{x}=12.15$. Assume the contents of cans of Coke have a normal distribution with standard deviation of $\\sigma=0.12.$ Find the value of the test statistic $z$ for the claim that the population mean is $\\mu=12.$ The test statistic is [ANS]", "answer_v1": [ "7.70552" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The contents of $31$ cans of Coke have a mean of $\\overline{x}=12.15$. Assume the contents of cans of Coke have a normal distribution with standard deviation of $\\sigma=0.09.$ Find the value of the test statistic $z$ for the claim that the population mean is $\\mu=12.$ The test statistic is [ANS]", "answer_v2": [ "9.27961" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The contents of $34$ cans of Coke have a mean of $\\overline{x}=12.15$. Assume the contents of cans of Coke have a normal distribution with standard deviation of $\\sigma=0.1.$ Find the value of the test statistic $z$ for the claim that the population mean is $\\mu=12.$ The test statistic is [ANS]", "answer_v3": [ "8.74643" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0237", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - z", "level": "3", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "(a) Find the P-value for the test statistic $z=1.24$ for the following null and alternative hypotheses: $H_0$: The population mean is 17. $H_a$: The population mean is less than 17. The P-value is [ANS]\n(b) Find the P-value for the test statistic $z=1.24$ for the following null and alternative hypotheses: $H_0$: The population mean is 17. $H_a$: The population mean is not equal to 17. The P-value is [ANS]", "answer_v1": [ "0.8925", "0.215" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "(a) Find the P-value for the test statistic $z=2.41$ for the following null and alternative hypotheses: $H_0$: The population mean is 6. $H_a$: The population mean is less than 6. The P-value is [ANS]\n(b) Find the P-value for the test statistic $z=2.41$ for the following null and alternative hypotheses: $H_0$: The population mean is 6. $H_a$: The population mean is not equal to 6. The P-value is [ANS]", "answer_v2": [ "0.992", "0.016" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "(a) Find the P-value for the test statistic $z=-1.24$ for the following null and alternative hypotheses: $H_0$: The population mean is 10. $H_a$: The population mean is less than 10. The P-value is [ANS]\n(b) Find the P-value for the test statistic $z=-1.24$ for the following null and alternative hypotheses: $H_0$: The population mean is 10. $H_a$: The population mean is not equal to 10. The P-value is [ANS]", "answer_v3": [ "0.1075", "0.215" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0238", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - z", "level": "3", "keywords": [ "hypothesis testing", "statistics" ], "problem_v1": "A new cream that advertises that it can reduce wrinkles and improve skin was subject to a recent study. A sample of $63$ women over the age of 50 used the new cream for 6 months. Of those $63$ women, $46$ of them reported skin improvement(as judged by a dermatologist). Is this evidence that the cream will improve the skin of more than $50$ \\% of women over the age of 50? Test using $\\alpha=0.05$.\n(a) Test statistic: $z=$ [ANS]\n(b) Critical Value: $z*=$ [ANS]\n(c) The final conclusion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that $p=0.5$. That is, there is not sufficient evidence to reject that the cream can improve the skin of more than $50$ \\% of women over 50. B. We can reject the null hypothesis that $p=0.5$ and accept that $p > 0.5$. That is, the cream can improve the skin of more than $50$ \\% of women over 50.", "answer_v1": [ "3.65365657242253", "1.64485", "B" ], "answer_type_v1": [ "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "A", "B" ] ], "problem_v2": "A new cream that advertises that it can reduce wrinkles and improve skin was subject to a recent study. A sample of $42$ women over the age of 50 used the new cream for 6 months. Of those $42$ women, $37$ of them reported skin improvement(as judged by a dermatologist). Is this evidence that the cream will improve the skin of more than $40$ \\% of women over the age of 50? Test using $\\alpha=0.01$.\n(a) Test statistic: $z=$ [ANS]\n(b) Critical Value: $z*=$ [ANS]\n(c) The final conclusion is [ANS] A. We can reject the null hypothesis that $p=0.4$ and accept that $p > 0.4$. That is, the cream can improve the skin of more than $40$ \\% of women over 50. B. There is not sufficient evidence to reject the null hypothesis that $p=0.4$. That is, there is not sufficient evidence to reject that the cream can improve the skin of more than $40$ \\% of women over 50.", "answer_v2": [ "6.36240196232199", "2.32635", "A" ], "answer_type_v2": [ "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "A", "B" ] ], "problem_v3": "A new cream that advertises that it can reduce wrinkles and improve skin was subject to a recent study. A sample of $49$ women over the age of 50 used the new cream for 6 months. Of those $49$ women, $39$ of them reported skin improvement(as judged by a dermatologist). Is this evidence that the cream will improve the skin of more than $40$ \\% of women over the age of 50? Test using $\\alpha=0.05$.\n(a) Test statistic: $z=$ [ANS]\n(b) Critical Value: $z*=$ [ANS]\n(c) The final conclusion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that $p=0.4$. That is, there is not sufficient evidence to reject that the cream can improve the skin of more than $40$ \\% of women over 50. B. We can reject the null hypothesis that $p=0.4$ and accept that $p > 0.4$. That is, the cream can improve the skin of more than $40$ \\% of women over 50.", "answer_v3": [ "5.6571548821421", "1.64485", "B" ], "answer_type_v3": [ "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "A", "B" ] ] }, { "id": "Statistics_0239", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - z", "level": "2", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "A random sample of 12 second-year university students enrolled in a business statistics course was drawn. At the course's completion, each student was asked how many hours he or she spent doing homework in statistics. The data are listed below.\n\\begin{array}{ccccccccccc} 59 & 59 & 61 &54 & 55 & 59 \\\\ 59 & 55 & 57 & 60 & 54 & 58 \\end{array} It is known that the population standard deviation is 8. The instructor has recommended that students devote 5 hours per week for the duration of the 12-week semester, for a total of 60 hours. Test to determine whether there is evidence at the 0.04 significance level that the average student spent less than the recommended amount of time. Fill in the requested information below. A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Reject $H_1$. B. Reject $H_0$. C. Do Not Reject $H_0$. D. Do Not Reject $H_1$.", "answer_v1": [ "-1.08253175473055", "(-infinity,-1.75069)", "0.139508155619101", "C" ], "answer_type_v1": [ "NV", "INT", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v2": "A random sample of 12 second-year university students enrolled in a business statistics course was drawn. At the course's completion, each student was asked how many hours he or she spent doing homework in statistics. The data are listed below.\n\\begin{array}{ccccccccccc} 28 & 16 & 19 &29 & 19 & 16 \\\\ 19 & 23 & 15 & 24 & 21 & 26 \\end{array} It is known that the population standard deviation is 6. The instructor has recommended that students devote 2 hours per week for the duration of the 12-week semester, for a total of 24 hours. Test to determine whether there is evidence at the 0.02 significance level that the average student spent less than the recommended amount of time. Fill in the requested information below. A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Do Not Reject $H_1$. B. Reject $H_1$. C. Reject $H_0$. D. Do Not Reject $H_0$.", "answer_v2": [ "-1.58771324027147", "(-infinity,-2.05375)", "0.0561755978586472", "D" ], "answer_type_v2": [ "NV", "INT", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v3": "A random sample of 12 second-year university students enrolled in a business statistics course was drawn. At the course's completion, each student was asked how many hours he or she spent doing homework in statistics. The data are listed below.\n\\begin{array}{ccccccccccc} 35 & 30 & 34 &29 & 31 & 38 \\\\ 40 & 39 & 29 & 30 & 30 & 26 \\end{array} It is known that the population standard deviation is 8. The instructor has recommended that students devote 3 hours per week for the duration of the 12-week semester, for a total of 36 hours. Test to determine whether there is evidence at the 0.09 significance level that the average student spent less than the recommended amount of time. Fill in the requested information below. A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Do Not Reject $H_0$. B. Do Not Reject $H_1$. C. Reject $H_1$. D. Reject $H_0$.", "answer_v3": [ "-1.47946006479841", "(-infinity,-1.34076)", "0.0695086975549463", "D" ], "answer_type_v3": [ "NV", "INT", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0240", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - z", "level": "3", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "Suppose that you are to conduct the following hypothesis test:\n$ \\qquad \\begin{array}{rrcl} H_0: & \\mu &=& 530 \\\\ H_1: & \\mu & \\not=& 530 \\end{array}$ Assume that you know that $\\sigma=105$, $n=46$, $\\bar{x}=506.9$, and take $\\alpha=0.01$. Draw the sampling distribution, and use it to determine each of the following: A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Reject $H_0$. B. Reject $H_1$. C. Do Not Reject $H_1$. D. Do Not Reject $H_0$.", "answer_v1": [ "-1.49211259628756", "(-infinity,-2.57583) U (2.57583,infinity)", "0.1356694", "D" ], "answer_type_v1": [ "NV", "INT", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v2": "Suppose that you are to conduct the following hypothesis test:\n$ \\qquad \\begin{array}{rrcl} H_0: & \\mu &=& 450 \\\\ H_1: & \\mu & \\not=& 450 \\end{array}$ Assume that you know that $\\sigma=80$, $n=50$, $\\bar{x}=435.6$, and take $\\alpha=0.1$. Draw the sampling distribution, and use it to determine each of the following: A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Reject $H_0$. B. Do Not Reject $H_0$. C. Do Not Reject $H_1$. D. Reject $H_1$.", "answer_v2": [ "-1.27279220613578", "(-infinity,-1.64485) U (1.64485,infinity)", "0.203092", "B" ], "answer_type_v2": [ "NV", "INT", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v3": "Suppose that you are to conduct the following hypothesis test:\n$ \\qquad \\begin{array}{rrcl} H_0: & \\mu &=& 480 \\\\ H_1: & \\mu & \\not=& 480 \\end{array}$ Assume that you know that $\\sigma=90$, $n=46$, $\\bar{x}=461.1$, and take $\\alpha=0.005$. Draw the sampling distribution, and use it to determine each of the following: A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Reject $H_1$. B. Do Not Reject $H_0$. C. Do Not Reject $H_1$. D. Reject $H_0$.", "answer_v3": [ "-1.4242892964563", "(-infinity,-2.80703) U (2.80703,infinity)", "0.1543624", "B" ], "answer_type_v3": [ "NV", "INT", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0241", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - z", "level": "3", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "Past experience indicates that the monthly long-distance telephone bill is normally distributed with a mean of 21.01 dollars and a standard deviation of 4 dollars. After an advertising campaign aimed at increasing long-distance telephone usage, a random sample of 25 household's bills was taken. The data (in dollars) is below.\n\\begin{array}{cccccccccccccccccccccccc} 23.61 & 24.25 & 25.93 & 18.99 & 19.47 & 23.37 & 23.49 \\\\ 20.13 & 22.17 & 24.78 & 18.46 & 22.38 & 21.07 & 20.56 \\\\ 21.18 & 22.5 & 16.22 & 16.62 & 20.51 & 22.46 & 26.99 \\\\ 21.6 & 19.33 & 21.67 & 28.68 \\\\ \\end{array} Can we conclude at the 3\\% significance level that the campaign was successful? Fill in the requested information below.\nA. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Do Not Reject $H_0$. B. Do Not Reject $H_1$. C. Reject $H_0$. D. Reject $H_1$.", "answer_v1": [ "1.0585", "(1.88079,infinity)", "0.14491377439927", "A" ], "answer_type_v1": [ "NV", "INT", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v2": "Past experience indicates that the monthly long-distance telephone bill is normally distributed with a mean of 18.33 dollars and a standard deviation of 3.87 dollars. After an advertising campaign aimed at increasing long-distance telephone usage, a random sample of 25 household's bills was taken. The data (in dollars) is below.\n\\begin{array}{cccccccccccccccccccccccc} 26.71 & 13.79 & 16.84 & 26.96 & 16.55 & 14.35 & 16.74 \\\\ 20.64 & 12.45 & 21.91 & 18.61 & 24.71 & 14.51 & 14.46 \\\\ 15.8 & 20.8 & 14.43 & 16.65 & 20.52 & 13.19 & 13.38 \\\\ 12.93 & 16.45 & 19.35 & 12.85 \\\\ \\end{array} Can we conclude at the 4\\% significance level that the campaign was successful? Fill in the requested information below.\nA. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Reject $H_1$. B. Do Not Reject $H_1$. C. Do Not Reject $H_0$. D. Reject $H_0$.", "answer_v2": [ "-1.17157622739018", "(1.75069,infinity)", "0.879316380492069", "C" ], "answer_type_v2": [ "NV", "INT", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v3": "Past experience indicates that the monthly long-distance telephone bill is normally distributed with a mean of 19.25 dollars and a standard deviation of 3.45 dollars. After an advertising campaign aimed at increasing long-distance telephone usage, a random sample of 25 household's bills was taken. The data (in dollars) is below.\n\\begin{array}{cccccccccccccccccccccccc} 22.24 & 16.84 & 21.3 & 15.65 & 17.96 & 25.6 & 27.29 \\\\ 26.66 & 15.55 & 17.03 & 16.39 & 12.95 & 21.7 & 28.42 \\\\ 25.32 & 22.39 & 13.99 & 17.08 & 22.83 & 23.86 & 20.88 \\\\ 19.17 & 27.47 & 25.3 & 26.18 \\\\ \\end{array} Can we conclude at the 6\\% significance level that the campaign was successful? Fill in the requested information below.\nA. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Reject $H_0$. B. Do Not Reject $H_0$. C. Reject $H_1$. D. Do Not Reject $H_1$.", "answer_v3": [ "2.82898550724637", "(1.55477,infinity)", "0.00233478968888659", "A" ], "answer_type_v3": [ "NV", "INT", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0242", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - z", "level": "3", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "A dean in the business school claims that GMAT scores of applicants to the school's MBA program have increased during the past 5 years. Five years ago, the mean and standard deviation of GMAT scores of MBA applicants were 530 and 50, respectively. 29 applications for this year's program were randomly selected and the GMAT scores recorded. If we assume that the distribution of GMAT scores of this year's applicants is the same as that of 5 years ago, find the probability of erroneously concluding that there is not enough evidence to supports the claim when, in fact, the true mean GMAT score is 560. Assume $\\alpha$ is 0.05. P(Type II Error)=[ANS]", "answer_v1": [ "0.0563414303158162" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A dean in the business school claims that GMAT scores of applicants to the school's MBA program have increased during the past 5 years. Five years ago, the mean and standard deviation of GMAT scores of MBA applicants were 530 and 45, respectively. 22 applications for this year's program were randomly selected and the GMAT scores recorded. If we assume that the distribution of GMAT scores of this year's applicants is the same as that of 5 years ago, find the probability of erroneously concluding that there is not enough evidence to supports the claim when, in fact, the true mean GMAT score is 570. Assume $\\alpha$ is 0.01. P(Type II Error)=[ANS]", "answer_v2": [ "0.0326711879565525" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A dean in the business school claims that GMAT scores of applicants to the school's MBA program have increased during the past 5 years. Five years ago, the mean and standard deviation of GMAT scores of MBA applicants were 580 and 40, respectively. 29 applications for this year's program were randomly selected and the GMAT scores recorded. If we assume that the distribution of GMAT scores of this year's applicants is the same as that of 5 years ago, find the probability of erroneously concluding that there is not enough evidence to supports the claim when, in fact, the true mean GMAT score is 610. Assume $\\alpha$ is 0.07. P(Type II Error)=[ANS]", "answer_v3": [ "0.00518735214642042" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0243", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - z", "level": "3", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "Determine the probability of making a Type II error for the following hypothesis test, given that $\\mu=211$.\n$ \\qquad \\begin{array}{rrcl} H_0: & \\mu &=& 210 \\\\ H_1: & \\mu & \\not=& 210 \\end{array}$ For this test, take $\\sigma=11$, $n=60$, and $\\alpha=0.05$. P(Type II Error)=[ANS]", "answer_v1": [ "0.891543061484441" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Determine the probability of making a Type II error for the following hypothesis test, given that $\\mu=182$.\n$ \\qquad \\begin{array}{rrcl} H_0: & \\mu &=& 183 \\\\ H_1: & \\mu & \\not=& 183 \\end{array}$ For this test, take $\\sigma=8$, $n=80$, and $\\alpha=0.1$. P(Type II Error)=[ANS]", "answer_v2": [ "0.697974662189261" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Determine the probability of making a Type II error for the following hypothesis test, given that $\\mu=191$.\n$ \\qquad \\begin{array}{rrcl} H_0: & \\mu &=& 192 \\\\ H_1: & \\mu & \\not=& 192 \\end{array}$ For this test, take $\\sigma=9$, $n=70$, and $\\alpha=0.05$. P(Type II Error)=[ANS]", "answer_v3": [ "0.846645487217304" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0244", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - z", "level": "3", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "Determine the probability of making a Type II error for the following hypothesis test, given that $\\mu=1089$.\n$ \\qquad \\begin{array}{rrcl} H_0: \\mu=1050 \\\\ H_1: \\mu > 1050 \\end{array}$ For this test, take $\\sigma=51$, $n=26$, and $\\alpha=0.04$. P(Type II Error)=[ANS]", "answer_v1": [ "0.0158346381172345" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Determine the probability of making a Type II error for the following hypothesis test, given that $\\mu=933$.\n$ \\qquad \\begin{array}{rrcl} H_0: \\mu=910 \\\\ H_1: \\mu > 910 \\end{array}$ For this test, take $\\sigma=46$, $n=30$, and $\\alpha=0.1$. P(Type II Error)=[ANS]", "answer_v2": [ "0.0725495236401302" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Determine the probability of making a Type II error for the following hypothesis test, given that $\\mu=992$.\n$ \\qquad \\begin{array}{rrcl} H_0: \\mu=960 \\\\ H_1: \\mu > 960 \\end{array}$ For this test, take $\\sigma=48$, $n=26$, and $\\alpha=0.03$. P(Type II Error)=[ANS]", "answer_v3": [ "0.0644371026702055" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0245", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - z", "level": "2", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "A machine that produces ball bearings is set so that the average diameter is 0.53 inch. A sample of 10 ball bearings was measured with the results shown below.\n\\begin{array}{ccccccccccc} 0.54 & 0.54 & 0.55 & 0.51 & 0.51 \\\\ 0.54 & 0.54 & 0.52 & 0.53 & 0.55 \\end{array} Assuming that the standard deviation is 0.04 inch, can we conclude at the 6\\% significance level that the mean diameter is not 0.53 inch? A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Reject $H_0$. B. Do Not Reject $H_1$. C. Do Not Reject $H_0$. D. Reject $H_1$.", "answer_v1": [ "0.237170824512629", "(-infinity,-1.88079) U (1.88079,infinity)", "0.812524267341679", "C" ], "answer_type_v1": [ "NV", "INT", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v2": "A machine that produces ball bearings is set so that the average diameter is 0.45 inch. A sample of 10 ball bearings was measured with the results shown below.\n\\begin{array}{ccccccccccc} 0.5 & 0.41 & 0.43 & 0.5 & 0.43 \\\\ 0.42 & 0.43 & 0.46 & 0.4 & 0.47 \\end{array} Assuming that the standard deviation is 0.05 inch, can we conclude at the 9\\% significance level that the mean diameter is not 0.45 inch? A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Reject $H_0$. B. Do Not Reject $H_1$. C. Do Not Reject $H_0$. D. Reject $H_1$.", "answer_v2": [ "-0.316227766016838", "(-infinity,-1.6954) U (1.6954,infinity)", "0.751829632072231", "C" ], "answer_type_v2": [ "NV", "INT", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v3": "A machine that produces ball bearings is set so that the average diameter is 0.48 inch. A sample of 10 ball bearings was measured with the results shown below.\n\\begin{array}{ccccccccccc} 0.49 & 0.46 & 0.49 & 0.45 & 0.46 \\\\ 0.51 & 0.53 & 0.52 & 0.45 & 0.46 \\end{array} Assuming that the standard deviation is 0.04 inch, can we conclude at the 1\\% significance level that the mean diameter is not 0.48 inch? A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Do Not Reject $H_0$. B. Reject $H_0$. C. Reject $H_1$. D. Do Not Reject $H_1$.", "answer_v3": [ "0.158113883008415", "(-infinity,-2.57583) U (2.57583,infinity)", "0.87436705918916", "A" ], "answer_type_v3": [ "NV", "INT", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0246", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - z", "level": "3", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "In an attempt to reduce the number of person-hours lost as a result of industrial accidents, a large production plant installed new safety equipment. In a test of the effectiveness of the equipment, a random sample of 55 departments was chosen. The number of person-hours lost in the month prior to and the month after the installation of the safety equipment was recorded and the percentage change was calculated. Find the probability that the test is unable to conclude that the new safety equipment is effective when the mean percent reduction is actually 1.9\\%. Assume that the population standard deviation is 10\\% and that $\\alpha$ is 0.08. P(Type II Error)=[ANS]", "answer_v1": [ "0.498401157310035" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "In an attempt to reduce the number of person-hours lost as a result of industrial accidents, a large production plant installed new safety equipment. In a test of the effectiveness of the equipment, a random sample of 41 departments was chosen. The number of person-hours lost in the month prior to and the month after the installation of the safety equipment was recorded and the percentage change was calculated. Find the probability that the test is unable to conclude that the new safety equipment is effective when the mean percent reduction is actually 0.5\\%. Assume that the population standard deviation is 12\\% and that $\\alpha$ is 0.04. P(Type II Error)=[ANS]", "answer_v2": [ "0.931081365195555" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "In an attempt to reduce the number of person-hours lost as a result of industrial accidents, a large production plant installed new safety equipment. In a test of the effectiveness of the equipment, a random sample of 46 departments was chosen. The number of person-hours lost in the month prior to and the month after the installation of the safety equipment was recorded and the percentage change was calculated. Find the probability that the test is unable to conclude that the new safety equipment is effective when the mean percent reduction is actually 0.9\\%. Assume that the population standard deviation is 10\\% and that $\\alpha$ is 0.06. P(Type II Error)=[ANS]", "answer_v3": [ "0.827507218685849" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0247", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - t", "level": "2", "keywords": [ "statistics", "hypothesis testing", "large sample" ], "problem_v1": "Test the claim that the population of sophomore college students has a mean grade point average greater than $2.3$. Sample statistics include $n=140$, $\\overline{x}=2.45$, and $s=0.6$. Use a significance level of $\\alpha=0.04$. The test statistic is [ANS]\nThe critical value is [ANS]\nThe P-Value is [ANS]\nThe final conclustion is [ANS] A. There is not sufficient evidence to support the claim that the mean grade point average is greater than 2.3. B. There is sufficient evidence to support the claim that the mean grade point average is greater than 2.3.", "answer_v1": [ "2.9580398915498", "1.75069", "0.00154801", "B" ], "answer_type_v1": [ "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B" ] ], "problem_v2": "Test the claim that the population of sophomore college students has a mean grade point average greater than $2.1$. Sample statistics include $n=190$, $\\overline{x}=2.38$, and $s=0.9$. Use a significance level of $\\alpha=0.02$. The test statistic is [ANS]\nThe critical value is [ANS]\nThe P-Value is [ANS]\nThe final conclustion is [ANS] A. There is sufficient evidence to support the claim that the mean grade point average is greater than 2.1. B. There is not sufficient evidence to support the claim that the mean grade point average is greater than 2.1.", "answer_v2": [ "4.28837072287251", "2.05375", "0.00000899943", "A" ], "answer_type_v2": [ "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B" ] ], "problem_v3": "Test the claim that the population of sophomore college students has a mean grade point average greater than $2.15$. Sample statistics include $n=140$, $\\overline{x}=2.4$, and $s=0.6$. Use a significance level of $\\alpha=0.03$. The test statistic is [ANS]\nThe critical value is [ANS]\nThe P-Value is [ANS]\nThe final conclustion is [ANS] A. There is not sufficient evidence to support the claim that the mean grade point average is greater than 2.15. B. There is sufficient evidence to support the claim that the mean grade point average is greater than 2.15.", "answer_v3": [ "4.93006648591635", "1.88079", "0.000000411008", "B" ], "answer_type_v3": [ "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B" ] ] }, { "id": "Statistics_0248", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - t", "level": "2", "keywords": [ "statistics", "inference", "hypothesis testing", "t score", "hypothesis testing" ], "problem_v1": "When a poultry farmer uses his regular feed, the newborn chickens have normally distributed weights with a mean of $62.5$ oz. In an experiment with an enriched feed mixture, ten chickens are born with the following weights (in ounces). 66.7, \\ 62.7, \\ 65, \\ 65.3, \\ 68.7, \\ 64.6, \\ 64.6, \\ 65.6, \\ 65.5, \\ 66 Use the $\\alpha=0.05$ significance level to test the claim that the mean weight is higher with the enriched feed.\n(a) The sample mean is $\\overline{x}=$ [ANS]\n(b) The sample standard deviation is $s=$ [ANS]\n(c) The test statistic is $t=$ [ANS]\n(d) The critical value is $t^*=$ [ANS]\n(e) The conclusion is [ANS] A. There is not sufficient evidence to support the claim that with the enriched feed, the mean weight is greater than 62.5. B. There is sufficient evidence to support the claim that with the enriched feed, the mean weight is greater than 62.5.", "answer_v1": [ "65.47", "1.5521", "6.051", "1.833", "B" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [], [ "A", "B" ] ], "problem_v2": "When a poultry farmer uses his regular feed, the newborn chickens have normally distributed weights with a mean of $61.1$ oz. In an experiment with an enriched feed mixture, ten chickens are born with the following weights (in ounces). 64.9, \\ 63.3, \\ 65.1, \\ 66.3, \\ 66, \\ 68, \\ 65, \\ 61.5, \\ 65.9, \\ 66 Use the $\\alpha=0.01$ significance level to test the claim that the mean weight is higher with the enriched feed.\n(a) The sample mean is $\\overline{x}=$ [ANS]\n(b) The sample standard deviation is $s=$ [ANS]\n(c) The test statistic is $t=$ [ANS]\n(d) The critical value is $t^*=$ [ANS]\n(e) The conclusion is [ANS] A. There is sufficient evidence to support the claim that with the enriched feed, the mean weight is greater than 61.1. B. There is not sufficient evidence to support the claim that with the enriched feed, the mean weight is greater than 61.1.", "answer_v2": [ "65.2", "1.772", "7.317", "2.821", "A" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [], [ "A", "B" ] ], "problem_v3": "When a poultry farmer uses his regular feed, the newborn chickens have normally distributed weights with a mean of $61.6$ oz. In an experiment with an enriched feed mixture, ten chickens are born with the following weights (in ounces). 66.6, \\ 64.6, \\ 69.8, \\ 64.8, \\ 68.1, \\ 63.5, \\ 65.3, \\ 65.2, \\ 64, \\ 65 Use the $\\alpha=0.05$ significance level to test the claim that the mean weight is higher with the enriched feed.\n(a) The sample mean is $\\overline{x}=$ [ANS]\n(b) The sample standard deviation is $s=$ [ANS]\n(c) The test statistic is $t=$ [ANS]\n(d) The critical value is $t^*=$ [ANS]\n(e) The conclusion is [ANS] A. There is not sufficient evidence to support the claim that with the enriched feed, the mean weight is greater than 61.6. B. There is sufficient evidence to support the claim that with the enriched feed, the mean weight is greater than 61.6.", "answer_v3": [ "65.69", "1.94448", "6.652", "1.833", "B" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [], [ "A", "B" ] ] }, { "id": "Statistics_0251", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - t", "level": "2", "keywords": [ "statistics", "inference", "hypothesis testing", "t score", "hypothesis testing", "statistics" ], "problem_v1": "One of the most feared predators in the ocean is the great white shark. It is known that the white shark grows to a mean length of $19$ feet; however, one marine biologist believes that great white sharks off the Bermuda coast grow much longer. To test this claim, full-grown white sharks were captured, measured, and then set free. However, this was a difficult, costly and very dangerous task, so only four sharks were actually sampled. Their lengths were $25, \\ 24, \\ 24, \\mbox{and} \\ 27$ feet. Do the data provide sufficient evidence to support the claim? Use $\\alpha=0.05$.\n(a) Calculate the test statistic: t=[ANS]\n(b) Find the critical value: t*=[ANS]\n(c) The final conclusion is [ANS] A. We can reject the null hypothesis that the average length of the shark is $19$. B. There is not sufficient evidence to reject the null hypothesis that the average length of the shark is $19$.", "answer_v1": [ "8.48528", "2.35336", "A" ], "answer_type_v1": [ "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "A", "B" ] ], "problem_v2": "One of the most feared predators in the ocean is the great white shark. It is known that the white shark grows to a mean length of $22$ feet; however, one marine biologist believes that great white sharks off the Bermuda coast grow much longer. To test this claim, full-grown white sharks were captured, measured, and then set free. However, this was a difficult, costly and very dangerous task, so only four sharks were actually sampled. Their lengths were $20, \\ 26, \\ 21, \\mbox{and} \\ 22$ feet. Do the data provide sufficient evidence to support the claim? Use $\\alpha=0.01$.\n(a) Calculate the test statistic: t=[ANS]\n(b) Find the critical value: t*=[ANS]\n(c) The final conclusion is [ANS] A. We can reject the null hypothesis that the average length of the shark is $22$. B. There is not sufficient evidence to reject the null hypothesis that the average length of the shark is $22$.", "answer_v2": [ "0.190117", "4.5407", "B" ], "answer_type_v2": [ "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "A", "B" ] ], "problem_v3": "One of the most feared predators in the ocean is the great white shark. It is known that the white shark grows to a mean length of $19$ feet; however, one marine biologist believes that great white sharks off the Bermuda coast grow much longer. To test this claim, full-grown white sharks were captured, measured, and then set free. However, this was a difficult, costly and very dangerous task, so only four sharks were actually sampled. Their lengths were $22, \\ 24, \\ 21, \\mbox{and} \\ 23$ feet. Do the data provide sufficient evidence to support the claim? Use $\\alpha=0.01$.\n(a) Calculate the test statistic: t=[ANS]\n(b) Find the critical value: t*=[ANS]\n(c) The final conclusion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that the average length of the shark is $19$. B. We can reject the null hypothesis that the average length of the shark is $19$.", "answer_v3": [ "5.42218", "4.5407", "B" ], "answer_type_v3": [ "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "A", "B" ] ] }, { "id": "Statistics_0253", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - t", "level": "3", "keywords": [ "statistics", "inference", "hypothesis testing", "t score" ], "problem_v1": "Justin is interested in buying a digital phone. He visited 16 stores at random and recorded the price of the particular phone he wants. The sample of prices had a mean of 295.4 and a standard deviation of 28.33.\n(a) What t-score should be used for a 95\\% confidence interval for the mean, $\\mu$, of the distribution? t*=[ANS]\n(b) Calculate a 95\\% confidence interval for the mean price of this model of digital phone: (Enter the smaller value in the left answer box.) [ANS] to [ANS]", "answer_v1": [ "2.131", "295.4-2.131*28.33/[sqrt(16)]", "295.4+2.131*28.33/[sqrt(16)]" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Justin is interested in buying a digital phone. He visited 9 stores at random and recorded the price of the particular phone he wants. The sample of prices had a mean of 382.92 and a standard deviation of 10.24.\n(a) What t-score should be used for a 95\\% confidence interval for the mean, $\\mu$, of the distribution? t*=[ANS]\n(b) Calculate a 95\\% confidence interval for the mean price of this model of digital phone: (Enter the smaller value in the left answer box.) [ANS] to [ANS]", "answer_v2": [ "2.306", "382.92-2.306*10.24/[sqrt(9)]", "382.92+2.306*10.24/[sqrt(9)]" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Justin is interested in buying a digital phone. He visited 11 stores at random and recorded the price of the particular phone he wants. The sample of prices had a mean of 301.36 and a standard deviation of 16.46.\n(a) What t-score should be used for a 95\\% confidence interval for the mean, $\\mu$, of the distribution? t*=[ANS]\n(b) Calculate a 95\\% confidence interval for the mean price of this model of digital phone: (Enter the smaller value in the left answer box.) [ANS] to [ANS]", "answer_v3": [ "2.228", "301.36-2.228*16.46/[sqrt(11)]", "301.36+2.228*16.46/[sqrt(11)]" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Statistics_0254", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - t", "level": "2", "keywords": [ "Statistics", "Hypothesis Testing", "small sample" ], "problem_v1": "The effectiveness of a new bug repellent is tested on $22$ subjects for a 10 hour period. (Assume normally distributed population.) Based on the number and location of the bug bites, the percentage of surface area exposed protected from bites was calculated for each of the subjects. The results were as follows: $\\overline{x}=94$, $\\ s=11$ The new repellent is considered effective if it provides a percent repellency of at least $91$. Using $\\alpha=0.01$, construct a hypothesis test with null hypothesis $\\mu=91$ and alternative hypothesis $\\mu>91$ to determine whether the mean repellency of the new bug repellent is greater than $91$ by computing the following:\n(a) $\\ $ the degree of freedom $\\ $ [ANS]\n(b) $\\ $ the critical $t$ value $\\ $ [ANS]\n(c) $\\ $ the test statistics $\\ $ [ANS]\nThe final conclusion is [ANS] A. We can reject the null hypothesis that $\\mu=91$. Our results indicate that the new bug repellent is effective. B. There is not sufficient evidence to reject the null hypothesis that $\\mu=91$. Our results do not provide enough evidence that the new bug repellent is effective.", "answer_v1": [ "21", "2.51765", "1.27920429813366", "B" ], "answer_type_v1": [ "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B" ] ], "problem_v2": "The effectiveness of a new bug repellent is tested on $12$ subjects for a 10 hour period. (Assume normally distributed population.) Based on the number and location of the bug bites, the percentage of surface area exposed protected from bites was calculated for each of the subjects. The results were as follows: $\\overline{x}=95$, $\\ s=6$ The new repellent is considered effective if it provides a percent repellency of at least $90$. Using $\\alpha=0.05$, construct a hypothesis test with null hypothesis $\\mu=90$ and alternative hypothesis $\\mu>90$ to determine whether the mean repellency of the new bug repellent is greater than $90$ by computing the following:\n(a) $\\ $ the degree of freedom $\\ $ [ANS]\n(b) $\\ $ the critical $t$ value $\\ $ [ANS]\n(c) $\\ $ the test statistics $\\ $ [ANS]\nThe final conclusion is [ANS] A. We can reject the null hypothesis that $\\mu=90$. Our results indicate that the new bug repellent is effective. B. There is not sufficient evidence to reject the null hypothesis that $\\mu=90$. Our results do not provide enough evidence that the new bug repellent is effective.", "answer_v2": [ "11", "1.79588", "2.88675134594813", "A" ], "answer_type_v2": [ "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B" ] ], "problem_v3": "The effectiveness of a new bug repellent is tested on $15$ subjects for a 10 hour period. (Assume normally distributed population.) Based on the number and location of the bug bites, the percentage of surface area exposed protected from bites was calculated for each of the subjects. The results were as follows: $\\overline{x}=94$, $\\ s=8$ The new repellent is considered effective if it provides a percent repellency of at least $90$. Using $\\alpha=0.01$, construct a hypothesis test with null hypothesis $\\mu=90$ and alternative hypothesis $\\mu>90$ to determine whether the mean repellency of the new bug repellent is greater than $90$ by computing the following:\n(a) $\\ $ the degree of freedom $\\ $ [ANS]\n(b) $\\ $ the critical $t$ value $\\ $ [ANS]\n(c) $\\ $ the test statistics $\\ $ [ANS]\nThe final conclusion is [ANS] A. We can reject the null hypothesis that $\\mu=90$. Our results indicate that the new bug repellent is effective. B. There is not sufficient evidence to reject the null hypothesis that $\\mu=90$. Our results do not provide enough evidence that the new bug repellent is effective.", "answer_v3": [ "14", "2.62449", "1.93649167310371", "B" ], "answer_type_v3": [ "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B" ] ] }, { "id": "Statistics_0258", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - t", "level": "3", "keywords": [ "statistics", "variance", "ANOVA", "2-sample t" ], "problem_v1": "Traffic police monitor the speed of vehicles as they travel over a new bridge. The average speed for a sample of $31$ vehicles was $82.72$ km/h, with the sample standard deviation being $4.61$ km/h. We will assume that the speeds are Normally distributed, and the police are interested in the mean speed. Part a) Since the variance of the underlying Normal distribution is not known, inference here would involve the t distribution. How many degrees of freedom would the relevant t distribution have? [ANS]\nPart b) Create a 95 \\% confidence interval for the mean speed of vehicles crossing the bridge. Give the upper and lower bounds to your interval, each to 2 decimal places. ([ANS])\nPart c) The police hypothesized that the mean speed of vehicles over the bridge would be the speed limit, 80 km/h. Taking a significance level of 5 \\%, what should infer about this hypothesis? [ANS] A. We should reject the hypothesis since the sample mean was not 80 km/h. B. We should not reject the hypothesis since 80 km/h is in the interval found in (b). C. We should reject the hypothesis since 80 km/h is not in the interval found in (b). D. We should reject the hypothesis since 80 km/h is in the interval found in (b). E. We should not reject the hypothesis since the sample mean is in the interval found in (b).\nPart d) Decreasing the significance level of the hypothesis test above would (select all that apply) [ANS] A. not change the Type I error probability. B. decrease the Type I error probability. C. increase the Type I error probability. D. either increase or decrease the Type I error probability. E. not change the Type II error probability.", "answer_v1": [ "30", "(81.03, 84.41)", "C", "B" ], "answer_type_v1": [ "NV", "INT", "MCS", "MCS" ], "options_v1": [ [], [], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Traffic police monitor the speed of vehicles as they travel over a new bridge. The average speed for a sample of $41$ vehicles was $81.35$ km/h, with the sample standard deviation being $4.52$ km/h. We will assume that the speeds are Normally distributed, and the police are interested in the mean speed. Part a) Since the variance of the underlying Normal distribution is not known, inference here would involve the t distribution. How many degrees of freedom would the relevant t distribution have? [ANS]\nPart b) Create a 95 \\% confidence interval for the mean speed of vehicles crossing the bridge. Give the upper and lower bounds to your interval, each to 2 decimal places. ([ANS])\nPart c) The police hypothesized that the mean speed of vehicles over the bridge would be the speed limit, 80 km/h. Taking a significance level of 5 \\%, what should infer about this hypothesis? [ANS] A. We should reject the hypothesis since 80 km/h is not in the interval found in (b). B. We should not reject the hypothesis since the sample mean is in the interval found in (b). C. We should reject the hypothesis since 80 km/h is in the interval found in (b). D. We should not reject the hypothesis since 80 km/h is in the interval found in (b). E. We should reject the hypothesis since the sample mean was not 80 km/h.\nPart d) Decreasing the significance level of the hypothesis test above would (select all that apply) [ANS] A. increase the Type I error probability. B. not change the Type I error probability. C. decrease the Type I error probability. D. not change the Type II error probability. E. either increase or decrease the Type I error probability.", "answer_v2": [ "40", "(79.92, 82.78)", "D", "C" ], "answer_type_v2": [ "NV", "INT", "MCS", "MCS" ], "options_v2": [ [], [], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Traffic police monitor the speed of vehicles as they travel over a new bridge. The average speed for a sample of $31$ vehicles was $82.72$ km/h, with the sample standard deviation being $4.61$ km/h. We will assume that the speeds are Normally distributed, and the police are interested in the mean speed. Part a) Since the variance of the underlying Normal distribution is not known, inference here would involve the t distribution. How many degrees of freedom would the relevant t distribution have? [ANS]\nPart b) Create a 95 \\% confidence interval for the mean speed of vehicles crossing the bridge. Give the upper and lower bounds to your interval, each to 2 decimal places. ([ANS])\nPart c) The police hypothesized that the mean speed of vehicles over the bridge would be the speed limit, 80 km/h. Taking a significance level of 5 \\%, what should infer about this hypothesis? [ANS] A. We should not reject the hypothesis since 80 km/h is in the interval found in (b). B. We should reject the hypothesis since 80 km/h is not in the interval found in (b). C. We should reject the hypothesis since the sample mean was not 80 km/h. D. We should reject the hypothesis since 80 km/h is in the interval found in (b). E. We should not reject the hypothesis since the sample mean is in the interval found in (b).\nPart d) Decreasing the significance level of the hypothesis test above would (select all that apply) [ANS] A. not change the Type I error probability. B. increase the Type I error probability. C. not change the Type II error probability. D. decrease the Type I error probability. E. either increase or decrease the Type I error probability.", "answer_v3": [ "30", "(81.03, 84.41)", "B", "D" ], "answer_type_v3": [ "NV", "INT", "MCS", "MCS" ], "options_v3": [ [], [], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Statistics_0259", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - t", "level": "2", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "A manufacturer of light bulbs advertises that, on average, its long-life bulb will last more than 5300 hours. To test this claim, a statistician took a random sample of 103 bulbs and measured the amount of time until each bulb burned out. The mean lifetime of the sample of bulbs is 5360 hours and has a standard deviation of 450 hours. Can we conclude with 93\\% confidence that the claim is true? Fill in the requested information below.\nA. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-inf, a), an answer of the form $(b, \\infty)$ is expressed (b, inf), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-inf, a)U(b, inf). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Reject $H_0$. B. Reject $H_1$. C. Do Not Reject $H_1$. D. Do Not Reject $H_0$.", "answer_v1": [ "1.3466", "(1.48738,infinity)", "0.0905468", "D" ], "answer_type_v1": [ "NV", "INT", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v2": "A manufacturer of light bulbs advertises that, on average, its long-life bulb will last more than 4800 hours. To test this claim, a statistician took a random sample of 96 bulbs and measured the amount of time until each bulb burned out. The mean lifetime of the sample of bulbs is 4822 hours and has a standard deviation of 410 hours. Can we conclude with 90\\% confidence that the claim is true? Fill in the requested information below.\nA. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-inf, a), an answer of the form $(b, \\infty)$ is expressed (b, inf), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-inf, a)U(b, inf). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Do Not Reject $H_1$. B. Reject $H_1$. C. Do Not Reject $H_0$. D. Reject $H_0$.", "answer_v2": [ "0.522999", "(1.29053,infinity)", "0.301096", "C" ], "answer_type_v2": [ "NV", "INT", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v3": "A manufacturer of light bulbs advertises that, on average, its long-life bulb will last more than 4800 hours. To test this claim, a statistician took a random sample of 95 bulbs and measured the amount of time until each bulb burned out. The mean lifetime of the sample of bulbs is 4863 hours and has a standard deviation of 410 hours. Can we conclude with 92\\% confidence that the claim is true? Fill in the requested information below.\nA. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-inf, a), an answer of the form $(b, \\infty)$ is expressed (b, inf), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-inf, a)U(b, inf). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Reject $H_1$. B. Reject $H_0$. C. Do Not Reject $H_1$. D. Do Not Reject $H_0$.", "answer_v3": [ "1.48977", "(1.41628,infinity)", "0.0698156", "B" ], "answer_type_v3": [ "NV", "INT", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0260", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - t", "level": "2", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "The hypothesis test $ \\qquad \\begin{array}{c} H_0: \\mu=31 \\\\ H_1: \\mu \\not=31\\\\ \\end{array}$ is to be carried out. A random sample is selected, and yields $\\bar{x}=35$ and s=12. If the value of the t statistic is $t=1.76383420737639$, what is the sample size? (If rounding is required, round to the nearest integer.) Sample Size=[ANS]", "answer_v1": [ "28" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The hypothesis test $ \\qquad \\begin{array}{c} H_0: \\mu=16 \\\\ H_1: \\mu \\not=16\\\\ \\end{array}$ is to be carried out. A random sample is selected, and yields $\\bar{x}=21$ and s=8. If the value of the t statistic is $t=2.65165042944955$, what is the sample size? (If rounding is required, round to the nearest integer.) Sample Size=[ANS]", "answer_v2": [ "18" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The hypothesis test $ \\qquad \\begin{array}{c} H_0: \\mu=22 \\\\ H_1: \\mu \\not=22\\\\ \\end{array}$ is to be carried out. A random sample is selected, and yields $\\bar{x}=26$ and s=11. If the value of the t statistic is $t=1.62623125636348$, what is the sample size? (If rounding is required, round to the nearest integer.) Sample Size=[ANS]", "answer_v3": [ "20" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0261", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample mean - t", "level": "4", "keywords": [ "statistics", "Inference about a population" ], "problem_v1": "A federal agency responsible for enforcing laws governing weights and measures routinely inspects packages to determine whether the weight of the contents is as advertised on the package. The agency will only take action if there is strong evidence that the packages are underfilled. A random sample of 14 containers was taken and the contents weighed, with the weights given below. The containers are labeled to have 10 ounces. In your role as a member of this crack federal team, you are to specify and carry out an appropriate hypothesis test to determine whether there is enough evidence to support this claim. (Use a 2\\% significance level.)\n\\begin{array}{ccccccc} 9.59 & 9.61 & 9.77 & 9.86 & 10.05 & 9.82 & 9.72 \\\\ 9.82 & 10.13 & 10.2 & 9.69 & 9.52 & 9.68 & 9.88 \\\\ \\end{array} A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Do Not Reject $H_1$. B. Reject $H_0$. C. Reject $H_1$. D. Do Not Reject $H_0$.", "answer_v1": [ "-3.51029992328117", "(-infinity,-2.2816)", "0.00191875", "B" ], "answer_type_v1": [ "NV", "INT", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v2": "A federal agency responsible for enforcing laws governing weights and measures routinely inspects packages to determine whether the weight of the contents is as advertised on the package. The agency will only take action if there is strong evidence that the packages are underfilled. A random sample of 14 containers was taken and the contents weighed, with the weights given below. The containers are labeled to have 8 ounces. In your role as a member of this crack federal team, you are to specify and carry out an appropriate hypothesis test to determine whether there is enough evidence to support this claim. (Use a 3\\% significance level.)\n\\begin{array}{ccccccc} 8.16 & 7.6 & 7.73 & 8.17 & 7.72 & 7.62 & 7.73 \\\\ 7.9 & 7.54 & 7.95 & 7.81 & 8.07 & 7.63 & 7.63 \\\\ \\end{array} A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Do Not Reject $H_1$. B. Reject $H_0$. C. Do Not Reject $H_0$. D. Reject $H_1$.", "answer_v2": [ "-3.44988581138629", "(-infinity,-2.06004)", "0.00215453", "B" ], "answer_type_v2": [ "NV", "INT", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v3": "A federal agency responsible for enforcing laws governing weights and measures routinely inspects packages to determine whether the weight of the contents is as advertised on the package. The agency will only take action if there is strong evidence that the packages are underfilled. A random sample of 14 containers was taken and the contents weighed, with the weights given below. The containers are labeled to have 9 ounces. In your role as a member of this crack federal team, you are to specify and carry out an appropriate hypothesis test to determine whether there is enough evidence to support this claim. (Use a 8\\% significance level.)\n\\begin{array}{ccccccc} 8.92 & 8.69 & 8.88 & 8.64 & 8.74 & 9.07 & 9.14 \\\\ 9.11 & 8.64 & 8.7 & 8.67 & 8.53 & 8.9 & 9.19 \\\\ \\end{array} A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Do Not Reject $H_0$. B. Reject $H_1$. C. Reject $H_0$. D. Do Not Reject $H_1$.", "answer_v3": [ "-2.6968811996332", "(-infinity,-1.49032)", "0.00915124", "C" ], "answer_type_v3": [ "NV", "INT", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0262", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample proportion", "level": "2", "keywords": [ "statistics", "hypothesis testing", "Inference", "one proportion" ], "problem_v1": "A survey of $1755$ people who took trips revealed that $163$ of them included a visit to a theme park. Based on those survery results, a management consultant claims that less than $10$ \\% of trips include a theme park visit. Test this claim using the $\\alpha=0.05$ significance level. The test statistic is [ANS]\nThe critical value is [ANS]\nThe conclusion is [ANS] A. There is sufficient evidence to support the claim that less than 10 \\% of trips include a theme park visit. B. There is not sufficient evidence to support the claim that less than 10 \\% of trips include a theme park visit.", "answer_v1": [ "-0.994603991721436", "-1.64485", "B" ], "answer_type_v1": [ "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "A", "B" ] ], "problem_v2": "A survey of $1080$ people who took trips revealed that $107$ of them included a visit to a theme park. Based on those survery results, a management consultant claims that less than $12$ \\% of trips include a theme park visit. Test this claim using the $\\alpha=0.01$ significance level. The test statistic is [ANS]\nThe critical value is [ANS]\nThe conclusion is [ANS] A. There is not sufficient evidence to support the claim that less than 12 \\% of trips include a theme park visit. B. There is sufficient evidence to support the claim that less than 12 \\% of trips include a theme park visit.", "answer_v2": [ "-2.11623845717629", "-2.32635", "A" ], "answer_type_v2": [ "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "A", "B" ] ], "problem_v3": "A survey of $1315$ people who took trips revealed that $124$ of them included a visit to a theme park. Based on those survery results, a management consultant claims that less than $11$ \\% of trips include a theme park visit. Test this claim using the $\\alpha=0.05$ significance level. The test statistic is [ANS]\nThe critical value is [ANS]\nThe conclusion is [ANS] A. There is not sufficient evidence to support the claim that less than 11 \\% of trips include a theme park visit. B. There is sufficient evidence to support the claim that less than 11 \\% of trips include a theme park visit.", "answer_v3": [ "-1.81997672368698", "-1.64485", "B" ], "answer_type_v3": [ "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "A", "B" ] ] }, { "id": "Statistics_0263", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample proportion", "level": "2", "keywords": [ "Test Statistic", "z Score", "hypothesis testing", "population proportion", "statistics" ], "problem_v1": "A random sample of $140$ observations is selected from a binomial population with unknown probability of success $p$. The computed value of $\\hat{p}$ is $0.72$. (1) $\\ $ Test $H_0: p=0.6$ against $H_a: p > 0.6$. Use $\\alpha=0.05$. test statistic $z=$ [ANS]\ncritical $z$ score $\\ $ [ANS]\nThe final conclusion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that $p=0.6$. B. We can reject the null hypothesis that $p=0.6$ and accept that $p > 0.6$.\n(2) $\\ $ Test $H_0: p=0.55$ against $H_a: p < 0.55$. Use $\\alpha=0.01$. test statistic $z=$ [ANS]\ncritical $z$ score $\\ $ [ANS]\nThe final conclusion is [ANS] A. We can reject the null hypothesis that $p=0.55$ and accept that $p < 0.55$. B. There is not sufficient evidence to reject the null hypothesis that $p=0.55$.\n(3) $\\ $ Test $H_0: p=0.6$ against $H_a: p \\ne 0.6$. Use $\\alpha=0.05$. test statistic $z=$ [ANS]\npositive critical $z$ score $\\ $ [ANS]\nnegative critical $z$ score $\\ $ [ANS]\nThe final conclusion is [ANS] A. We can reject the null hypothesis that $p=0.6$ and accept that $p \\ne 0.6$. B. There is not sufficient evidence to reject the null hypothesis that $p=0.6$.", "answer_v1": [ "2.89827534923789", "1.64485", "B", "4.04320105207183", "-2.32635", "B", "2.89827534923789", "1.95996", "-1.95996", "A" ], "answer_type_v1": [ "NV", "NV", "MCS", "NV", "NV", "MCS", "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "A", "B" ], [], [], [ "A", "B" ], [], [], [], [ "A", "B" ] ], "problem_v2": "A random sample of $90$ observations is selected from a binomial population with unknown probability of success $p$. The computed value of $\\hat{p}$ is $0.79$. (1) $\\ $ Test $H_0: p=0.5$ against $H_a: p > 0.5$. Use $\\alpha=0.01$. test statistic $z=$ [ANS]\ncritical $z$ score $\\ $ [ANS]\nThe final conclusion is [ANS] A. We can reject the null hypothesis that $p=0.5$ and accept that $p > 0.5$. B. There is not sufficient evidence to reject the null hypothesis that $p=0.5$.\n(2) $\\ $ Test $H_0: p=0.65$ against $H_a: p < 0.65$. Use $\\alpha=0.01$. test statistic $z=$ [ANS]\ncritical $z$ score $\\ $ [ANS]\nThe final conclusion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that $p=0.65$. B. We can reject the null hypothesis that $p=0.65$ and accept that $p < 0.65$.\n(3) $\\ $ Test $H_0: p=0.5$ against $H_a: p \\ne 0.5$. Use $\\alpha=0.01$. test statistic $z=$ [ANS]\npositive critical $z$ score $\\ $ [ANS]\nnegative critical $z$ score $\\ $ [ANS]\nThe final conclusion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that $p=0.5$. B. We can reject the null hypothesis that $p=0.5$ and accept that $p \\ne 0.5$.", "answer_v2": [ "5.50236312869298", "2.32635", "A", "2.78457288535354", "-2.32635", "A", "5.50236312869298", "2.57583", "-2.57583", "B" ], "answer_type_v2": [ "NV", "NV", "MCS", "NV", "NV", "MCS", "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "A", "B" ], [], [], [ "A", "B" ], [], [], [], [ "A", "B" ] ], "problem_v3": "A random sample of $110$ observations is selected from a binomial population with unknown probability of success $p$. The computed value of $\\hat{p}$ is $0.72$. (1) $\\ $ Test $H_0: p=0.55$ against $H_a: p > 0.55$. Use $\\alpha=0.05$. test statistic $z=$ [ANS]\ncritical $z$ score $\\ $ [ANS]\nThe final conclusion is [ANS] A. We can reject the null hypothesis that $p=0.55$ and accept that $p > 0.55$. B. There is not sufficient evidence to reject the null hypothesis that $p=0.55$.\n(2) $\\ $ Test $H_0: p=0.5$ against $H_a: p < 0.5$. Use $\\alpha=0.01$. test statistic $z=$ [ANS]\ncritical $z$ score $\\ $ [ANS]\nThe final conclusion is [ANS] A. We can reject the null hypothesis that $p=0.5$ and accept that $p < 0.5$. B. There is not sufficient evidence to reject the null hypothesis that $p=0.5$.\n(3) $\\ $ Test $H_0: p=0.65$ against $H_a: p \\ne 0.65$. Use $\\alpha=0.05$. test statistic $z=$ [ANS]\npositive critical $z$ score $\\ $ [ANS]\nnegative critical $z$ score $\\ $ [ANS]\nThe final conclusion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that $p=0.65$. B. We can reject the null hypothesis that $p=0.65$ and accept that $p \\ne 0.65$.", "answer_v3": [ "3.58391468152416", "1.64485", "A", "4.61475893194867", "-2.32635", "B", "1.53923057701917", "1.95996", "-1.95996", "A" ], "answer_type_v3": [ "NV", "NV", "MCS", "NV", "NV", "MCS", "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "A", "B" ], [], [], [ "A", "B" ], [], [], [], [ "A", "B" ] ] }, { "id": "Statistics_0264", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample proportion", "level": "3", "keywords": [ "statistics", "Inference", "one proportion" ], "problem_v1": "Physicians at a clinic gave what they thought were drugs to $950$ asthma, ulcer, and herpes patients. Although the doctors later learned that the drugs were really placebos, $55$ \\% of the patients reported an improved condition. Assume that if the placebo is ineffective, the probability of a patients condition improving is $0.53$. For the hypotheses that the proportion of improving is $0.53$ against that it is $>0.53$, find the p-value. $p=$ [ANS]", "answer_v1": [ "0.108395" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Physicians at a clinic gave what they thought were drugs to $810$ asthma, ulcer, and herpes patients. Although the doctors later learned that the drugs were really placebos, $59$ \\% of the patients reported an improved condition. Assume that if the placebo is ineffective, the probability of a patients condition improving is $0.54$. For the hypotheses that the proportion of improving is $0.54$ against that it is $>0.54$, find the p-value. $p=$ [ANS]", "answer_v2": [ "0.00215048" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Physicians at a clinic gave what they thought were drugs to $860$ asthma, ulcer, and herpes patients. Although the doctors later learned that the drugs were really placebos, $56$ \\% of the patients reported an improved condition. Assume that if the placebo is ineffective, the probability of a patients condition improving is $0.52$. For the hypotheses that the proportion of improving is $0.52$ against that it is $>0.52$, find the p-value. $p=$ [ANS]", "answer_v3": [ "0.00943879" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0265", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample proportion", "level": "2", "keywords": [], "problem_v1": "According to a recent marketing campaign, $120$ drinkers of either Diet Coke or Diet Pepsi participated in a blind taste test to see which of the drinks was their favorite. In one Pepsi television commercial, an anouncer states that \"in recent blind taste tests, more than one half of the surveyed preferred Diet Pepsi over Diet Coke.\" Suppose that out of those $120$, $52$ preferred Diet Pepsi. Test the hypothesis, using $\\alpha=0.05$ that more than half of all participants will select Diet Pepsi in a blind taste test by giving the following:\n(a) $\\ $ the test statistic $\\ $ [ANS]\n(b) $\\ $ the critical $z$ score $\\ $ [ANS]\nThe final conclustion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that $p \\le 0.5$. B. We can reject the null hypothesis that $p \\le 0.5$ and accept that $p > 0.5$.", "answer_v1": [ "-1.46059348668044", "1.64485", "A" ], "answer_type_v1": [ "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "A", "B" ] ], "problem_v2": "According to a recent marketing campaign, $90$ drinkers of either Diet Coke or Diet Pepsi participated in a blind taste test to see which of the drinks was their favorite. In one Pepsi television commercial, an anouncer states that \"in recent blind taste tests, more than one half of the surveyed preferred Diet Pepsi over Diet Coke.\" Suppose that out of those $90$, $59$ preferred Diet Pepsi. Test the hypothesis, using $\\alpha=0.01$ that more than half of all participants will select Diet Pepsi in a blind taste test by giving the following:\n(a) $\\ $ the test statistic $\\ $ [ANS]\n(b) $\\ $ the critical $z$ score $\\ $ [ANS]\nThe final conclustion is [ANS] A. We can reject the null hypothesis that $p \\le 0.5$ and accept that $p > 0.5$. B. There is not sufficient evidence to reject the null hypothesis that $p \\le 0.5$.", "answer_v2": [ "2.95145914949049", "2.32635", "A" ], "answer_type_v2": [ "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "A", "B" ] ], "problem_v3": "According to a recent marketing campaign, $100$ drinkers of either Diet Coke or Diet Pepsi participated in a blind taste test to see which of the drinks was their favorite. In one Pepsi television commercial, an anouncer states that \"in recent blind taste tests, more than one half of the surveyed preferred Diet Pepsi over Diet Coke.\" Suppose that out of those $100$, $52$ preferred Diet Pepsi. Test the hypothesis, using $\\alpha=0.01$ that more than half of all participants will select Diet Pepsi in a blind taste test by giving the following:\n(a) $\\ $ the test statistic $\\ $ [ANS]\n(b) $\\ $ the critical $z$ score $\\ $ [ANS]\nThe final conclustion is [ANS] A. We can reject the null hypothesis that $p \\le 0.5$ and accept that $p > 0.5$. B. There is not sufficient evidence to reject the null hypothesis that $p \\le 0.5$.", "answer_v3": [ "0.4", "2.32635", "B" ], "answer_type_v3": [ "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "A", "B" ] ] }, { "id": "Statistics_0267", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample proportion", "level": "3", "keywords": [ "statistics", "Inference", "one proportion" ], "problem_v1": "A newspaper conducted a statewide survey concerning the 1998 race for state senator. The newspaper took a SRS of $n=1300$ registered voters and found that 670 would vote for the Republican candidate. Let $p$ represent the proportion of registered voters in the state who would vote for the Republican candidate. We test $H_0: p=.50$ $H_a: p >.50$\n(a) What is the $z$-statistic for this test? [ANS]\n(b) What is the P-value of the test? [ANS]", "answer_v1": [ "1.11", "0.1335" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A newspaper conducted a statewide survey concerning the 1998 race for state senator. The newspaper took a SRS of $n=1000$ registered voters and found that 520 would vote for the Republican candidate. Let $p$ represent the proportion of registered voters in the state who would vote for the Republican candidate. We test $H_0: p=.50$ $H_a: p >.50$\n(a) What is the $z$-statistic for this test? [ANS]\n(b) What is the P-value of the test? [ANS]", "answer_v2": [ "1.26", "0.1038" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A newspaper conducted a statewide survey concerning the 1998 race for state senator. The newspaper took a SRS of $n=1100$ registered voters and found that 570 would vote for the Republican candidate. Let $p$ represent the proportion of registered voters in the state who would vote for the Republican candidate. We test $H_0: p=.50$ $H_a: p >.50$\n(a) What is the $z$-statistic for this test? [ANS]\n(b) What is the P-value of the test? [ANS]", "answer_v3": [ "1.21", "0.1131" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0268", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample proportion", "level": "3", "keywords": [ "statistics", "Inference", "one proportion" ], "problem_v1": "A noted psychic was tested for ESP. The psychic was presented with 240 cards face down and was asked to determine if the card was one of 5 symbols: a star, cross, circle, square, or three wavy lines. The psychic was correct in 65 cases. Let $p$ represent the probability that the psychic correctly identifies the symbol on the card in a random trial. Assume the 240 trials can be treated as an SRS from the population of all guesses. To see if there is evidence that the psychic is doing better than just guessing, we test $H_0: p=.2$ $H_a: p >.2$\n(a) What is the $z$-statistic for this test? [ANS]\n(b) What is the P-value of the test? [ANS]", "answer_v1": [ "2.74336", "0.0031" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A noted psychic was tested for ESP. The psychic was presented with 180 cards face down and was asked to determine if the card was one of 5 symbols: a star, cross, circle, square, or three wavy lines. The psychic was correct in 43 cases. Let $p$ represent the probability that the psychic correctly identifies the symbol on the card in a random trial. Assume the 180 trials can be treated as an SRS from the population of all guesses. To see if there is evidence that the psychic is doing better than just guessing, we test $H_0: p=.2$ $H_a: p >.2$\n(a) What is the $z$-statistic for this test? [ANS]\n(b) What is the P-value of the test? [ANS]", "answer_v2": [ "1.30437", "0.0968" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A noted psychic was tested for ESP. The psychic was presented with 200 cards face down and was asked to determine if the card was one of 5 symbols: a star, cross, circle, square, or three wavy lines. The psychic was correct in 49 cases. Let $p$ represent the probability that the psychic correctly identifies the symbol on the card in a random trial. Assume the 200 trials can be treated as an SRS from the population of all guesses. To see if there is evidence that the psychic is doing better than just guessing, we test $H_0: p=.2$ $H_a: p >.2$\n(a) What is the $z$-statistic for this test? [ANS]\n(b) What is the P-value of the test? [ANS]", "answer_v3": [ "1.59099", "0.0559" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0269", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample proportion", "level": "2", "keywords": [ "hypothesis testing", "population proportion", "statistics" ], "problem_v1": "A recent poll of $2400$ randomly selected 18-25-year-olds revealed that $278$ currently use marijuana or hashish. According to a publication, $12.7$ \\% of 18-25-year-olds were current users of marijuana or hashish in 1997. Do the data provide sufficient evidence to conclude that the percentage of 18-25-year-olds who currently use marijuana or hashish has changed from the 1997 percentage of $12.7$ \\%? Use $\\alpha=0.05$ significance level. test statistic $z=$ [ANS]\npositive critical $z$ score $\\ $ [ANS]\nnegative critical $z$ score $\\ $ [ANS]\nThe final conclusion is [ANS] A. There is not sufficient evidence to conclude that the percentage of 18-25-year-olds who currently use marijuana or hashish has changed from the 1997 percentage of $12.7$ \\%. B. There is sufficient evidence to conclude that the percentage of 18-25-year-olds who currently use marijuana or hashish has changed from the 1997 percentage of $12.7$ \\%.", "answer_v1": [ "-1.64293427823668", "1.95996", "-1.95996", "A" ], "answer_type_v1": [ "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B" ] ], "problem_v2": "A recent poll of $1900$ randomly selected 18-25-year-olds revealed that $220$ currently use marijuana or hashish. According to a publication, $12.5$ \\% of 18-25-year-olds were current users of marijuana or hashish in 1997. Do the data provide sufficient evidence to conclude that the percentage of 18-25-year-olds who currently use marijuana or hashish has changed from the 1997 percentage of $12.5$ \\%? Use $\\alpha=0.05$ significance level. test statistic $z=$ [ANS]\npositive critical $z$ score $\\ $ [ANS]\nnegative critical $z$ score $\\ $ [ANS]\nThe final conclusion is [ANS] A. There is not sufficient evidence to conclude that the percentage of 18-25-year-olds who currently use marijuana or hashish has changed from the 1997 percentage of $12.5$ \\%. B. There is sufficient evidence to conclude that the percentage of 18-25-year-olds who currently use marijuana or hashish has changed from the 1997 percentage of $12.5$ \\%.", "answer_v2": [ "-1.21395395733377", "1.95996", "-1.95996", "A" ], "answer_type_v2": [ "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B" ] ], "problem_v3": "A recent poll of $2100$ randomly selected 18-25-year-olds revealed that $243$ currently use marijuana or hashish. According to a publication, $12.6$ \\% of 18-25-year-olds were current users of marijuana or hashish in 1997. Do the data provide sufficient evidence to conclude that the percentage of 18-25-year-olds who currently use marijuana or hashish has changed from the 1997 percentage of $12.6$ \\%? Use $\\alpha=0.05$ significance level. test statistic $z=$ [ANS]\npositive critical $z$ score $\\ $ [ANS]\nnegative critical $z$ score $\\ $ [ANS]\nThe final conclusion is [ANS] A. There is sufficient evidence to conclude that the percentage of 18-25-year-olds who currently use marijuana or hashish has changed from the 1997 percentage of $12.6$ \\%. B. There is not sufficient evidence to conclude that the percentage of 18-25-year-olds who currently use marijuana or hashish has changed from the 1997 percentage of $12.6$ \\%.", "answer_v3": [ "-1.42037531142713", "1.95996", "-1.95996", "B" ], "answer_type_v3": [ "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B" ] ] }, { "id": "Statistics_0270", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample proportion", "level": "3", "keywords": [ "statistics", "statistical inference", "hypothesis test", "p-value", "interpretatin" ], "problem_v1": "Prospective drivers who enrol in Smart Driver Driving School have always been taught by a conventional teaching method. The driving school has many branches across provinces. Last year, among all students that took driving lessons from the school in a certain province, 80\\% passed the provincial road test. This year, the teaching committee came up with a new teaching method. The committee randomly assigned half of its 2400 students enrolled this year to receive the conventional teaching method and the remaining half to receive the new teaching method. In a random sample of 100 students who received the conventional teaching method, 78\\% passed the road test.\nPart i) To test if the passing rate has decreased from last year for students who received the conventional teaching method, what will be the null hypothesis? [ANS] A. The proportion of 1200 students who received the conventional teaching method and subsequently passed the road test this year equals 0.78. B. The proportion of 100 students who received the conventional teaching method and subsequently passed the road test this year is lower than 0.80. C. The proportion of 1200 students who received the conventional teaching method and subsequently passed the road test this year equals 0.80. D. The proportion of 100 students who received the conventional teaching method and subsequently passed the road test this year equals 0.78. E. The proportion of 100 students who received the conventional teaching method and subsequently passed the road test this year equals 0.80. F. The proportion of 1200 students who received the conventional teaching method and subsequently passed the road test this year is lower than 0.80.\nPart ii) For the test mentioned in the previous part, what will be the alternative hypothesis? [ANS] A. The proportion of 1200 students who received the conventional teaching method and subsequently passed the road test this year equals 0.80. B. The proportion of 100 students who received the conventional teaching method and subsequently passed the road test this year equals 0.78. C. The proportion of 100 students who received the conventional teaching method and subsequently passed the road test this year equals 0.80. D. The proportion of 1200 students who received the conventional teaching method and subsequently passed the road test this year equals 0.78. E. The proportion of 1200 students who received the conventional teaching method and subsequently passed the road test this year is lower than 0.80. F. The proportion of 100 students who received the conventional teaching method and subsequently passed the road test this year is lower than 0.80.\nPart iii) What is the approximate null model for the sample proportion of the conventional teaching group who passed the road test? [ANS] A. $N\\left(0.78, \\sqrt{\\frac{0.78 \\,\\cdot\\, 0.22}{1200}} \\right)$. B. $N\\left(0.80, \\sqrt{\\frac{0.78 \\,\\cdot\\, 0.22}{100}} \\right)$. C. $N\\left(0.80, \\sqrt{\\frac{0.8 \\,\\cdot\\, 0.2}{100}} \\right)$. D. $N\\left(0.78, \\sqrt{\\frac{0.78 \\,\\cdot\\, 0.22}{100}} \\right)$. E. $N\\left(0.78, \\sqrt{\\frac{0.8 \\cdot 0.2}{100}} \\right)$. F. $N\\left(0.80, \\sqrt{\\frac{0.8 \\,\\cdot\\, 0.2}{1200}} \\right)$.\nPart iv) Compute the $P$-value: [ANS] (your answer must be expressed as a proportion and rounded to 4 decimal places.)\nPart v) What is an appropriate conclusion for the hypothesis test at the 2\\% significance level? [ANS] A. The passing rate for students taught using the conventional method this year is significantly lower than last year\u2019s. B. The passing rate for students taught using the conventional method this year is not significantly lower than last year\u2019s. C. The passing rate for students taught using the conventional method this year is the same as last year\u2019s. D. Both (b) and (c).", "answer_v1": [ "C", "E", "C", "0.308538", "B" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "NV", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [], [ "A", "B", "C", "D" ] ], "problem_v2": "Prospective drivers who enrol in Smart Driver Driving School have always been taught by a conventional teaching method. The driving school has many branches across provinces. Last year, among all students that took driving lessons from the school in a certain province, 80\\% passed the provincial road test. This year, the teaching committee came up with a new teaching method. The committee randomly assigned half of its 2400 students enrolled this year to receive the conventional teaching method and the remaining half to receive the new teaching method. In a random sample of 100 students who received the conventional teaching method, 75\\% passed the road test.\nPart i) To test if the passing rate has decreased from last year for students who received the conventional teaching method, what will be the null hypothesis? [ANS] A. The proportion of 1200 students who received the conventional teaching method and subsequently passed the road test this year equals 0.80. B. The proportion of 100 students who received the conventional teaching method and subsequently passed the road test this year is lower than 0.80. C. The proportion of 1200 students who received the conventional teaching method and subsequently passed the road test this year is lower than 0.80. D. The proportion of 100 students who received the conventional teaching method and subsequently passed the road test this year equals 0.80. E. The proportion of 1200 students who received the conventional teaching method and subsequently passed the road test this year equals 0.75. F. The proportion of 100 students who received the conventional teaching method and subsequently passed the road test this year equals 0.75.\nPart ii) For the test mentioned in the previous part, what will be the alternative hypothesis? [ANS] A. The proportion of 100 students who received the conventional teaching method and subsequently passed the road test this year is lower than 0.80. B. The proportion of 1200 students who received the conventional teaching method and subsequently passed the road test this year equals 0.75. C. The proportion of 1200 students who received the conventional teaching method and subsequently passed the road test this year equals 0.80. D. The proportion of 100 students who received the conventional teaching method and subsequently passed the road test this year equals 0.75. E. The proportion of 1200 students who received the conventional teaching method and subsequently passed the road test this year is lower than 0.80. F. The proportion of 100 students who received the conventional teaching method and subsequently passed the road test this year equals 0.80.\nPart iii) What is the approximate null model for the sample proportion of the conventional teaching group who passed the road test? [ANS] A. $N\\left(0.80, \\sqrt{\\frac{0.8 \\,\\cdot\\, 0.2}{1200}} \\right)$. B. $N\\left(0.80, \\sqrt{\\frac{0.75 \\,\\cdot\\, 0.25}{100}} \\right)$. C. $N\\left(0.80, \\sqrt{\\frac{0.8 \\,\\cdot\\, 0.2}{100}} \\right)$. D. $N\\left(0.75, \\sqrt{\\frac{0.8 \\cdot 0.2}{100}} \\right)$. E. $N\\left(0.75, \\sqrt{\\frac{0.75 \\,\\cdot\\, 0.25}{1200}} \\right)$. F. $N\\left(0.75, \\sqrt{\\frac{0.75 \\,\\cdot\\, 0.25}{100}} \\right)$.\nPart iv) Compute the $P$-value: [ANS] (your answer must be expressed as a proportion and rounded to 4 decimal places.)\nPart v) What is an appropriate conclusion for the hypothesis test at the 2\\% significance level? [ANS] A. The passing rate for students taught using the conventional method this year is significantly lower than last year\u2019s. B. The passing rate for students taught using the conventional method this year is not significantly lower than last year\u2019s. C. The passing rate for students taught using the conventional method this year is the same as last year\u2019s. D. Both (b) and (c).", "answer_v2": [ "A", "E", "C", "0.10565", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "NV", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [], [ "A", "B", "C", "D" ] ], "problem_v3": "Prospective drivers who enrol in Smart Driver Driving School have always been taught by a conventional teaching method. The driving school has many branches across provinces. Last year, among all students that took driving lessons from the school in a certain province, 80\\% passed the provincial road test. This year, the teaching committee came up with a new teaching method. The committee randomly assigned half of its 2400 students enrolled this year to receive the conventional teaching method and the remaining half to receive the new teaching method. In a random sample of 100 students who received the conventional teaching method, 76\\% passed the road test.\nPart i) To test if the passing rate has decreased from last year for students who received the conventional teaching method, what will be the null hypothesis? [ANS] A. The proportion of 1200 students who received the conventional teaching method and subsequently passed the road test this year is lower than 0.80. B. The proportion of 100 students who received the conventional teaching method and subsequently passed the road test this year is lower than 0.80. C. The proportion of 100 students who received the conventional teaching method and subsequently passed the road test this year equals 0.80. D. The proportion of 1200 students who received the conventional teaching method and subsequently passed the road test this year equals 0.80. E. The proportion of 1200 students who received the conventional teaching method and subsequently passed the road test this year equals 0.76. F. The proportion of 100 students who received the conventional teaching method and subsequently passed the road test this year equals 0.76.\nPart ii) For the test mentioned in the previous part, what will be the alternative hypothesis? [ANS] A. The proportion of 1200 students who received the conventional teaching method and subsequently passed the road test this year equals 0.76. B. The proportion of 100 students who received the conventional teaching method and subsequently passed the road test this year is lower than 0.80. C. The proportion of 1200 students who received the conventional teaching method and subsequently passed the road test this year is lower than 0.80. D. The proportion of 100 students who received the conventional teaching method and subsequently passed the road test this year equals 0.76. E. The proportion of 1200 students who received the conventional teaching method and subsequently passed the road test this year equals 0.80. F. The proportion of 100 students who received the conventional teaching method and subsequently passed the road test this year equals 0.80.\nPart iii) What is the approximate null model for the sample proportion of the conventional teaching group who passed the road test? [ANS] A. $N\\left(0.76, \\sqrt{\\frac{0.76 \\,\\cdot\\, 0.24}{1200}} \\right)$. B. $N\\left(0.76, \\sqrt{\\frac{0.8 \\cdot 0.2}{100}} \\right)$. C. $N\\left(0.80, \\sqrt{\\frac{0.8 \\,\\cdot\\, 0.2}{100}} \\right)$. D. $N\\left(0.76, \\sqrt{\\frac{0.76 \\,\\cdot\\, 0.24}{100}} \\right)$. E. $N\\left(0.80, \\sqrt{\\frac{0.8 \\,\\cdot\\, 0.2}{1200}} \\right)$. F. $N\\left(0.80, \\sqrt{\\frac{0.76 \\,\\cdot\\, 0.24}{100}} \\right)$.\nPart iv) Compute the $P$-value: [ANS] (your answer must be expressed as a proportion and rounded to 4 decimal places.)\nPart v) What is an appropriate conclusion for the hypothesis test at the 2\\% significance level? [ANS] A. The passing rate for students taught using the conventional method this year is significantly lower than last year\u2019s. B. The passing rate for students taught using the conventional method this year is not significantly lower than last year\u2019s. C. The passing rate for students taught using the conventional method this year is the same as last year\u2019s. D. Both (b) and (c).", "answer_v3": [ "D", "C", "C", "0.158655", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "NV", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0271", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample proportion", "level": "4", "keywords": [ "statistics", "Inference about a population" ], "problem_v1": "An Office of Admission document claims that 55.9\\% of UVA undergraduates are female. To test this claim, a random sample of 225 UVA undergraduates was selected. In this sample, 56.4\\% were female. Is there sufficient evidence to conclude that the document's claim is false? Carry out a hypothesis test at a 8\\% significance level. A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Do Not Reject $H_1$. B. Reject $H_0$. C. Reject $H_1$. D. Do Not Reject $H_0$.", "answer_v1": [ "0.151055333727787", "(-infinity,-1.75069) U (1.75069,infinity)", "0.879932", "D" ], "answer_type_v1": [ "NV", "INT", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v2": "An Office of Admission document claims that 54.2\\% of UVA undergraduates are female. To test this claim, a random sample of 205 UVA undergraduates was selected. In this sample, 56.8\\% were female. Is there sufficient evidence to conclude that the document's claim is false? Carry out a hypothesis test at a 4\\% significance level. A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Do Not Reject $H_0$. B. Reject $H_1$. C. Do Not Reject $H_1$. D. Reject $H_0$.", "answer_v2": [ "0.747167368158261", "(-infinity,-2.05375) U (2.05375,infinity)", "0.454962", "A" ], "answer_type_v2": [ "NV", "INT", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v3": "An Office of Admission document claims that 54.8\\% of UVA undergraduates are female. To test this claim, a random sample of 210 UVA undergraduates was selected. In this sample, 55.4\\% were female. Is there sufficient evidence to conclude that the document's claim is false? Carry out a hypothesis test at a 6\\% significance level. A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Do Not Reject $H_0$. B. Reject $H_1$. C. Reject $H_0$. D. Do Not Reject $H_1$.", "answer_v3": [ "0.174703417696311", "(-infinity,-1.88079) U (1.88079,infinity)", "0.861312", "A" ], "answer_type_v3": [ "NV", "INT", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0272", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Two sample proportion", "level": "2", "keywords": [ "Test Statistic", "Confidence Interval" ], "problem_v1": "In a study of red/green color blindness, $900$ men and $2650$ women are randomly selected and tested. Among the men, $82$ have red/green color blindness. Among the women, $7$ have red/green color blindness. Test the claim that men have a higher rate of red/green color blindness. The test statistic is [ANS]\nIs there sufficient evidence to support the claim that men have a higher rate of red/green color blindness than women? [ANS] A. No B. Yes\nConstruct the $96$ \\% confidence interval for the difference between the color blindness rates of men and women. [ANS] $< (p_1-p_2) <$ [ANS]", "answer_v1": [ "14.6675182469538", "B", "0.0686634069466562", "0.108275796407642" ], "answer_type_v1": [ "NV", "MCS", "NV", "NV" ], "options_v1": [ [], [ "A", "B" ], [], [] ], "problem_v2": "In a study of red/green color blindness, $500$ men and $2150$ women are randomly selected and tested. Among the men, $48$ have red/green color blindness. Among the women, $5$ have red/green color blindness. Test the claim that men have a higher rate of red/green color blindness. The test statistic is [ANS]\nIs there sufficient evidence to support the claim that men have a higher rate of red/green color blindness than women? [ANS] A. Yes B. No\nConstruct the $96$ \\% confidence interval for the difference between the color blindness rates of men and women. [ANS] $< (p_1-p_2) <$ [ANS]", "answer_v2": [ "13.4764115670049", "A", "0.0665332634174966", "0.120815573791806" ], "answer_type_v2": [ "NV", "MCS", "NV", "NV" ], "options_v2": [ [], [ "A", "B" ], [], [] ], "problem_v3": "In a study of red/green color blindness, $650$ men and $2250$ women are randomly selected and tested. Among the men, $59$ have red/green color blindness. Among the women, $6$ have red/green color blindness. Test the claim that men have a higher rate of red/green color blindness. The test statistic is [ANS]\nIs there sufficient evidence to support the claim that men have a higher rate of red/green color blindness than women? [ANS] A. No B. Yes\nConstruct the $96$ \\% confidence interval for the difference between the color blindness rates of men and women. [ANS] $< (p_1-p_2) <$ [ANS]", "answer_v3": [ "13.3660251669466", "B", "0.0648532860351268", "0.111351842170001" ], "answer_type_v3": [ "NV", "MCS", "NV", "NV" ], "options_v3": [ [], [ "A", "B" ], [], [] ] }, { "id": "Statistics_0273", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Two sample proportion", "level": "2", "keywords": [ "statistics", "inferences", "population proportion", "paired differences" ], "problem_v1": "Suppose a group of $1000$ smokers (who all wanted to give up smoking) were randomly assigned to receive an antidepressant drug or a placebo for six weeks. Of the $391$ patients who received the antidepressant drug, $125$ were not smoking one year later. Of the $609$ patients who received the placebo, $223$ were not smoking one year later. Given the null hypothesis $H_0: (p_1-p_2)=0$ and the alternative hypothesis $H_a: (p_1-p_2) \\ne 0$, conduct a test to see if taking an antidepressant drug can help smokers stop smoking. Use $\\alpha=0.04$\n(a) $\\ $ The rejection region is $|z| >$ [ANS]\n(b) $\\ $ The test statistic is $z=$ [ANS]\nThe final conclustion is [ANS] A. We can reject the null hypothesis that $(p_1-p_2)=0$ and accept that $(p_1-p_2) \\ne 0$. B. There is not sufficient evidence to reject the null hypothesis that $(p_1-p_2)=0$.", "answer_v1": [ "2.05375", "-1.5057702265112", "B" ], "answer_type_v1": [ "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "A", "B" ] ], "problem_v2": "Suppose a group of $700$ smokers (who all wanted to give up smoking) were randomly assigned to receive an antidepressant drug or a placebo for six weeks. Of the $566$ patients who received the antidepressant drug, $50$ were not smoking one year later. Of the $134$ patients who received the placebo, $29$ were not smoking one year later. Given the null hypothesis $H_0: (p_1-p_2)=0$ and the alternative hypothesis $H_a: (p_1-p_2) \\ne 0$, conduct a test to see if taking an antidepressant drug can help smokers stop smoking. Use $\\alpha=0.1$\n(a) $\\ $ The rejection region is $|z| >$ [ANS]\n(b) $\\ $ The test statistic is $z=$ [ANS]\nThe final conclustion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that $(p_1-p_2)=0$. B. We can reject the null hypothesis that $(p_1-p_2)=0$ and accept that $(p_1-p_2) \\ne 0$.", "answer_v2": [ "1.64485", "-4.21334632769257", "A" ], "answer_type_v2": [ "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "A", "B" ] ], "problem_v3": "Suppose a group of $800$ smokers (who all wanted to give up smoking) were randomly assigned to receive an antidepressant drug or a placebo for six weeks. Of the $403$ patients who received the antidepressant drug, $63$ were not smoking one year later. Of the $397$ patients who received the placebo, $113$ were not smoking one year later. Given the null hypothesis $H_0: (p_1-p_2)=0$ and the alternative hypothesis $H_a: (p_1-p_2) \\ne 0$, conduct a test to see if taking an antidepressant drug can help smokers stop smoking. Use $\\alpha=0.03$\n(a) $\\ $ The rejection region is $|z| >$ [ANS]\n(b) $\\ $ The test statistic is $z=$ [ANS]\nThe final conclustion is [ANS] A. We can reject the null hypothesis that $(p_1-p_2)=0$ and accept that $(p_1-p_2) \\ne 0$. B. There is not sufficient evidence to reject the null hypothesis that $(p_1-p_2)=0$.", "answer_v3": [ "2.17009", "-4.38021297514545", "B" ], "answer_type_v3": [ "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "A", "B" ] ] }, { "id": "Statistics_0274", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Two sample proportion", "level": "3", "keywords": [ "percent" ], "problem_v1": "Industry Research polled teenagers on sunscreen use. The survey revealed that 46\\% of teenage girls and 30\\% of teenage boys regularly use sunscreen before going out in the sun.\nidentify the two populations [ANS] A. teenage girls and teenage boys who use sunscreen regularly B. all teenagers C. teenage girls and teenage boys D. None of the above\nidentify the specified attribute [ANS] A. uses sunscreen before going out in the sun B. being a teenage girl or a teenage boy C. being a teenager D. None of the above\nare the proportions 0.46 (46\\%) and 0.30 (30\\%) population proportions or a sample proportions? [ANS] A. sample proportions B. population proportions C. None of the above", "answer_v1": [ "C", "A", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C" ] ], "problem_v2": "Industry Research polled teenagers on sunscreen use. The survey revealed that 46\\% of teenage girls and 30\\% of teenage boys regularly use sunscreen before going out in the sun.\nidentify the two populations [ANS] A. teenage girls and teenage boys B. all teenagers C. teenage girls and teenage boys who use sunscreen regularly D. None of the above\nidentify the specified attribute [ANS] A. uses sunscreen before going out in the sun B. being a teenager C. being a teenage girl or a teenage boy D. None of the above\nare the proportions 0.46 (46\\%) and 0.30 (30\\%) population proportions or a sample proportions? [ANS] A. population proportions B. sample proportions C. None of the above", "answer_v2": [ "A", "A", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C" ] ], "problem_v3": "Industry Research polled teenagers on sunscreen use. The survey revealed that 46\\% of teenage girls and 30\\% of teenage boys regularly use sunscreen before going out in the sun.\nidentify the two populations [ANS] A. teenage girls and teenage boys who use sunscreen regularly B. teenage girls and teenage boys C. all teenagers D. None of the above\nidentify the specified attribute [ANS] A. being a teenage girl or a teenage boy B. uses sunscreen before going out in the sun C. being a teenager D. None of the above\nare the proportions 0.46 (46\\%) and 0.30 (30\\%) population proportions or a sample proportions? [ANS] A. population proportions B. sample proportions C. None of the above", "answer_v3": [ "B", "B", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C" ] ] }, { "id": "Statistics_0277", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Two sample proportion", "level": "4", "keywords": [ "statistics", "Inference about a population" ], "problem_v1": "A marketing research firm suspects that a particular product has higher name recognition among college graduates than among high school graduates. A sample from each population is selected, and each asked if they have heard of the product in question. A summary of the sample sizes and number of each group answering ``yes'' are given below:\n\\begin{array}{lll} \\mbox{College Grads (Pop. 1)}: & n_1=95, & x_1=64 \\\\ \\mbox{High School Grads (Pop. 2)}: & n_2=92, & x_2=45 \\\\ \\end{array} The company making the product is willing to increase marketing targeted at high school graduates if the difference between the two groups is at least 5\\%. Is there evidence, at an $\\alpha=0.037$ level of significance, to support such an increase in marketing? Carry out an appropriate hypothesis test, filling in the information requested. A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Do Not Reject $H_0$. B. Reject $H_1$. C. Do Not Reject $H_1$. D. Reject $H_0$.", "answer_v1": [ "1.89716694364141", "(1.78661,infinity)", "0.0289031", "D" ], "answer_type_v1": [ "NV", "INT", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v2": "A marketing research firm suspects that a particular product has higher name recognition among college graduates than among high school graduates. A sample from each population is selected, and each asked if they have heard of the product in question. A summary of the sample sizes and number of each group answering ``yes'' are given below:\n\\begin{array}{lll} \\mbox{College Grads (Pop. 1)}: & n_1=81, & x_1=55 \\\\ \\mbox{High School Grads (Pop. 2)}: & n_2=99, & x_2=56 \\\\ \\end{array} The company making the product is willing to increase marketing targeted at high school graduates if the difference between the two groups is at least 5\\%. Is there evidence, at an $\\alpha=0.096$ level of significance, to support such an increase in marketing? Carry out an appropriate hypothesis test, filling in the information requested. A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Reject $H_0$. B. Do Not Reject $H_0$. C. Do Not Reject $H_1$. D. Reject $H_1$.", "answer_v2": [ "0.880918475134752", "(1.30469,infinity)", "0.189181", "B" ], "answer_type_v2": [ "NV", "INT", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v3": "A marketing research firm suspects that a particular product has higher name recognition among college graduates than among high school graduates. A sample from each population is selected, and each asked if they have heard of the product in question. A summary of the sample sizes and number of each group answering ``yes'' are given below:\n\\begin{array}{lll} \\mbox{College Grads (Pop. 1)}: & n_1=86, & x_1=58 \\\\ \\mbox{High School Grads (Pop. 2)}: & n_2=92, & x_2=46 \\\\ \\end{array} The company making the product is willing to increase marketing targeted at high school graduates if the difference between the two groups is at least 5\\%. Is there evidence, at an $\\alpha=0.028$ level of significance, to support such an increase in marketing? Carry out an appropriate hypothesis test, filling in the information requested. A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Reject $H_1$. B. Do Not Reject $H_0$. C. Do Not Reject $H_1$. D. Reject $H_0$.", "answer_v3": [ "1.71377673333331", "(1.91104,infinity)", "0.0432849", "B" ], "answer_type_v3": [ "NV", "INT", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0278", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Paired samples", "level": "2", "keywords": [ "statistics", "inferences", "two samples", "2 samples" ], "problem_v1": "Suppose you want to test the claim the paired sample data given below come from a population for which the mean difference is $\\mu_d=0$. \\begin{array}{c|ccccccc} x & 80 & 75 & 62 & 73 & 65 & 76 & 70 \\cr \\hline y & 78 & 84 & 68 & 78 & 75 & 66 & 72 \\cr \\end{array} Use a $0.01$ significance level to find the following:\n(a) $\\ $ The mean value of the differnces $d$ for the paired sample data $\\overline{d}=$ [ANS]\n(b) $\\ $ The standard deviation of the differences $d$ for the paired sample data $s_d=$ [ANS]\n(c) $\\ $ The t test statistic $t=$ [ANS]\n(d) $\\ $ The positive critical value $t=$ [ANS]\n(e) $\\ $ The negative critical value $t=$ [ANS]\n(f) $\\ $ Does the test statistic fall in the critical region? [ANS] A. No B. Yes\n(g) $\\ $ Construct a $99$ \\% conficence interval for the population mean of all differences $x-y$. [ANS] $< \\mu_d <$ [ANS]", "answer_v1": [ "-2.85714285714286", "6.98638131005772", "-1.08200356160092", "3.70728", "-3.70728", "A", "-12.6466009951146", "6.93231528082893" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "MCS", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [ "A", "B" ], [], [] ], "problem_v2": "Suppose you want to test the claim the paired sample data given below come from a population for which the mean difference is $\\mu_d=0$. \\begin{array}{c|ccccccc} x & 53 & 56 & 88 & 57 & 73 & 76 & 83 \\cr \\hline y & 93 & 68 & 67 & 68 & 57 & 73 & 62 \\cr \\end{array} Use a $0.01$ significance level to find the following:\n(a) $\\ $ The mean value of the differnces $d$ for the paired sample data $\\overline{d}=$ [ANS]\n(b) $\\ $ The standard deviation of the differences $d$ for the paired sample data $s_d=$ [ANS]\n(c) $\\ $ The t test statistic $t=$ [ANS]\n(d) $\\ $ The positive critical value $t=$ [ANS]\n(e) $\\ $ The negative critical value $t=$ [ANS]\n(f) $\\ $ Does the test statistic fall in the critical region? [ANS] A. No B. Yes\n(g) $\\ $ Construct a $99$ \\% conficence interval for the population mean of all differences $x-y$. [ANS] $< \\mu_d <$ [ANS]", "answer_v2": [ "-0.285714285714286", "22.403231059487", "-0.0337419608810554", "3.70728", "-3.70728", "A", "-31.6775726568608", "31.1061440854322" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "MCS", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [ "A", "B" ], [], [] ], "problem_v3": "Suppose you want to test the claim the paired sample data given below come from a population for which the mean difference is $\\mu_d=0$. \\begin{array}{c|ccccccc} x & 62 & 61 & 58 & 83 & 85 & 61 & 51 \\cr \\hline y & 79 & 77 & 69 & 92 & 63 & 65 & 78 \\cr \\end{array} Use a $0.05$ significance level to find the following:\n(a) $\\ $ The mean value of the differnces $d$ for the paired sample data $\\overline{d}=$ [ANS]\n(b) $\\ $ The standard deviation of the differences $d$ for the paired sample data $s_d=$ [ANS]\n(c) $\\ $ The t test statistic $t=$ [ANS]\n(d) $\\ $ The positive critical value $t=$ [ANS]\n(e) $\\ $ The negative critical value $t=$ [ANS]\n(f) $\\ $ Does the test statistic fall in the critical region? [ANS] A. Yes B. No\n(g) $\\ $ Construct a $95$ \\% conficence interval for the population mean of all differences $x-y$. [ANS] $< \\mu_d <$ [ANS]", "answer_v3": [ "-8.85714285714286", "15.4210740160834", "-1.51959567162", "2.44690", "-2.4469", "B", "-23.1191885196172", "5.40490280533149" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "MCS", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [ "A", "B" ], [], [] ] }, { "id": "Statistics_0279", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Paired samples", "level": "2", "keywords": [ "statistics", "inferences", "paired difference" ], "problem_v1": "A paired difference experiment produced the following results:\nn_D=48, \\ \\overline{x}_1=158, \\ \\overline{x}_2=155, \\ \\overline{x}_D=3, \\ s_D=65,\n(a) $\\ $ Determine the rejection region for the hypothesis $H_0: \\mu_D=0$ if $H_a: \\mu_D >0$. Use $\\alpha=0.04$. $t >$ [ANS]\n(b) $\\ $ Conduct a paired difference test described above. The test statistic is [ANS]\nThe final conclusion is [ANS] A. We can reject the null hypothesis that $\\mu_D=0$ and accept that $\\mu_D > 0$. B. There is not sufficient evidence to reject the null hypothesis that $\\mu_D=0$.", "answer_v1": [ "1.78937", "0.319763226012716", "B" ], "answer_type_v1": [ "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "A", "B" ] ], "problem_v2": "A paired difference experiment produced the following results:\nn_D=40, \\ \\overline{x}_1=194, \\ \\overline{x}_2=201, \\ \\overline{x}_D=-7, \\ s_D=57,\n(a) $\\ $ Determine the rejection region for the hypothesis $H_0: \\mu_D=0$ if $H_a: \\mu_D >0$. Use $\\alpha=0.1$. $t >$ [ANS]\n(b) $\\ $ Conduct a paired difference test described above. The test statistic is [ANS]\nThe final conclusion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that $\\mu_D=0$. B. We can reject the null hypothesis that $\\mu_D=0$ and accept that $\\mu_D > 0$.", "answer_v2": [ "1.30364", "-0.776699776181707", "A" ], "answer_type_v2": [ "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "A", "B" ] ], "problem_v3": "A paired difference experiment produced the following results:\nn_D=43, \\ \\overline{x}_1=161, \\ \\overline{x}_2=166, \\ \\overline{x}_D=-5, \\ s_D=61,\n(a) $\\ $ Determine the rejection region for the hypothesis $H_0: \\mu_D=0$ if $H_a: \\mu_D >0$. Use $\\alpha=0.03$. $t >$ [ANS]\n(b) $\\ $ Conduct a paired difference test described above. The test statistic is [ANS]\nThe final conclusion is [ANS] A. We can reject the null hypothesis that $\\mu_D=0$ and accept that $\\mu_D > 0$. B. There is not sufficient evidence to reject the null hypothesis that $\\mu_D=0$.", "answer_v3": [ "1.93298", "-0.537494961008361", "B" ], "answer_type_v3": [ "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "A", "B" ] ] }, { "id": "Statistics_0280", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Paired samples", "level": "3", "keywords": [ "Statistics" ], "problem_v1": "Suppose you want to use a paired sample to compare the mean TV viewing times of married men and married women.\nWhat is the variable under consideration? [ANS] A. marital status B. mean TV viewing time C. TV viewing time D. None of the above\nWhat are the two populations under consideration? [ANS] A. married men and married women B. men and women C. married people and single people D. None of the above\nWhat are the pairs? [ANS] A. married couples B. men and women C. None of the above\nWhat is the paired-difference variable? [ANS] A. the difference between the TV viewing times of a married couple B. the difference between the TV viewing times of men and women C. None of the above", "answer_v1": [ "C", "A", "A", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v2": "Suppose you want to use a paired sample to compare the mean TV viewing times of married men and married women.\nWhat is the variable under consideration? [ANS] A. TV viewing time B. mean TV viewing time C. marital status D. None of the above\nWhat are the two populations under consideration? [ANS] A. married men and married women B. married people and single people C. men and women D. None of the above\nWhat are the pairs? [ANS] A. men and women B. married couples C. None of the above\nWhat is the paired-difference variable? [ANS] A. the difference between the TV viewing times of a married couple B. the difference between the TV viewing times of men and women C. None of the above", "answer_v2": [ "A", "A", "B", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v3": "Suppose you want to use a paired sample to compare the mean TV viewing times of married men and married women.\nWhat is the variable under consideration? [ANS] A. marital status B. TV viewing time C. mean TV viewing time D. None of the above\nWhat are the two populations under consideration? [ANS] A. men and women B. married men and married women C. married people and single people D. None of the above\nWhat are the pairs? [ANS] A. men and women B. married couples C. None of the above\nWhat is the paired-difference variable? [ANS] A. the difference between the TV viewing times of men and women B. the difference between the TV viewing times of a married couple C. None of the above", "answer_v3": [ "B", "B", "B", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ] }, { "id": "Statistics_0281", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Paired samples", "level": "3", "keywords": [ "hypothesis testing", "t distribution" ], "problem_v1": "It is considered quite common to have feet of unequal length. In a sample of 10 healthy college students the right-foot and left-foot lengths are given (in mm). Test the claim that, on average, there is a measurable difference between left and right foot length.\n$\\begin{array}{ccccccccccccc}\\hline & Length in mm & mean & s \\\\ \\hline Left foot (x) & 267 & 255 & 268 & 253 & 270 & 269 & 273 & 251 & 254 & 265 & 262.5 & 8.27647267862342 \\\\ \\hline Right foot (y) & 271 & 255 & 260 & 255 & 271 & 273 & 272 & 257 & 249 & 265 & 262.8 & 8.70249006510957 \\\\ \\hline d=x-y &-4 & 0 & 8 &-2 &-1 &-4 & 1 &-6 & 5 & 0 &-0.3 & 4.24394994210713 \\\\ \\hline \\end{array}$ 4.24394994210713\n(a) Find the test statistic. [ANS]\n(b) Test the claim at the 0.05 significance level. Positive critical value: [ANS]\nNegative critical value: [ANS]\nIs there sufficient data to support the claim? [ANS] [ANS]", "answer_v1": [ "-0.3/(4.24395/[sqrt(10)])", "2.262", "-2.262", "No" ], "answer_type_v1": [ "NV", "NV", "NV", "TF" ], "options_v1": [ [], [], [], [] ], "problem_v2": "It is considered quite common to have feet of unequal length. In a sample of 10 healthy college students the right-foot and left-foot lengths are given (in mm). Test the claim that, on average, there is a measurable difference between left and right foot length.\n$\\begin{array}{ccccccccccccc}\\hline & Length in mm & mean & s \\\\ \\hline Left foot (x) & 268 & 265 & 278 & 261 & 271 & 266 & 259 & 252 & 275 & 275 & 267 & 8.13770374382247 \\\\ \\hline Right foot (y) & 271 & 264 & 278 & 261 & 273 & 266 & 260 & 253 & 275 & 275 & 267.6 & 8.08565258824421 \\\\ \\hline d=x-y &-3 & 1 & 0 & 0 &-2 & 0 &-1 &-1 & 0 & 0 &-0.6 & 1.17378779077727 \\\\ \\hline \\end{array}$ 1.17378779077727\n(a) Find the test statistic. [ANS]\n(b) Test the claim at the 0.05 significance level. Positive critical value: [ANS]\nNegative critical value: [ANS]\nIs there sufficient data to support the claim? [ANS] [ANS]", "answer_v2": [ "-0.6/(1.17379/[sqrt(10)])", "2.262", "-2.262", "No" ], "answer_type_v2": [ "NV", "NV", "NV", "TF" ], "options_v2": [ [], [], [], [] ], "problem_v3": "It is considered quite common to have feet of unequal length. In a sample of 10 healthy college students the right-foot and left-foot lengths are given (in mm). Test the claim that, on average, there is a measurable difference between left and right foot length.\n$\\begin{array}{ccccccccccccc}\\hline & Length in mm & mean & s \\\\ \\hline Left foot (x) & 254 & 245 & 262 & 264 & 268 & 270 & 253 & 255 & 262 & 256 & 258.9 & 7.65143414298568 \\\\ \\hline Right foot (y) & 253 & 241 & 266 & 261 & 267 & 271 & 254 & 255 & 266 & 256 & 259 & 8.94427190999916 \\\\ \\hline d=x-y & 1 & 4 &-4 & 3 & 1 &-1 &-1 & 0 &-4 & 0 &-0.1 & 2.60128173535022 \\\\ \\hline \\end{array}$ 2.60128173535022\n(a) Find the test statistic. [ANS]\n(b) Test the claim at the 0.05 significance level. Positive critical value: [ANS]\nNegative critical value: [ANS]\nIs there sufficient data to support the claim? [ANS] [ANS]", "answer_v3": [ "-0.1/(2.60128/[sqrt(10)])", "2.262", "-2.262", "No" ], "answer_type_v3": [ "NV", "NV", "NV", "TF" ], "options_v3": [ [], [], [], [] ] }, { "id": "Statistics_0284", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Paired samples", "level": "3", "keywords": [ "hypothesis testing", "t distribution" ], "problem_v1": "It is widely accepted that people are a little taller in the morning than at night. Here we perform a test on how big the difference is. In a sample of 35 adults, the mean difference between morning height and evening height was 5.6 millimeters (mm) with a standard deviation of 1.54333 mm. Test the claim that, on average, people are more than 5 mm taller in the morning than at night. Test this claim at the 0.05 significance level.\n(a) Find the test statistic. [ANS]\n(b) Find the critical value. [ANS]\n(c) Is there sufficient data to support the claim? [ANS] [ANS]", "answer_v1": [ "2.3", "1.691", "Yes" ], "answer_type_v1": [ "NV", "NV", "TF" ], "options_v1": [ [], [], [] ], "problem_v2": "It is widely accepted that people are a little taller in the morning than at night. Here we perform a test on how big the difference is. In a sample of 35 adults, the mean difference between morning height and evening height was 6 millimeters (mm) with a standard deviation of 3.36141 mm. Test the claim that, on average, people are more than 5 mm taller in the morning than at night. Test this claim at the 0.05 significance level.\n(a) Find the test statistic. [ANS]\n(b) Find the critical value. [ANS]\n(c) Is there sufficient data to support the claim? [ANS] [ANS]", "answer_v2": [ "1.76", "1.691", "Yes" ], "answer_type_v2": [ "NV", "NV", "TF" ], "options_v2": [ [], [], [] ], "problem_v3": "It is widely accepted that people are a little taller in the morning than at night. Here we perform a test on how big the difference is. In a sample of 35 adults, the mean difference between morning height and evening height was 5.7 millimeters (mm) with a standard deviation of 2.12372 mm. Test the claim that, on average, people are more than 5 mm taller in the morning than at night. Test this claim at the 0.05 significance level.\n(a) Find the test statistic. [ANS]\n(b) Find the critical value. [ANS]\n(c) Is there sufficient data to support the claim? [ANS] [ANS]", "answer_v3": [ "1.95", "1.691", "Yes" ], "answer_type_v3": [ "NV", "NV", "TF" ], "options_v3": [ [], [], [] ] }, { "id": "Statistics_0285", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Paired samples", "level": "3", "keywords": [ "\"Statistics\"", "\"Statistical inference\"", "\"hypothesis testing\"", "\"parameter identification\"", "\"errors\"", "\"P-value\"", "\"interpretation\"" ], "problem_v1": "In which of the following scenarios will conducting a paired $t$-test for means be appropriate? CHECK ALL THAT APPLY. [ANS] A. To test if there is a difference between the mean annual income of male British Columbians and that of female British Columbians. B. To test if the mean annual income of Ontarians is higher than that of British Columbians. C. To test if there is a difference between the mean annual income of husbands and that of their wives in Canada. D. To test if there is a difference between the mean number of antibodies in patients before surgery and after surgery. E. To test if the proportion of low-income families is higher than that of high-income families in British Columbia. F. To test if there is a difference between the mean number of CD4 T cells in healthy patients and patients with cancer. G. None of the above", "answer_v1": [ "CD" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v2": "In which of the following scenarios will conducting a paired $t$-test for means be appropriate? CHECK ALL THAT APPLY. [ANS] A. To test if the mean annual income of Ontarians is higher than that of British Columbians. B. To test if the proportion of low-income families is higher than that of high-income families in British Columbia. C. To test if there is a difference between the mean annual income of male British Columbians and that of female British Columbians. D. To test if there is a difference between the mean number of antibodies in patients before surgery and after surgery. E. To test if there is a difference between the mean number of CD4 T cells in healthy patients and patients with cancer. F. To test if there is a difference between the mean annual income of husbands and that of their wives in Canada. G. None of the above", "answer_v2": [ "DF" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v3": "In which of the following scenarios will conducting a paired $t$-test for means be appropriate? CHECK ALL THAT APPLY. [ANS] A. To test if the proportion of low-income families is higher than that of high-income families in British Columbia. B. To test if the mean annual income of Ontarians is higher than that of British Columbians. C. To test if there is a difference between the mean number of antibodies in patients before surgery and after surgery. D. To test if there is a difference between the mean annual income of male British Columbians and that of female British Columbians. E. To test if there is a difference between the mean annual income of husbands and that of their wives in Canada. F. To test if there is a difference between the mean number of CD4 T cells in healthy patients and patients with cancer. G. None of the above", "answer_v3": [ "CE" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G" ] ] }, { "id": "Statistics_0286", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Paired samples", "level": "3", "keywords": [ "statistics", "Two Populations", "Inference" ], "problem_v1": "The number of degrees of freedom associated with the t test, when the data are gathered from a matched pairs experiment with 10 pairs, is: [ANS] A. 9 B. 18 C. 20 D. 10\nThe quantity $s^2_p$ is called the pooled variance estimate of the common variance of two unknown but equal population variances. It is the weighted average of the two sample variances, where the weights represent the: [ANS] A. sample sizes B. sample variances C. degrees of freedom for each sample D. sample standard deviations", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "The number of degrees of freedom associated with the t test, when the data are gathered from a matched pairs experiment with 10 pairs, is: [ANS] A. 20 B. 18 C. 10 D. 9\nThe symbol $\\bar{x}_D$ refers to: [ANS] A. the mean difference in the pairs of observations taken from dependent samples B. the matched pairs differences C. the difference in the means of two dependent populations D. the difference in the means of the two independent populations", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "The number of degrees of freedom associated with the t test, when the data are gathered from a matched pairs experiment with 10 pairs, is: [ANS] A. 20 B. 9 C. 18 D. 10\nTwo independent samples of sizes 40 and 50 are randomly selected from two populations to test the difference between the population means $\\mu_1-\\mu_2$. The sampling distribution of the sample mean difference $\\bar{x}_1-\\bar{x}_2$ is: [ANS] A. approximately normal B. t distributed with 88 degrees of freedom C. normally distributed D. chi-square distributed with 90 degrees of freedom", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0287", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Paired samples", "level": "4", "keywords": [ "statistics", "Inference about a population" ], "problem_v1": "A company that sponsors LSAT prep courses would like to be able to claim that their courses improve scores by at least 3 percentage points. To test this, they take a sample of 8 people, have each take an initial diagnostic test, then take the prep course, and then take a post-test after the course. The test results are below (scores are out of 100\\%):\n\\begin{array}{ccc} \\mbox{Person} & \\mbox{Initial Test} & \\mbox{Post-Test} \\\\ 1 & 75 & 77 \\\\ 2 & 72 & 77 \\\\ 3 & 73 & 81 \\\\ 4 & 75 & 75 \\\\ 5 & 66 & 71 \\\\ 6 & 66 & 69 \\\\ 7 & 71 & 74 \\\\ 8 & 72 & 76 \\\\ \\end{array}\nIs there evidence, at an $\\alpha=0.055$ level of significance, to conclude that the prep course improves scores by at least 3 percentage points? Carry out an appropriate hypothesis test, filling in the information requested. (Arrange your data so that the standardized test statistic is for the change from the initial test to the post-test.) A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Do Not Reject $H_0$. B. Reject $H_1$. C. Do Not Reject $H_1$. D. Reject $H_0$.", "answer_v1": [ "0.893010836681381", "(1.82966,infinity)", "0.200754", "A" ], "answer_type_v1": [ "NV", "INT", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v2": "A company that sponsors LSAT prep courses would like to be able to claim that their courses improve scores by at least 3 percentage points. To test this, they take a sample of 8 people, have each take an initial diagnostic test, then take the prep course, and then take a post-test after the course. The test results are below (scores are out of 100\\%):\n\\begin{array}{ccc} \\mbox{Person} & \\mbox{Initial Test} & \\mbox{Post-Test} \\\\ 1 & 61 & 67 \\\\ 2 & 79 & 75 \\\\ 3 & 63 & 71 \\\\ 4 & 67 & 71 \\\\ 5 & 79 & 91 \\\\ 6 & 66 & 65 \\\\ 7 & 63 & 61 \\\\ 8 & 66 & 66 \\\\ \\end{array}\nIs there evidence, at an $\\alpha=0.06$ level of significance, to conclude that the prep course improves scores by at least 3 percentage points? Carry out an appropriate hypothesis test, filling in the information requested. (Arrange your data so that the standardized test statistic is for the change from the initial test to the post-test.) A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Do Not Reject $H_0$. B. Reject $H_0$. C. Reject $H_1$. D. Do Not Reject $H_1$.", "answer_v2": [ "-0.0638132820201823", "(1.77021,infinity)", "0.524549", "A" ], "answer_type_v2": [ "NV", "INT", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v3": "A company that sponsors LSAT prep courses would like to be able to claim that their courses improve scores by at least 3 percentage points. To test this, they take a sample of 8 people, have each take an initial diagnostic test, then take the prep course, and then take a post-test after the course. The test results are below (scores are out of 100\\%):\n\\begin{array}{ccc} \\mbox{Person} & \\mbox{Initial Test} & \\mbox{Post-Test} \\\\ 1 & 66 & 79 \\\\ 2 & 72 & 71 \\\\ 3 & 65 & 66 \\\\ 4 & 71 & 71 \\\\ 5 & 64 & 59 \\\\ 6 & 67 & 74 \\\\ 7 & 76 & 91 \\\\ 8 & 79 & 90 \\\\ \\end{array}\nIs there evidence, at an $\\alpha=0.065$ level of significance, to conclude that the prep course improves scores by at least 3 percentage points? Carry out an appropriate hypothesis test, filling in the information requested. (Arrange your data so that the standardized test statistic is for the change from the initial test to the post-test.) A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Do Not Reject $H_1$. B. Do Not Reject $H_0$. C. Reject $H_0$. D. Reject $H_1$.", "answer_v3": [ "0.814820063654487", "(1.71532,infinity)", "0.221005", "B" ], "answer_type_v3": [ "NV", "INT", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0288", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Independent samples - z", "level": "2", "keywords": [ "inferences", "statistics", "differences" ], "problem_v1": "Two independent samples have been selected, $88$ observations from population 1 and $79$ observations from population 2. The sample means have been calculated to be $\\overline{x}_1=11.2$ and $\\overline{x}_2=12.2$. From previous experience with these populations, it is known that the variances are $\\sigma_1^2=26$ and $\\sigma_2^2=31$.\n(a) $\\ $ Find $\\sigma_{(\\overline{x}_1-\\overline{x}_2)}$. answer: [ANS]\n(b) $\\ $ Determine the rejection region for the test of $H_0:(\\mu_1-\\mu_2)=3.15$ and $H_a:(\\mu_1-\\mu_2) > 3.15$ Use $\\alpha=0.02$. $z >$ [ANS]\n(c) $\\ $ Compute the test statistic. $z=$ [ANS]\nThe final conclustion is [ANS] A. We can reject the null hypothesis that $(\\mu_1-\\mu_2)=3.15$ and accept that $(\\mu_1-\\mu_2) > 3.15$. B. There is not sufficient evidence to reject the null hypothesis that $(\\mu_1-\\mu_2)=3.15$.\n(d) $\\ $ Construct a $98$ \\% confidence interval for $(\\mu_1-\\mu_2)$. [ANS] $\\leq (\\mu_1-\\mu_2) \\leq$ [ANS]", "answer_v1": [ "0.829373021472054", "2.05375", "-5.00377983435507", "B", "-2.92941192850151", "0.929411928501514" ], "answer_type_v1": [ "NV", "NV", "NV", "MCS", "NV", "NV" ], "options_v1": [ [], [], [], [ "A", "B" ], [], [] ], "problem_v2": "Two independent samples have been selected, $54$ observations from population 1 and $97$ observations from population 2. The sample means have been calculated to be $\\overline{x}_1=6.5$ and $\\overline{x}_2=8.3$. From previous experience with these populations, it is known that the variances are $\\sigma_1^2=39$ and $\\sigma_2^2=26$.\n(a) $\\ $ Find $\\sigma_{(\\overline{x}_1-\\overline{x}_2)}$. answer: [ANS]\n(b) $\\ $ Determine the rejection region for the test of $H_0:(\\mu_1-\\mu_2)=2.36$ and $H_a:(\\mu_1-\\mu_2) > 2.36$ Use $\\alpha=0.02$. $z >$ [ANS]\n(c) $\\ $ Compute the test statistic. $z=$ [ANS]\nThe final conclustion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that $(\\mu_1-\\mu_2)=2.36$. B. We can reject the null hypothesis that $(\\mu_1-\\mu_2)=2.36$ and accept that $(\\mu_1-\\mu_2) > 2.36$.\n(d) $\\ $ Construct a $98$ \\% confidence interval for $(\\mu_1-\\mu_2)$. [ANS] $\\leq (\\mu_1-\\mu_2) \\leq$ [ANS]", "answer_v2": [ "0.995119821597191", "2.05375", "-4.18040110317881", "A", "-4.11499699697262", "0.514996996972624" ], "answer_type_v2": [ "NV", "NV", "NV", "MCS", "NV", "NV" ], "options_v2": [ [], [], [], [ "A", "B" ], [], [] ], "problem_v3": "Two independent samples have been selected, $65$ observations from population 1 and $80$ observations from population 2. The sample means have been calculated to be $\\overline{x}_1=7.8$ and $\\overline{x}_2=10.5$. From previous experience with these populations, it is known that the variances are $\\sigma_1^2=24$ and $\\sigma_2^2=27$.\n(a) $\\ $ Find $\\sigma_{(\\overline{x}_1-\\overline{x}_2)}$. answer: [ANS]\n(b) $\\ $ Determine the rejection region for the test of $H_0:(\\mu_1-\\mu_2)=3.62$ and $H_a:(\\mu_1-\\mu_2) > 3.62$ Use $\\alpha=0.05$. $z >$ [ANS]\n(c) $\\ $ Compute the test statistic. $z=$ [ANS]\nThe final conclustion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that $(\\mu_1-\\mu_2)=3.62$. B. We can reject the null hypothesis that $(\\mu_1-\\mu_2)=3.62$ and accept that $(\\mu_1-\\mu_2) > 3.62$.\n(d) $\\ $ Construct a $95$ \\% confidence interval for $(\\mu_1-\\mu_2)$. [ANS] $\\leq (\\mu_1-\\mu_2) \\leq$ [ANS]", "answer_v3": [ "0.840672807476707", "1.64485", "-7.5177880666434", "A", "-4.34768507574205", "-1.05231492425795" ], "answer_type_v3": [ "NV", "NV", "NV", "MCS", "NV", "NV" ], "options_v3": [ [], [], [], [ "A", "B" ], [], [] ] }, { "id": "Statistics_0289", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Independent samples - t", "level": "2", "keywords": [ "statistics", "inferences", "two samples", "2 samples" ], "problem_v1": "Test the given claim using the $\\alpha=0.05$ significance level and assuming that the populations are normally distributed. Claim: The treatment population and the placebo population have different variances. Treatment group: $n=9,$ $\\overline{x}=117.8,$ $s=17.$ Placebo group: $n=11,$ $\\overline{x}=109.8,$ $s=12.3.$ The test statistic is [ANS]\nThe larger critical value is [ANS]\nWhat is your conclusion? [ANS] A. There is not sufficient evidence to support the claim that the treatment and placebo populations have different variances. B. There is sufficient evidence to support the claim that the treatment and placebo populations have different variances.", "answer_v1": [ "1.91023861458127", "3.85489", "A" ], "answer_type_v1": [ "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "A", "B" ] ], "problem_v2": "Test the given claim using the $\\alpha=0.02$ significance level and assuming that the populations are normally distributed. Claim: The treatment population and the placebo population have different variances. Treatment group: $n=11,$ $\\overline{x}=86.7,$ $s=16.9.$ Placebo group: $n=6,$ $\\overline{x}=104.6,$ $s=10.7.$ The test statistic is [ANS]\nThe larger critical value is [ANS]\nWhat is your conclusion? [ANS] A. There is not sufficient evidence to support the claim that the treatment and placebo populations have different variances. B. There is sufficient evidence to support the claim that the treatment and placebo populations have different variances.", "answer_v2": [ "2.49462835182112", "10.0512", "A" ], "answer_type_v2": [ "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "A", "B" ] ], "problem_v3": "Test the given claim using the $\\alpha=0.02$ significance level and assuming that the populations are normally distributed. Claim: The treatment population and the placebo population have different variances. Treatment group: $n=9,$ $\\overline{x}=103.9,$ $s=17.1.$ Placebo group: $n=8,$ $\\overline{x}=92.1,$ $s=13.3.$ The test statistic is [ANS]\nThe larger critical value is [ANS]\nWhat is your conclusion? [ANS] A. There is not sufficient evidence to support the claim that the treatment and placebo populations have different variances. B. There is sufficient evidence to support the claim that the treatment and placebo populations have different variances.", "answer_v3": [ "1.6530612244898", "6.84006", "A" ], "answer_type_v3": [ "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "A", "B" ] ] }, { "id": "Statistics_0290", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Independent samples - t", "level": "3", "keywords": [ "statistics", "inferences", "two samples", "2 samples" ], "problem_v1": "Randomly selected $130$ student cars have ages with a mean of $7.6$ years and a standard deviation of $3.6$ years, while randomly selected $95$ faculty cars have ages with a mean of $5.3$ years and a standard deviation of $3.3$ years. 1. $\\ $ Use a $0.03$ significance level to test the claim that student cars are older than faculty cars. The test statistic is [ANS]\nThe critical value is [ANS]\nIs there sufficient evidence to support the claim that student cars are older than faculty cars? [ANS] A. Yes B. No\n2. $\\ $ Construct a $97$ \\% confidence interval estimate of the difference $\\mu_1-\\mu_2$, where $\\mu_1$ is the mean age of student cars and $\\mu_2$ is the mean age of faculty cars. [ANS] $< (\\mu_1-\\mu_2) <$ [ANS]", "answer_v1": [ "4.96812538338943", "1.88079", "A", "1.29535405916128", "3.30464594083872" ], "answer_type_v1": [ "NV", "NV", "MCS", "NV", "NV" ], "options_v1": [ [], [], [ "A", "B" ], [], [] ], "problem_v2": "Randomly selected $60$ student cars have ages with a mean of $8$ years and a standard deviation of $3.4$ years, while randomly selected $65$ faculty cars have ages with a mean of $6$ years and a standard deviation of $3.3$ years. 1. $\\ $ Use a $0.01$ significance level to test the claim that student cars are older than faculty cars. The test statistic is [ANS]\nThe critical value is [ANS]\nIs there sufficient evidence to support the claim that student cars are older than faculty cars? [ANS] A. Yes B. No\n2. $\\ $ Construct a $99$ \\% confidence interval estimate of the difference $\\mu_1-\\mu_2$, where $\\mu_1$ is the mean age of student cars and $\\mu_2$ is the mean age of faculty cars. [ANS] $< (\\mu_1-\\mu_2) <$ [ANS]", "answer_v2": [ "3.33238407136525", "2.32635", "A", "0.454061749884248", "3.54593825011575" ], "answer_type_v2": [ "NV", "NV", "MCS", "NV", "NV" ], "options_v2": [ [], [], [ "A", "B" ], [], [] ], "problem_v3": "Randomly selected $90$ student cars have ages with a mean of $7.6$ years and a standard deviation of $3.4$ years, while randomly selected $75$ faculty cars have ages with a mean of $5.2$ years and a standard deviation of $3.5$ years. 1. $\\ $ Use a $0.05$ significance level to test the claim that student cars are older than faculty cars. The test statistic is [ANS]\nThe critical value is [ANS]\nIs there sufficient evidence to support the claim that student cars are older than faculty cars? [ANS] A. Yes B. No\n2. $\\ $ Construct a $95$ \\% confidence interval estimate of the difference $\\mu_1-\\mu_2$, where $\\mu_1$ is the mean age of student cars and $\\mu_2$ is the mean age of faculty cars. [ANS] $< (\\mu_1-\\mu_2) <$ [ANS]", "answer_v3": [ "4.44309025658716", "1.64485", "A", "1.341299021998", "3.458700978002" ], "answer_type_v3": [ "NV", "NV", "MCS", "NV", "NV" ], "options_v3": [ [], [], [ "A", "B" ], [], [] ] }, { "id": "Statistics_0291", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Independent samples - t", "level": "2", "keywords": [ "statistics", "inferences", "two samples", "2 samples" ], "problem_v1": "Test the claim that the two samples described below come from populations with the same mean. Assume that the samples are independent simple random samples. Use a significance level of $0.03$. Sample 1: $n_1=88, \\ \\overline{x}_1=16, \\ s_1=2$. Sample 2: $n_2=79, \\ \\overline{x}_2=17, \\ s_2=2$. The test statistic is [ANS]\nThe P-Value is [ANS]\nThe conclusion is [ANS] A. There is sufficient evidence to warrant rejection of the claim that the two populations have the same mean. B. There is not sufficient evidence to warrant rejection of the claim that the two populations have the same mean.", "answer_v1": [ "-3.22601699139086", "0.00125526", "A" ], "answer_type_v1": [ "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "A", "B" ] ], "problem_v2": "Test the claim that the two samples described below come from populations with the same mean. Assume that the samples are independent simple random samples. Use a significance level of $0.01$. Sample 1: $n_1=54, \\ \\overline{x}_1=11, \\ s_1=5$. Sample 2: $n_2=97, \\ \\overline{x}_2=13, \\ s_2=2$. The test statistic is [ANS]\nThe P-Value is [ANS]\nThe conclusion is [ANS] A. There is sufficient evidence to warrant rejection of the claim that the two populations have the same mean. B. There is not sufficient evidence to warrant rejection of the claim that the two populations have the same mean.", "answer_v2": [ "-2.81662183764467", "0.00485316", "A" ], "answer_type_v2": [ "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "A", "B" ] ], "problem_v3": "Test the claim that the two samples described below come from populations with the same mean. Assume that the samples are independent simple random samples. Use a significance level of $0.05$. Sample 1: $n_1=65, \\ \\overline{x}_1=13, \\ s_1=1.5$. Sample 2: $n_2=80, \\ \\overline{x}_2=16, \\ s_2=2$. The test statistic is [ANS]\nThe P-Value is [ANS]\nThe conclusion is [ANS] A. There is sufficient evidence to warrant rejection of the claim that the two populations have the same mean. B. There is not sufficient evidence to warrant rejection of the claim that the two populations have the same mean.", "answer_v3": [ "-10.3132747642849", "6.13736E-25", "A" ], "answer_type_v3": [ "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "A", "B" ] ] }, { "id": "Statistics_0292", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Independent samples - t", "level": "3", "keywords": [ "statistics", "two sample", "inference", "t score" ], "problem_v1": "Samples were collected from two ponds in the Bahamas to compare salinity values (in parts per thousand). Several samples were drawn at each site. Pond 1: 36.75, 37.03, 37.71, 37.36, 36.72, 37.03, 37.02 Pond 2: 38.71, 38.24, 38.53, 38.66, 39.21 Use a $0.05$ significance level to test the claim that the two ponds have the same mean salinity value.\n(a) The test statistic is [ANS]. (b) The conclusion is [ANS] A. There is sufficient evidence to indicate that the two ponds have different salinity values. B. There is not sufficient evidence to indicate that the two ponds have different salinity values.\n(c) We should [ANS] A. not take the results too seriously since neither sample is big enough to be meaningful. B. check to see if the data appear close to Normal since the sum of the sample sizes is less than 15. C. remove the largest and smallest values from the larger data set and only test equal size samples. D. All of the above.", "answer_v1": [ "-7.70945", "A", "B" ], "answer_type_v1": [ "NV", "MCS", "MCS" ], "options_v1": [ [], [ "A", "B" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Samples were collected from two ponds in the Bahamas to compare salinity values (in parts per thousand). Several samples were drawn at each site. Pond 1: 37.54, 38.85, 36.72, 37.02, 37.45, 37.32, 37.03 Pond 2: 38.24, 38.89, 39.04, 38.66, 38.53 Use a $0.05$ significance level to test the claim that the two ponds have the same mean salinity value.\n(a) The test statistic is [ANS]. (b) The conclusion is [ANS] A. There is sufficient evidence to indicate that the two ponds have different salinity values. B. There is not sufficient evidence to indicate that the two ponds have different salinity values.\n(c) We should [ANS] A. remove the largest and smallest values from the larger data set and only test equal size samples. B. check to see if the data appear close to Normal since the sum of the sample sizes is less than 15. C. not take the results too seriously since neither sample is big enough to be meaningful. D. All of the above.", "answer_v2": [ "-4.22815", "A", "B" ], "answer_type_v2": [ "NV", "MCS", "MCS" ], "options_v2": [ [], [ "A", "B" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Samples were collected from two ponds in the Bahamas to compare salinity values (in parts per thousand). Several samples were drawn at each site. Pond 1: 37.03, 37.71, 36.72, 37.03, 37.01, 37.02, 37.45 Pond 2: 40.08, 38.66, 39.21, 38.24, 39.05 Use a $0.05$ significance level to test the claim that the two ponds have the same mean salinity value.\n(a) The test statistic is [ANS]. (b) The conclusion is [ANS] A. There is not sufficient evidence to indicate that the two ponds have different salinity values. B. There is sufficient evidence to indicate that the two ponds have different salinity values.\n(c) We should [ANS] A. remove the largest and smallest values from the larger data set and only test equal size samples. B. not take the results too seriously since neither sample is big enough to be meaningful. C. check to see if the data appear close to Normal since the sum of the sample sizes is less than 15. D. All of the above.", "answer_v3": [ "-5.74986", "B", "C" ], "answer_type_v3": [ "NV", "MCS", "MCS" ], "options_v3": [ [], [ "A", "B" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0293", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Independent samples - t", "level": "3", "keywords": [ "statistics", "two sample", "inference", "t score" ], "problem_v1": "A market research firm supplies manufacturers with estimates of the retail sales of their products from samples of retail stores. Marketing managers are prone to look at the estimate and ignore sampling error. An SRS of $26$ stores this year shows mean sales of $70$ units of a small appliance, with a standard deviation of $11.2$ units. During the same point in time last year, an SRS of $25$ stores had mean sales of $80.416$ units, with standard deviation $11.5$ units. A decrease from $80.416$ to $70$ is a drop of about 15\\%.\n1. Construct a 95\\% confidence interval estimate of the difference $\\mu_1-\\mu_2$, where $\\mu_1$ is the mean of this year's sales and $\\mu_2$ is the mean of last year's sales.\n(a) [ANS] $< (\\mu_1-\\mu_2) <$ [ANS]\n(b) The margin of error is [ANS].\n2. At a $0.05$ significance level, is there sufficient evidence to show that sales this year are different from last year? [ANS] A. No B. Yes", "answer_v1": [ "-16.9799", "-3.85208", "6.56392", "B" ], "answer_type_v1": [ "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B" ] ], "problem_v2": "A market research firm supplies manufacturers with estimates of the retail sales of their products from samples of retail stores. Marketing managers are prone to look at the estimate and ignore sampling error. An SRS of $11$ stores this year shows mean sales of $83$ units of a small appliance, with a standard deviation of $6.4$ units. During the same point in time last year, an SRS of $17$ stores had mean sales of $77.048$ units, with standard deviation $5.7$ units. An increase from $77.048$ to $83$ is a rise of about 7\\%.\n1. Construct a 99\\% confidence interval estimate of the difference $\\mu_1-\\mu_2$, where $\\mu_1$ is the mean of this year's sales and $\\mu_2$ is the mean of last year's sales.\n(a) [ANS] $< (\\mu_1-\\mu_2) <$ [ANS]\n(b) The margin of error is [ANS].\n2. At a $0.01$ significance level, is there sufficient evidence to show that sales this year are different from last year? [ANS] A. No B. Yes", "answer_v2": [ "-1.57114", "13.4751", "7.52314", "A" ], "answer_type_v2": [ "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B" ] ], "problem_v3": "A market research firm supplies manufacturers with estimates of the retail sales of their products from samples of retail stores. Marketing managers are prone to look at the estimate and ignore sampling error. An SRS of $16$ stores this year shows mean sales of $71$ units of a small appliance, with a standard deviation of $7.8$ units. During the same point in time last year, an SRS of $22$ stores had mean sales of $78.566$ units, with standard deviation $15.2$ units. A decrease from $78.566$ to $71$ is a drop of about 11\\%.\n1. Construct a 95\\% confidence interval estimate of the difference $\\mu_1-\\mu_2$, where $\\mu_1$ is the mean of this year's sales and $\\mu_2$ is the mean of last year's sales.\n(a) [ANS] $< (\\mu_1-\\mu_2) <$ [ANS]\n(b) The margin of error is [ANS].\n2. At a $0.05$ significance level, is there sufficient evidence to show that sales this year are different from last year? [ANS] A. No B. Yes", "answer_v3": [ "-15.6274", "0.495368", "8.06137", "A" ], "answer_type_v3": [ "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B" ] ] }, { "id": "Statistics_0296", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Independent samples - t", "level": "3", "keywords": [ "hypothesis testing", "t distribution" ], "problem_v1": "There are several sections of statistics, some in the morning (AM) and some in the afternoon (PM). We want to see if afternoon sections do better. We randomly select 24 students from the AM sections and 28 students from the PM sections. Their final averages (out of 100) are given in the table with other relevant statistics. The degrees of freedom (d.f.) is given to save calculation time if you are not using software.\n$\\begin{array}{ccccc}\\hline & n & \\bar x & s^2 & s \\\\ \\hline PM & 28 & 78.6 & 277.2225 & 16.65 \\\\ \\hline AM & 24 & 72.5 & 272.5801 & 16.51 \\\\ \\hline degrees of freedom: d.f.=49 \\\\ \\hline \\end{array}$ degrees of freedom: d.f.=49 Test the claim that the average for all students in the PM sections is greater than the AM sections. Use a 0.05 significance level.\n(a) Find the test statistic. [ANS]\n(b) Find the critical value. [ANS]\n(c) Is there sufficient data to support the claim? [ANS] [ANS]", "answer_v1": [ "(78.6-72.5)/[sqrt(11.3575+9.9008)]", "1.67655", "No" ], "answer_type_v1": [ "EX", "NV", "TF" ], "options_v1": [ [], [], [] ], "problem_v2": "There are several sections of statistics, some in the morning (AM) and some in the afternoon (PM). We want to see if afternoon sections do better. We randomly select 20 students from the AM sections and 30 students from the PM sections. Their final averages (out of 100) are given in the table with other relevant statistics. The degrees of freedom (d.f.) is given to save calculation time if you are not using software.\n$\\begin{array}{ccccc}\\hline & n & \\bar x & s^2 & s \\\\ \\hline PM & 30 & 76.7 & 274.8964 & 16.58 \\\\ \\hline AM & 20 & 70.6 & 389.6676 & 19.74 \\\\ \\hline degrees of freedom: d.f.=36 \\\\ \\hline \\end{array}$ degrees of freedom: d.f.=36 Test the claim that the average for all students in the PM sections is greater than the AM sections. Use a 0.05 significance level.\n(a) Find the test statistic. [ANS]\n(b) Find the critical value. [ANS]\n(c) Is there sufficient data to support the claim? [ANS] [ANS]", "answer_v2": [ "(76.7-70.6)/[sqrt(19.4834+9.16321)]", "1.6883", "No" ], "answer_type_v2": [ "EX", "NV", "TF" ], "options_v2": [ [], [], [] ], "problem_v3": "There are several sections of statistics, some in the morning (AM) and some in the afternoon (PM). We want to see if afternoon sections do better. We randomly select 21 students from the AM sections and 28 students from the PM sections. Their final averages (out of 100) are given in the table with other relevant statistics. The degrees of freedom (d.f.) is given to save calculation time if you are not using software.\n$\\begin{array}{ccccc}\\hline & n & \\bar x & s^2 & s \\\\ \\hline PM & 28 & 77.7 & 279.8929 & 16.73 \\\\ \\hline AM & 21 & 71.1 & 256.9609 & 16.03 \\\\ \\hline degrees of freedom: d.f.=44 \\\\ \\hline \\end{array}$ degrees of freedom: d.f.=44 Test the claim that the average for all students in the PM sections is greater than the AM sections. Use a 0.05 significance level.\n(a) Find the test statistic. [ANS]\n(b) Find the critical value. [ANS]\n(c) Is there sufficient data to support the claim? [ANS] [ANS]", "answer_v3": [ "(77.7-71.1)/[sqrt(12.2362+9.99617)]", "1.68023", "No" ], "answer_type_v3": [ "EX", "NV", "TF" ], "options_v3": [ [], [], [] ] }, { "id": "Statistics_0297", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Independent samples - t", "level": "3", "keywords": [ "hypothesis testing", "t distribution" ], "problem_v1": "There is a lot of interest in the relationship between studying music and studying math. We will look at some sample data that investigates this relationship. Here are the Math SAT scores from 9 students who studied music through high school and 11 students who did not. The degrees of freedom (d.f.) is given to save calculation time if you are not using software.\n$\\begin{array}{ccccccccccccccc}\\hline & Math SAT Scores & mean & s^2 & s \\\\ \\hline Music (x_1) & 556 & 585 & 642 & 564 & 574 & 556 & 593 & 539 & 626 & & 581.666666666667 & 1156.75 & 34.0110276234047 \\\\ \\hline No Music (x_2) & 539 & 490 & 540 & 484 & 550 & 547 & 560 & 475 & 488 & 531 & 558 & 523.818181818182 & 1063.96363636362 & 32.6184554564379 \\\\ \\hline degrees of freedom: d.f.=17 \\\\ \\hline \\end{array}$ degrees of freedom: d.f.=17 Test the claim that students who study music in high school have a higher average Math SAT score than those who do not. Use a 0.01 significance level.\n(a) Find the test statistic. [ANS]\n(b) Find the critical value. [ANS]\n(c) Is there sufficient data to support the claim? [ANS] [ANS]", "answer_v1": [ "(581.667-523.818)/[sqrt(128.528+96.724)]", "2.56693", "Yes" ], "answer_type_v1": [ "EX", "NV", "TF" ], "options_v1": [ [], [], [] ], "problem_v2": "There is a lot of interest in the relationship between studying music and studying math. We will look at some sample data that investigates this relationship. Here are the Math SAT scores from 7 students who studied music through high school and 12 students who did not. The degrees of freedom (d.f.) is given to save calculation time if you are not using software.\n$\\begin{array}{cccccccccccccccc}\\hline & Math SAT Scores & mean & s^2 & s \\\\ \\hline Music (x_1) & 654 & 555 & 603 & 587 & 652 & 559 & 608 & & 602.571428571429 & 1586.95238095235 & 39.8365708985142 \\\\ \\hline No Music (x_2) & 534 & 495 & 542 & 530 & 580 & 516 & 554 & 536 & 506 & 479 & 569 & 570 & 534.25 & 979.477272727273 & 31.2966016162661 \\\\ \\hline degrees of freedom: d.f.=10 \\\\ \\hline \\end{array}$ degrees of freedom: d.f.=10 Test the claim that students who study music in high school have a higher average Math SAT score than those who do not. Use a 0.01 significance level.\n(a) Find the test statistic. [ANS]\n(b) Find the critical value. [ANS]\n(c) Is there sufficient data to support the claim? [ANS] [ANS]", "answer_v2": [ "(602.571-534.25)/[sqrt(226.707+81.6231)]", "2.76377", "Yes" ], "answer_type_v2": [ "EX", "NV", "TF" ], "options_v2": [ [], [], [] ], "problem_v3": "There is a lot of interest in the relationship between studying music and studying math. We will look at some sample data that investigates this relationship. Here are the Math SAT scores from 7 students who studied music through high school and 11 students who did not. The degrees of freedom (d.f.) is given to save calculation time if you are not using software.\n$\\begin{array}{ccccccccccccccc}\\hline & Math SAT Scores & mean & s^2 & s \\\\ \\hline Music (x_1) & 574 & 535 & 646 & 554 & 579 & 612 & 610 & & 587.142857142857 & 1443.47619047621 & 37.9931071442731 \\\\ \\hline No Music (x_2) & 573 & 523 & 488 & 452 & 517 & 525 & 542 & 547 & 485 & 492 & 519 & 514.818181818182 & 1142.76363636362 & 33.8047871811615 \\\\ \\hline degrees of freedom: d.f.=12 \\\\ \\hline \\end{array}$ degrees of freedom: d.f.=12 Test the claim that students who study music in high school have a higher average Math SAT score than those who do not. Use a 0.01 significance level.\n(a) Find the test statistic. [ANS]\n(b) Find the critical value. [ANS]\n(c) Is there sufficient data to support the claim? [ANS] [ANS]", "answer_v3": [ "(587.143-514.818)/[sqrt(206.211+103.888)]", "2.681", "Yes" ], "answer_type_v3": [ "EX", "NV", "TF" ], "options_v3": [ [], [], [] ] }, { "id": "Statistics_0298", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Independent samples - t", "level": "3", "keywords": [ "statistics", "Two Populations", "Inference" ], "problem_v1": "When the necessary conditions are met, a two-tail test is being conducted to test the difference between two population proportions. If the value of the test statistic $z$ is 2.05, the $p-$ value is: [ANS] A. 0.0404 B. 0.0202 C. 0.2399 D. 0.4798\nWhich of the following statements is not correct for an F-distribution? [ANS] A. Degrees of freedom for the numerator can be larger, smaller, or equal to the degrees of freedom for the denominator. B. Variables that are F-distributed range from 0 to 100 C. Degrees of freedom for the denominator are always smaller than the degrees of freedom for the numerator D. Exact shape of the distribution is determined by two numbers of degrees of freedom", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "When the necessary conditions are met, a two-tail test is being conducted to test the difference between two population proportions. If the value of the test statistic $z$ is 2.05, the $p-$ value is: [ANS] A. 0.2399 B. 0.0202 C. 0.4798 D. 0.0404\nTwo independent samples of sizes 25 and 35 are randomly selected from two normal populations with equal variances. In order to test difference between the population means, the test statistic is: [ANS] A. Student t distributed with 58 degrees of freedom B. Student t distributed with 33 degrees of freedom C. a standard normal random variable D. approximately standard normal random variable", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "When the necessary conditions are met, a two-tail test is being conducted to test the difference between two population proportions. If the value of the test statistic $z$ is 2.05, the $p-$ value is: [ANS] A. 0.2399 B. 0.0404 C. 0.0202 D. 0.4798\nIn testing the difference between two population means using two independent samples, the sampling distribution of the sample mean difference $\\bar{x}_1-\\bar{x}_2$ is normal if the: [ANS] A. populations are non-normal and the sample sizes are large B. populations are normal C. sample sizes are both greater than 30 D. all of the above are required conditions", "answer_v3": [ "B", "B" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0299", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Independent samples - t", "level": "3", "keywords": [ "statistics", "Two Populations", "Inference" ], "problem_v1": "Select True or False, depending on whether the corresponding statement is true or false.\n[ANS] 1. Independent samples are those for which the selection process for one is not related to the selection process for the other. [ANS] 2. We say that two samples are dependent when the selection process for one is related to the selection process for the other. [ANS] 3. We can use either the $z-$ test ot the $t-$ test to determine whether two population variances are equal. [ANS] 4. In testing the difference between two population means using two independent samples, we use the pooled variance in estimating the standard error of the sampling distribution of the sample mean difference $\\bar{x}_1-\\bar{x}_2$ if the populations are normal with equal variances.", "answer_v1": [ "T", "T", "F", "T" ], "answer_type_v1": [ "TF", "TF", "TF", "TF" ], "options_v1": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v2": "Select True or False, depending on whether the corresponding statement is true or false.\n[ANS] 1. We say that two samples are dependent when the selection process for one is related to the selection process for the other. [ANS] 2. In testing the difference between two population means using two independent samples, we use the pooled variance in estimating the standard error of the sampling distribution of the sample mean difference $\\bar{x}_1-\\bar{x}_2$ if the populations are normal with equal variances. [ANS] 3. Independent samples are those for which the selection process for one is not related to the selection process for the other. [ANS] 4. When comparing two population variances, we use the ratio $\\frac{\\sigma^2_1}{\\sigma^2_2}$ rather than the difference $\\sigma^2_1-\\sigma^2_2$.", "answer_v2": [ "T", "T", "T", "T" ], "answer_type_v2": [ "TF", "TF", "TF", "TF" ], "options_v2": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v3": "Select True or False, depending on whether the corresponding statement is true or false.\n[ANS] 1. In comparing two population means when the samples are dependent, the variable under consideration is $\\hat{p}_1-\\hat{p}_2$. [ANS] 2. When comparing two population variances, we use the ratio $\\frac{\\sigma^2_1}{\\sigma^2_2}$ rather than the difference $\\sigma^2_1-\\sigma^2_2$. [ANS] 3. Independent samples are those for which the selection process for one is not related to the selection process for the other. [ANS] 4. The pooled variances $t-$ test requires that the two population variances are not the same.", "answer_v3": [ "F", "T", "T", "F" ], "answer_type_v3": [ "TF", "TF", "TF", "TF" ], "options_v3": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ] }, { "id": "Statistics_0300", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Independent samples - t", "level": "3", "keywords": [ "statistics", "Two Populations", "Inference" ], "problem_v1": "The expected value of the difference of two sample means equals the difference of the corresponding population means: [ANS] A. the statement is correct under all circumstances B. only if the populations are approximately normal and the sample sizes are large C. only if the samples are independent D. only if the populations are normally distributed\nIn constructing a confidence interval estimate for the difference between two population proportions, we: [ANS] A. pool the population proportions when they are equal B. pool the population proportions when the populations are normally distributed C. never pool the population proportions D. pool the population proportions when the population means are equal", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "The expected value of the difference of two sample means equals the difference of the corresponding population means: [ANS] A. only if the samples are independent B. only if the populations are approximately normal and the sample sizes are large C. only if the populations are normally distributed D. the statement is correct under all circumstances\nTwo independent samples are drawn from two normal populations where the population variances are assumed to be equal. The sampling distribution of the ratio of the two sample variances is: [ANS] A. an F-distribution B. a chi-squared distribution C. a normal distribution D. a Student t-distribution", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "The expected value of the difference of two sample means equals the difference of the corresponding population means: [ANS] A. only if the samples are independent B. the statement is correct under all circumstances C. only if the populations are approximately normal and the sample sizes are large D. only if the populations are normally distributed\nFor testing the difference between two population proportions, the pooled proportion estimate should be used to compute the value of the test statistic when the: [ANS] A. null hypothesis states that the two population proportions are equal B. sample sizes are small C. populations are normally distributed D. samples are independently drawn from the populations", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0301", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Independent samples - t", "level": "4", "keywords": [ "statistics", "Inference about a population" ], "problem_v1": "Random samples of female and male drivers are asked to estimate the number of miles that each drives in a year. The data is below: Females (Population 1): 9300, 8800, 8600, 9700, 8400, 8500, 8000, 8700, 9600 Males (Population 2): 8800, 10700, 9900, 10000, 9800, 9500, 8900 Is there evidence, at an $\\alpha=0.045$ level of significance, to conclude that there is a difference in the number of miles driven between females and males? (Assume that the population variances are equal.) Carry out an appropriate hypothesis test, filling in the information requested. A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Reject $H_1$. B. Do Not Reject $H_1$. C. Reject $H_0$. D. Do Not Reject $H_0$.", "answer_v1": [ "-2.63611518103772", "(-infinity,-2.20119) U (2.20119,infinity)", "0.01955184", "C" ], "answer_type_v1": [ "NV", "INT", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v2": "Random samples of female and male drivers are asked to estimate the number of miles that each drives in a year. The data is below: Females (Population 1): 7800, 8300, 9200, 9100, 8900, 9400, 9000, 8400, 8900 Males (Population 2): 10000, 10200, 8800, 9300, 9400, 9300, 9800 Is there evidence, at an $\\alpha=0.03$ level of significance, to conclude that there is a difference in the number of miles driven between females and males? (Assume that the population variances are equal.) Carry out an appropriate hypothesis test, filling in the information requested. A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Reject $H_0$. B. Do Not Reject $H_1$. C. Do Not Reject $H_0$. D. Reject $H_1$.", "answer_v2": [ "-3.04822665031225", "(-infinity,-2.4149) U (2.4149,infinity)", "0.0086803", "A" ], "answer_type_v2": [ "NV", "INT", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v3": "Random samples of female and male drivers are asked to estimate the number of miles that each drives in a year. The data is below: Females (Population 1): 8200, 7900, 9000, 9500, 8900, 9200, 8900, 9700, 9100 Males (Population 2): 10800, 10500, 8800, 9400, 9900, 9900, 10000 Is there evidence, at an $\\alpha=0.03$ level of significance, to conclude that there is a difference in the number of miles driven between females and males? (Assume that the population variances are equal.) Carry out an appropriate hypothesis test, filling in the information requested. A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Reject $H_1$. B. Do Not Reject $H_1$. C. Reject $H_0$. D. Do Not Reject $H_0$.", "answer_v3": [ "-3.1293790610894", "(-infinity,-2.4149) U (2.4149,infinity)", "0.00738832", "C" ], "answer_type_v3": [ "NV", "INT", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0302", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Independent samples - t", "level": "4", "keywords": [ "statistics", "Inference about a population" ], "problem_v1": "Random samples of resting heart rates are taken from two groups. Population 1 exercises regularly, and Population 2 does not. The data from these two samples is given below: Population 1: 68, 70, 69, 73, 72, 67, 70 Population 2: 69, 76, 73, 73, 78, 72, 72, 69 Is there evidence, at an $\\alpha=0.05$ level of significance, to conclude that there those who exercise regularly have lower resting heart rates? (Assume that the population variances are equal.) Carry out an appropriate hypothesis test, filling in the information requested. A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Reject $H_0$. B. Do Not Reject $H_0$. C. Do Not Reject $H_1$. D. Reject $H_1$.", "answer_v1": [ "-2.07472077962287", "(-infinity,-1.77093)", "0.0292144", "A" ], "answer_type_v1": [ "NV", "INT", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v2": "Random samples of resting heart rates are taken from two groups. Population 1 exercises regularly, and Population 2 does not. The data from these two samples is given below: Population 1: 71, 72, 65, 67, 69, 65, 62 Population 2: 73, 70, 77, 69, 68, 75, 70, 71 Is there evidence, at an $\\alpha=0.075$ level of significance, to conclude that there those who exercise regularly have lower resting heart rates? (Assume that the population variances are equal.) Carry out an appropriate hypothesis test, filling in the information requested. A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Do Not Reject $H_1$. B. Do Not Reject $H_0$. C. Reject $H_1$. D. Reject $H_0$.", "answer_v2": [ "-2.50784194681066", "(-infinity,-1.52992)", "0.0130992", "D" ], "answer_type_v2": [ "NV", "INT", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v3": "Random samples of resting heart rates are taken from two groups. Population 1 exercises regularly, and Population 2 does not. The data from these two samples is given below: Population 1: 72, 70, 71, 71, 71, 71, 70 Population 2: 74, 71, 78, 70, 77, 76, 72, 70 Is there evidence, at an $\\alpha=0.04$ level of significance, to conclude that there those who exercise regularly have lower resting heart rates? (Assume that the population variances are equal.) Carry out an appropriate hypothesis test, filling in the information requested. A. The value of the standardized test statistic: [ANS]\nNote: For the next part, your answer should use interval notation. An answer of the form $(-\\infty, a)$ is expressed (-infty, a), an answer of the form $(b, \\infty)$ is expressed (b, infty), and an answer of the form $(-\\infty, a) \\cup (b, \\infty)$ is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: [ANS]\nC. The p-value is [ANS]\nD. Your decision for the hypothesis test: [ANS] A. Do Not Reject $H_1$. B. Reject $H_1$. C. Do Not Reject $H_0$. D. Reject $H_0$.", "answer_v3": [ "-2.128026956534", "(-infinity,-1.89887)", "0.0265199", "D" ], "answer_type_v3": [ "NV", "INT", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0303", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Type I/type II errors and power", "level": "1", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "Assume you are using a significance level of $\\alpha=0.05$ to test the claim that $\\mu < 17$ and that your sample is a random sample of $45$ values. Find $\\beta$, the probability of making a type II error (failing to reject a false null hypothesis), given that the population actually has a normal distribution with $\\mu=13$ and $\\sigma=7.$ $\\beta=$ [ANS]", "answer_v1": [ "0.0143254" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Assume you are using a significance level of $\\alpha=0.05$ to test the claim that $\\mu < 7$ and that your sample is a random sample of $37$ values. Find $\\beta$, the probability of making a type II error (failing to reject a false null hypothesis), given that the population actually has a normal distribution with $\\mu=5$ and $\\sigma=9.$ $\\beta=$ [ANS]", "answer_v2": [ "0.615344" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Assume you are using a significance level of $\\alpha=0.05$ to test the claim that $\\mu < 10$ and that your sample is a random sample of $41$ values. Find $\\beta$, the probability of making a type II error (failing to reject a false null hypothesis), given that the population actually has a normal distribution with $\\mu=8$ and $\\sigma=7.$ $\\beta=$ [ANS]", "answer_v3": [ "0.426825" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0304", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Type I/type II errors and power", "level": "1", "keywords": [ "Statistics", "Hypothesis Testing" ], "problem_v1": "Type I error is: [ANS] A. Deciding the null hypothesis is false when it is true B. Deciding the alternative hypothesis is true when it is true C. Deciding the null hypothesis is true when it is false D. Deciding the alternative hypothesis is true when it is false E. All of the above F. None of the above\nType II error is: [ANS] A. Deciding the null hypothesis is true when it is false B. Deciding the alternative hypothesis is true when it is true C. Deciding the alternative hypothesis is false when it is true D. Deciding the null hypothesis is false when it is true E. All of the above F. None of the above", "answer_v1": [ "A", "A" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Type I error is: [ANS] A. Deciding the null hypothesis is false when it is true B. Deciding the alternative hypothesis is true when it is true C. Deciding the null hypothesis is true when it is false D. Deciding the alternative hypothesis is true when it is false E. All of the above F. None of the above\nType II error is: [ANS] A. Deciding the alternative hypothesis is false when it is true B. Deciding the alternative hypothesis is true when it is true C. Deciding the null hypothesis is false when it is true D. Deciding the null hypothesis is true when it is false E. All of the above F. None of the above", "answer_v2": [ "A", "D" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Type I error is: [ANS] A. Deciding the null hypothesis is false when it is true B. Deciding the null hypothesis is true when it is false C. Deciding the alternative hypothesis is true when it is true D. Deciding the alternative hypothesis is true when it is false E. All of the above F. None of the above\nType II error is: [ANS] A. Deciding the alternative hypothesis is false when it is true B. Deciding the null hypothesis is true when it is false C. Deciding the alternative hypothesis is true when it is true D. Deciding the null hypothesis is false when it is true E. All of the above F. None of the above", "answer_v3": [ "A", "B" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Statistics_0305", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Type I/type II errors and power", "level": "2", "keywords": [ "Statistics", "Hypothesis Testing" ], "problem_v1": "Consider a hypothesis test to decide whether the mean annual consumption of beer in the nation's capital is less than the national mean. Answer the following questions. a. \"The mean annual consumption of beer in the nation's captial is not less than the national mean but the result of the sampling leads to the conclusion that the mean annual consumption of beer in the nation's capital is less than the national mean\" is a [ANS] A. Type I error B. Correct decision C. Type II error\nb. \"The mean annual consumption of beer in the nation's captial is not less than the national mean and the result of the sampling does not lead to the conclusion that the mean annual consumption of beer in the nation's capital is less than the national mean\" is a [ANS] A. Type I error B. Type II error C. Correct decision\nc. \"The mean annual consumption of beer in the nation's captial is less than the national mean and the result of the sampling leads to the conclusion that the mean annul consumption of beer in the nation's capital is less than the national mean\" is a [ANS] A. Correct decision B. Type II error C. Type I error\nd. \"The mean annual consumption of beer in the nation's captial is less than the national mean but the result of the sampling does not lead to the conclusion that the mean annual consumption of beer in the nation's capital is less than the national mean\" is a [ANS] A. Type II error B. Correct decision C. Type I error", "answer_v1": [ "A", "C", "A", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v2": "Consider a hypothesis test to decide whether the mean annual consumption of beer in the nation's capital is less than the national mean. Answer the following questions. a. \"The mean annual consumption of beer in the nation's captial is less than the national mean and the result of the sampling leads to the conclusion that the mean annul consumption of beer in the nation's capital is less than the national mean\" is a [ANS] A. Type I error B. Type II error C. Correct decision\nb. \"The mean annual consumption of beer in the nation's captial is not less than the national mean and the result of the sampling does not lead to the conclusion that the mean annual consumption of beer in the nation's capital is less than the national mean\" is a [ANS] A. Correct decision B. Type II error C. Type I error\nc. \"The mean annual consumption of beer in the nation's captial is not less than the national mean but the result of the sampling leads to the conclusion that the mean annual consumption of beer in the nation's capital is less than the national mean\" is a [ANS] A. Type I error B. Type II error C. Correct decision\nd. \"The mean annual consumption of beer in the nation's captial is less than the national mean but the result of the sampling does not lead to the conclusion that the mean annual consumption of beer in the nation's capital is less than the national mean\" is a [ANS] A. Type I error B. Type II error C. Correct decision", "answer_v2": [ "C", "A", "A", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v3": "Consider a hypothesis test to decide whether the mean annual consumption of beer in the nation's capital is less than the national mean. Answer the following questions. a. \"The mean annual consumption of beer in the nation's captial is not less than the national mean and the result of the sampling does not lead to the conclusion that the mean annual consumption of beer in the nation's capital is less than the national mean\" is a [ANS] A. Type I error B. Correct decision C. Type II error\nb. \"The mean annual consumption of beer in the nation's captial is less than the national mean but the result of the sampling does not lead to the conclusion that the mean annual consumption of beer in the nation's capital is less than the national mean\" is a [ANS] A. Type II error B. Type I error C. Correct decision\nc. \"The mean annual consumption of beer in the nation's captial is less than the national mean and the result of the sampling leads to the conclusion that the mean annul consumption of beer in the nation's capital is less than the national mean\" is a [ANS] A. Correct decision B. Type II error C. Type I error\nd. \"The mean annual consumption of beer in the nation's captial is not less than the national mean but the result of the sampling leads to the conclusion that the mean annual consumption of beer in the nation's capital is less than the national mean\" is a [ANS] A. Correct decision B. Type I error C. Type II error", "answer_v3": [ "B", "A", "A", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ] }, { "id": "Statistics_0306", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Type I/type II errors and power", "level": "1", "keywords": [ "Statistics", "Hypothesis Testing" ], "problem_v1": "Consider a hypothesis test to decide whether the mean cost to community hospitals per patient per day in Ohio exceed the national mean. Answer the following questions. a. \" The mean cost to community hospitals per patient per day in Ohio does not exceed the national mean but the result of the sampling leads to the conclusion that the mean cost to community hospitals per patient per day in Ohio exceeds the national mean \" is a [ANS] A. Type II error B. correct decision C. Type I error\nb. \" The mean cost to community hospitals per patient per day in Ohio does not exceed the national mean and the result of the sampling does not lead to the conclusion that the mean cost to community hospitals per patient per day in Ohio exceeds the national mean \" is a [ANS] A. Type I error B. Type II error C. correct decision\nc. \" The mean cost to community hospitals per patient per day in Ohio exceeds the national mean and the result of the sampling leads to the conclusion that the mean cost to community hospitals per patient per day in Ohio exceeds the national mean \" is a [ANS] A. correct decision B. Type II error C. Type I error\nd. \" The mean cost to community hospitals per patient per day in Ohio exceeds the national mean but the result of the sampling does not lead to the conclusion that the mean cost to community hospitals per patient per day in Ohio exceeds the national mean \" is a [ANS] A. Type I error B. correct decision C. Type II error", "answer_v1": [ "A", "C", "A", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v2": "Consider a hypothesis test to decide whether the mean cost to community hospitals per patient per day in Ohio exceed the national mean. Answer the following questions. a. \" The mean cost to community hospitals per patient per day in Ohio exceeds the national mean and the result of the sampling leads to the conclusion that the mean cost to community hospitals per patient per day in Ohio exceeds the national mean \" is a [ANS] A. Type I error B. Type II error C. correct decision\nb. \" The mean cost to community hospitals per patient per day in Ohio does not exceed the national mean and the result of the sampling does not lead to the conclusion that the mean cost to community hospitals per patient per day in Ohio exceeds the national mean \" is a [ANS] A. correct decision B. Type II error C. Type I error\nc. \" The mean cost to community hospitals per patient per day in Ohio does not exceed the national mean but the result of the sampling leads to the conclusion that the mean cost to community hospitals per patient per day in Ohio exceeds the national mean \" is a [ANS] A. Type II error B. Type I error C. correct decision\nd. \" The mean cost to community hospitals per patient per day in Ohio exceeds the national mean but the result of the sampling does not lead to the conclusion that the mean cost to community hospitals per patient per day in Ohio exceeds the national mean \" is a [ANS] A. Type II error B. Type I error C. correct decision", "answer_v2": [ "C", "A", "A", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v3": "Consider a hypothesis test to decide whether the mean cost to community hospitals per patient per day in Ohio exceed the national mean. Answer the following questions. a. \" The mean cost to community hospitals per patient per day in Ohio does not exceed the national mean and the result of the sampling does not lead to the conclusion that the mean cost to community hospitals per patient per day in Ohio exceeds the national mean \" is a [ANS] A. Type I error B. correct decision C. Type II error\nb. \" The mean cost to community hospitals per patient per day in Ohio exceeds the national mean but the result of the sampling does not lead to the conclusion that the mean cost to community hospitals per patient per day in Ohio exceeds the national mean \" is a [ANS] A. Type I error B. Type II error C. correct decision\nc. \" The mean cost to community hospitals per patient per day in Ohio exceeds the national mean and the result of the sampling leads to the conclusion that the mean cost to community hospitals per patient per day in Ohio exceeds the national mean \" is a [ANS] A. correct decision B. Type II error C. Type I error\nd. \" The mean cost to community hospitals per patient per day in Ohio does not exceed the national mean but the result of the sampling leads to the conclusion that the mean cost to community hospitals per patient per day in Ohio exceeds the national mean \" is a [ANS] A. correct decision B. Type II error C. Type I error", "answer_v3": [ "B", "A", "A", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ] }, { "id": "Statistics_0307", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Type I/type II errors and power", "level": "1", "keywords": [ "Statistics", "Hypothesis Testing" ], "problem_v1": "Consider the following hypothesis test. The null hypothesis is \"The mean body temperature for humans is 98.6 degrees Farenheit.\" and the alternative hypothesis is \"The mean body temperature for humans differs from 98.6 degrees Farenheit.\" Answer the following questions. a. \"The mean body temperature for humans in fact is 98.6 degrees Farenheit but the result of the sampling lead to the conclusion that the mean body temprature for humans differ from 98.6 degrees Farenheit\" is a [ANS] A. Type I error B. correct decision C. Type II error\nb. \"The mean body temperature for humans in fact is 98.6 degrees Farenheit and the result of the sampling do not lead to the rejection of the fact that the mean body temprature is 98.6 degrees Farenheit\" is a [ANS] A. Type I error B. Type II error C. correct decision\nc. \"The mean body temperature for humans in fact differs from 98.6 degrees Farenheit and the result of the sampling lead to that conclusion\" is a [ANS] A. correct decision B. Type II error C. Type I error\nd. \"The mean body temperature for humans in fact differs from 98.6 degrees Farenheit but the result of the sampling fail to lead that conclusion\" is a [ANS] A. Type II error B. correct decision C. Type I error", "answer_v1": [ "A", "C", "A", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v2": "Consider the following hypothesis test. The null hypothesis is \"The mean body temperature for humans is 98.6 degrees Farenheit.\" and the alternative hypothesis is \"The mean body temperature for humans differs from 98.6 degrees Farenheit.\" Answer the following questions. a. \"The mean body temperature for humans in fact differs from 98.6 degrees Farenheit and the result of the sampling lead to that conclusion\" is a [ANS] A. Type I error B. Type II error C. correct decision\nb. \"The mean body temperature for humans in fact is 98.6 degrees Farenheit and the result of the sampling do not lead to the rejection of the fact that the mean body temprature is 98.6 degrees Farenheit\" is a [ANS] A. correct decision B. Type II error C. Type I error\nc. \"The mean body temperature for humans in fact is 98.6 degrees Farenheit but the result of the sampling lead to the conclusion that the mean body temprature for humans differ from 98.6 degrees Farenheit\" is a [ANS] A. Type I error B. Type II error C. correct decision\nd. \"The mean body temperature for humans in fact differs from 98.6 degrees Farenheit but the result of the sampling fail to lead that conclusion\" is a [ANS] A. Type I error B. Type II error C. correct decision", "answer_v2": [ "C", "A", "A", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v3": "Consider the following hypothesis test. The null hypothesis is \"The mean body temperature for humans is 98.6 degrees Farenheit.\" and the alternative hypothesis is \"The mean body temperature for humans differs from 98.6 degrees Farenheit.\" Answer the following questions. a. \"The mean body temperature for humans in fact is 98.6 degrees Farenheit and the result of the sampling do not lead to the rejection of the fact that the mean body temprature is 98.6 degrees Farenheit\" is a [ANS] A. Type I error B. correct decision C. Type II error\nb. \"The mean body temperature for humans in fact differs from 98.6 degrees Farenheit but the result of the sampling fail to lead that conclusion\" is a [ANS] A. Type II error B. Type I error C. correct decision\nc. \"The mean body temperature for humans in fact differs from 98.6 degrees Farenheit and the result of the sampling lead to that conclusion\" is a [ANS] A. correct decision B. Type II error C. Type I error\nd. \"The mean body temperature for humans in fact is 98.6 degrees Farenheit but the result of the sampling lead to the conclusion that the mean body temprature for humans differ from 98.6 degrees Farenheit\" is a [ANS] A. correct decision B. Type I error C. Type II error", "answer_v3": [ "B", "A", "A", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ] }, { "id": "Statistics_0308", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Type I/type II errors and power", "level": "3", "keywords": [ "statistics", "hypothesis testing", "hypothesis tests" ], "problem_v1": "In a given hypothesis test, the null hypothesis can be rejected at the 0.10 and the 0.05 level of significance, but cannot be rejected at the 0.01 level. The most accurate statement about the $p-$ value for this test is: [ANS] A. $0.01 < p\\mbox{-value} < 0.05$ B. $0.05 < p\\mbox{-value} < 0.10$ C. $p\\mbox{-value}=0.10$ D. $p\\mbox{-value}=0.01$\nIf we do not reject the null hypothesis, we conclude that: [ANS] A. the test is statistically insignificant at whatever level of significance the tested B. there is enough statistical evidence to infer that the alternative hypothesis is true C. there is not enough statistical evidence to infer that the alternative hypothesis is true D. there is enough statistical evidence to infer that the null hypothesis is true", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "In a given hypothesis test, the null hypothesis can be rejected at the 0.10 and the 0.05 level of significance, but cannot be rejected at the 0.01 level. The most accurate statement about the $p-$ value for this test is: [ANS] A. $p\\mbox{-value}=0.10$ B. $0.05 < p\\mbox{-value} < 0.10$ C. $p\\mbox{-value}=0.01$ D. $0.01 < p\\mbox{-value} < 0.05$\nIf we reject the null hypothesis, we conclude that: [ANS] A. there is enough statistical evidence to infer that the alternative hypothesis is true B. the test is statistically insignificant at whatever level of significance is specified C. there is not enough statistical evidence to infer that the alternative hypothesis is true D. there is enough statistical evidence to infer that the null hypothesis is true", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "In a given hypothesis test, the null hypothesis can be rejected at the 0.10 and the 0.05 level of significance, but cannot be rejected at the 0.01 level. The most accurate statement about the $p-$ value for this test is: [ANS] A. $p\\mbox{-value}=0.10$ B. $0.01 < p\\mbox{-value} < 0.05$ C. $0.05 < p\\mbox{-value} < 0.10$ D. $p\\mbox{-value}=0.01$\nIn a criminal trial, a Type II error is made when: [ANS] A. a guilty defendant is acquitted B. a guilty defendant is convicted C. an innocent person is convicted D. an innocent person is acquitted", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0309", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Type I/type II errors and power", "level": "3", "keywords": [ "statistics", "hypothesis testing", "hypothesis tests" ], "problem_v1": "The probability of a Type I error is denoted by: [ANS] A. $\\alpha$ B. $1-\\beta$ C. $1-\\alpha$ D. $\\beta$\nA Type I error occurs when we: [ANS] A. do not reject a true null hypothesis B. reject a false null hypothesis C. reject a true null hypothesis D. do not reject a false null hypothesis", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "The probability of a Type I error is denoted by: [ANS] A. $1-\\alpha$ B. $1-\\beta$ C. $\\beta$ D. $\\alpha$\nWhenever the null hypothesis is not rejected, the alternative hypothesis: [ANS] A. is rejected B. is true C. is not rejected D. must be modified", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "The probability of a Type I error is denoted by: [ANS] A. $1-\\alpha$ B. $\\alpha$ C. $1-\\beta$ D. $\\beta$\nThe power of a test is the probability that it will lead us to: [ANS] A. reject the null hypothesis when it is false B. fail to reject the null hypothesis when it is true C. reject the null hypothesis when it is true D. fail to reject the null hypothesis when it is false", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0310", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Type I/type II errors and power", "level": "4", "keywords": [ "statistics", "hypothesis testing", "introduction to hypothesis testing" ], "problem_v1": "Select True or False, depending on whether the corresponding statement is true or false.\n[ANS] 1. The $p$-value is usually 0.05. [ANS] 2. An alternative or research hypothesis is an assertion that holds if the null hypothesis is false. [ANS] 3. A two-tail test is a test in which a null hypothesis can be rejected by an extreme result occurring in only one direction. [ANS] 4. The probability of a Type I error is represented by $\\beta$, and is the probability of failing to reject a false null hypothesis.", "answer_v1": [ "F", "T", "F", "F" ], "answer_type_v1": [ "TF", "TF", "TF", "TF" ], "options_v1": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v2": "Select True or False, depending on whether the corresponding statement is true or false.\n[ANS] 1. The power of a test is the probability that a true null hypothesis will be rejected. [ANS] 2. The probability of a Type I error is represented by $\\beta$, and is the probability of failing to reject a false null hypothesis. [ANS] 3. The $p$-value is usually 0.05. [ANS] 4. The $p$-value of a test is the probability of observing a test statistic at least as extreme as the one computed given that the null hypothesis is true.", "answer_v2": [ "F", "F", "F", "T" ], "answer_type_v2": [ "TF", "TF", "TF", "TF" ], "options_v2": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v3": "Select True or False, depending on whether the corresponding statement is true or false.\n[ANS] 1. A one-tail $p$-value is twice the size of a two-tail test. [ANS] 2. The $p$-value of a test is the probability of observing a test statistic at least as extreme as the one computed given that the null hypothesis is true. [ANS] 3. The power of a test is the probability that a true null hypothesis will be rejected. [ANS] 4. An alternative or research hypothesis is an assertion that holds if the null hypothesis is false.", "answer_v3": [ "F", "T", "F", "T" ], "answer_type_v3": [ "TF", "TF", "TF", "TF" ], "options_v3": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ] }, { "id": "Statistics_0311", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Type I/type II errors and power", "level": "3", "keywords": [ "statistics", "Two Populations", "Inference" ], "problem_v1": "In testing the null hypothesis $H_0:p_1-p_2=0$, if $H_0$ is false, the test could lead to: [ANS] A. a Type II error B. either a Type I error or a Type II error C. a Type I error D. none of the above\nTwo samples of size 25 and 35 are independently drawn from two normal populations where the unknown population variances are assumed to be equal. The number of degrees of freedom of the equal-variances t test statistic is: [ANS] A. 35 B. 60 C. 58 D. 59", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "In testing the null hypothesis $H_0:p_1-p_2=0$, if $H_0$ is false, the test could lead to: [ANS] A. a Type II error B. a Type I error C. either a Type I error or a Type II error D. none of the above\nThe F-distribution is the sampling distribution of the ratio of: [ANS] A. two sample variances provided that the samples are independently drawn from two normal populations with equal variances B. two sample variances provided that the sample sizes are large C. two normal population variances D. two normal population means", "answer_v2": [ "A", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "In testing the null hypothesis $H_0:p_1-p_2=0$, if $H_0$ is false, the test could lead to: [ANS] A. either a Type I error or a Type II error B. a Type II error C. a Type I error D. none of the above\nIn testing the difference between the means of two normally distributed populations, the number of degrees of freedom associated with the unequal-variances t test statistic usually results in a non-integer number. It is recommended that you: [ANS] A. round down to the nearest integer B. change the sample sizes until the number of degrees of freedom becomes an integer C. round up to the nearest integer D. assume that the population variances are equal and then use $d.f.=n_1+n_2-2$", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0312", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample variance", "level": "2", "keywords": [ "Test Statistic", "Critical Value" ], "problem_v1": "Randomly selected students were given five seconds to estimate the value of a product of numbers with the results shown below. Estimates from students given $1 \\times 2 \\times 3 \\times 4 \\times 5 \\times 6 \\times 7 \\times 8$: 175, \\ 200, \\ 1000, \\ 2000, \\ 40320, \\ 40320, \\ 200, \\ 10000, \\ 5000, \\ 1000 Estimates from students given $8 \\times 7 \\times 6 \\times 5 \\times 4 \\times 3 \\times 2 \\times 1$: 10000, \\ 50000, \\ 225, \\ 350, \\ 400, \\ 300, \\ 500, \\ 2500, \\ 40000, \\ 377 Use a $0.05$ significance level to test the following claims: 1. $\\ $ Claim: the two populations have equal variances. The test statistic is [ANS]\nThe larger critical value is [ANS]\nThe conclusion is [ANS] A. There is not sufficient evidence to warrant the rejection of the claim that the two populations have equal variances B. There is sufficient evidence to warrant the rejection of the claim that the two populations have equal variances 2. $\\ $ Claim: the two populations have the same mean.\nThe test statistic is [ANS]\nThe positive critical value is [ANS]\nThe negative critical value is [ANS]\nThe conclusion is [ANS] A. There is sufficient evidence to warrant the rejection of the claim that the two populations have the same mean B. There is not sufficient evidence to warrant the rejection of the claim that the two populations have the same mean", "answer_v1": [ "1.00077404296556", "4.02599", "A", "-0.10288385057259", "2.10092", "-2.10092", "B" ], "answer_type_v1": [ "NV", "NV", "MCS", "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "A", "B" ], [], [], [], [ "A", "B" ] ], "problem_v2": "Randomly selected students were given five seconds to estimate the value of a product of numbers with the results shown below. Estimates from students given $1 \\times 2 \\times 3 \\times 4 \\times 5 \\times 6 \\times 7 \\times 8$: 200, \\ 600, \\ 25, \\ 5000, \\ 800, \\ 5000, \\ 50, \\ 500, \\ 1252, \\ 1500 Estimates from students given $8 \\times 7 \\times 6 \\times 5 \\times 4 \\times 3 \\times 2 \\times 1$: 10000, \\ 350, \\ 2000, \\ 1500, \\ 1500, \\ 2050, \\ 550, \\ 40000, \\ 400, \\ 1200 Use a $0.05$ significance level to test the following claims: 1. $\\ $ Claim: the two populations have equal variances. The test statistic is [ANS]\nThe larger critical value is [ANS]\nThe conclusion is [ANS] A. There is not sufficient evidence to warrant the rejection of the claim that the two populations have equal variances B. There is sufficient evidence to warrant the rejection of the claim that the two populations have equal variances 2. $\\ $ Claim: the two populations have the same mean.\nThe test statistic is [ANS]\nThe positive critical value is [ANS]\nThe negative critical value is [ANS]\nThe conclusion is [ANS] A. There is sufficient evidence to warrant the rejection of the claim that the two populations have the same mean B. There is not sufficient evidence to warrant the rejection of the claim that the two populations have the same mean", "answer_v2": [ "1.17818671219622", "4.02599", "A", "-6.69118512357507", "2.10092", "-2.10092", "A" ], "answer_type_v2": [ "NV", "NV", "MCS", "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "A", "B" ], [], [], [], [ "A", "B" ] ], "problem_v3": "Randomly selected students were given five seconds to estimate the value of a product of numbers with the results shown below. Estimates from students given $1 \\times 2 \\times 3 \\times 4 \\times 5 \\times 6 \\times 7 \\times 8$: 40320, \\ 1000, \\ 40320, \\ 200, \\ 50, \\ 500, \\ 1000, \\ 600, \\ 45000, \\ 6000 Estimates from students given $8 \\times 7 \\times 6 \\times 5 \\times 4 \\times 3 \\times 2 \\times 1$: 2050, \\ 52836, \\ 100000, \\ 550, \\ 25000, \\ 1876, \\ 1200, \\ 42000, \\ 428, \\ 400 Use a $0.05$ significance level to test the following claims: 1. $\\ $ Claim: the two populations have equal variances. The test statistic is [ANS]\nThe larger critical value is [ANS]\nThe conclusion is [ANS] A. There is not sufficient evidence to warrant the rejection of the claim that the two populations have equal variances B. There is sufficient evidence to warrant the rejection of the claim that the two populations have equal variances 2. $\\ $ Claim: the two populations have the same mean.\nThe test statistic is [ANS]\nThe positive critical value is [ANS]\nThe negative critical value is [ANS]\nThe conclusion is [ANS] A. There is sufficient evidence to warrant the rejection of the claim that the two populations have the same mean B. There is not sufficient evidence to warrant the rejection of the claim that the two populations have the same mean", "answer_v3": [ "1.10848576369171", "4.02599", "A", "-1.61654090788817", "2.10092", "-2.10092", "B" ], "answer_type_v3": [ "NV", "NV", "MCS", "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "A", "B" ], [], [], [], [ "A", "B" ] ] }, { "id": "Statistics_0313", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One sample variance", "level": "2", "keywords": [ "Sigma", "Chi Squared", "Test Statistic", "Critical Value" ], "problem_v1": "A random sample of $n=8$ observations from a normal population produced the following measurements: $\\begin{array}{llllllll} 7& 3& 3& 5& 5& 3& 4& 6 \\cr \\end{array}$ Do the data provide sufficient evidence to indicate that $\\sigma^2 < 7$? Use $\\alpha=0.05$, and compute the following:\n(a) $\\ $ sample standard deviation $s=$ [ANS]\n(b) $\\ $ test statistic $\\chi^2=$ [ANS]\n(c) $\\ $ critical $\\chi^2_{\\alpha}=$ [ANS]\nThe final conclusion is [ANS] A. We can reject the null hypothesis that $\\sigma^2=7$ in favor of the alternative $\\sigma^2 < 7$. B. There is not sufficient evidence to reject the null hypothesis that $\\sigma^2 \\geq 7$.", "answer_v1": [ "1.51186", "2.28571", "2.16735", "B" ], "answer_type_v1": [ "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B" ] ], "problem_v2": "A random sample of $n=5$ observations from a normal population produced the following measurements: $\\begin{array}{lllll} 3& 9& 3& 1& 3 \\cr \\end{array}$ Do the data provide sufficient evidence to indicate that $\\sigma^2 < 1$? Use $\\alpha=0.05$, and compute the following:\n(a) $\\ $ sample standard deviation $s=$ [ANS]\n(b) $\\ $ test statistic $\\chi^2=$ [ANS]\n(c) $\\ $ critical $\\chi^2_{\\alpha}=$ [ANS]\nThe final conclusion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that $\\sigma^2 \\geq 1$. B. We can reject the null hypothesis that $\\sigma^2=1$ in favor of the alternative $\\sigma^2 < 1$.", "answer_v2": [ "3.03315", "36.8", "0.71072", "A" ], "answer_type_v2": [ "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B" ] ], "problem_v3": "A random sample of $n=6$ observations from a normal population produced the following measurements: $\\begin{array}{llllll} 5& 2& 3& 8& 9& 8 \\cr \\end{array}$ Do the data provide sufficient evidence to indicate that $\\sigma^2 < 3$? Use $\\alpha=0.05$, and compute the following:\n(a) $\\ $ sample standard deviation $s=$ [ANS]\n(b) $\\ $ test statistic $\\chi^2=$ [ANS]\n(c) $\\ $ critical $\\chi^2_{\\alpha}=$ [ANS]\nThe final conclusion is [ANS] A. We can reject the null hypothesis that $\\sigma^2=3$ in favor of the alternative $\\sigma^2 < 3$. B. There is not sufficient evidence to reject the null hypothesis that $\\sigma^2 \\geq 3$.", "answer_v3": [ "2.92689", "14.2778", "1.14548", "B" ], "answer_type_v3": [ "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B" ] ] }, { "id": "Statistics_0314", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Chi-squared test for independence", "level": "2", "keywords": [ "Multinomial", "Contingency", "Hypothesis", "Test Statistic", "Critical Value" ], "problem_v1": "The number of men and women among professors in Math, Physics, Chemistry, Linguistics, and English departments of a certain college were counted, and the results are shown in the table below. $\\begin{array}{cccccc}\\hline Dept. & Math & Physics & Chemistry & Linguistics & English \\\\ \\hline Men & 50 & 77 & 29 & 20 & 31 \\\\ \\hline Women & 3 & 3 & 3 & 3 & 21 \\\\ \\hline \\end{array}$ Test the claim that the gender of a professor is independent of the department. Use the significance level $\\alpha=0.01$ The test statistic is $\\chi^2=$ [ANS]\nThe critical value is $\\chi^2=$ [ANS]\nIs there sufficient evidence to warrant the rejection of the claim that the gender of a professor is independent of the department? [ANS] A. Yes B. No", "answer_v1": [ "41.3017962527539", "13.2767", "A" ], "answer_type_v1": [ "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "A", "B" ] ], "problem_v2": "The number of men and women among professors in Math, Physics, Chemistry, Linguistics, and English departments of a certain college were counted, and the results are shown in the table below. $\\begin{array}{cccccc}\\hline Dept. & Math & Physics & Chemistry & Linguistics & English \\\\ \\hline Men & 28 & 32 & 13 & 7 & 11 \\\\ \\hline Women & 2 & 5 & 2 & 3 & 9 \\\\ \\hline \\end{array}$ Test the claim that the gender of a professor is independent of the department. Use the significance level $\\alpha=0.01$ The test statistic is $\\chi^2=$ [ANS]\nThe critical value is $\\chi^2=$ [ANS]\nIs there sufficient evidence to warrant the rejection of the claim that the gender of a professor is independent of the department? [ANS] A. Yes B. No", "answer_v2": [ "13.7069993069993", "13.2767", "A" ], "answer_type_v2": [ "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "A", "B" ] ], "problem_v3": "The number of men and women among professors in Math, Physics, Chemistry, Linguistics, and English departments of a certain college were counted, and the results are shown in the table below. $\\begin{array}{cccccc}\\hline Dept. & Math & Physics & Chemistry & Linguistics & English \\\\ \\hline Men & 33 & 47 & 19 & 16 & 17 \\\\ \\hline Women & 2 & 2 & 4 & 4 & 10 \\\\ \\hline \\end{array}$ Test the claim that the gender of a professor is independent of the department. Use the significance level $\\alpha=0.01$ The test statistic is $\\chi^2=$ [ANS]\nThe critical value is $\\chi^2=$ [ANS]\nIs there sufficient evidence to warrant the rejection of the claim that the gender of a professor is independent of the department? [ANS] A. No B. Yes", "answer_v3": [ "18.3947396672034", "13.2767", "B" ], "answer_type_v3": [ "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "A", "B" ] ] }, { "id": "Statistics_0315", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Chi-squared test for independence", "level": "3", "keywords": [ "statistics", "chi-square", "contingency tables", "Hypothesis Testing", "Multinomial", "Contingency", "Hypothesis", "Test Statistic", "Critical Value", "Chi Squared", "Rejection", "Region", "Estimated Expected Value" ], "problem_v1": "It has been suggusted that the highest priority of retirees is travel. Thus, a study was conducted to investigate the differences in the length of stay of a trip for pre-and post-retirees. A sample of $704$ travelers were asked how long they stayed on a typical trip. The observed results of the study are found below.\n$\\begin{array}{cccc}\\hline Number of Nights & Pre-retirement & Post-retirement & Total \\\\ \\hline 4-7 & 245 & 164 & 409 \\\\ \\hline 8-13 & 81 & 65 & 146 \\\\ \\hline 14-21 & 36 & 56 & 92 \\\\ \\hline 22or more & 21 & 36 & 57 \\\\ \\hline Total & 383 & 321 & 704 \\\\ \\hline \\end{array}$\nWith this information, construct a table of estimated expected values.\n$\\begin{array}{ccc}\\hline Number of Nights & Pre-retirement & Post-retirement \\\\ \\hline 4-7 & [ANS] & [ANS] \\\\ \\hline 8-13 & [ANS] & [ANS] \\\\ \\hline 14-21 & [ANS] & [ANS] \\\\ \\hline 22or more & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nNow, with that information, determine whether the length of stay is independent of retirement using $\\alpha=0.01$.\n(a) $\\chi^2=$ [ANS]\n(b) Find the degrees of freedom: [ANS]\n(c) Find the critical value: [ANS]\n(d) The final conclusion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that the length of stay is independent of retirement. B. We can reject the null hypothesis that the length of stay is independent of retirement and accept the alternative hypothesis that the two are dependent.", "answer_v1": [ "222.51", "186.49", "79.429", "66.571", "50.0511", "41.9489", "31.0099", "25.9901", "20.7912", "3", "11.3449", "B" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [ "A", "B" ] ], "problem_v2": "It has been suggusted that the highest priority of retirees is travel. Thus, a study was conducted to investigate the differences in the length of stay of a trip for pre-and post-retirees. A sample of $687$ travelers were asked how long they stayed on a typical trip. The observed results of the study are found below.\n$\\begin{array}{cccc}\\hline Number of Nights & Pre-retirement & Post-retirement & Total \\\\ \\hline 4-7 & 231 & 175 & 406 \\\\ \\hline 8-13 & 85 & 65 & 150 \\\\ \\hline 14-21 & 31 & 52 & 83 \\\\ \\hline 22or more & 15 & 33 & 48 \\\\ \\hline Total & 362 & 325 & 687 \\\\ \\hline \\end{array}$\nWith this information, construct a table of estimated expected values.\n$\\begin{array}{ccc}\\hline Number of Nights & Pre-retirement & Post-retirement \\\\ \\hline 4-7 & [ANS] & [ANS] \\\\ \\hline 8-13 & [ANS] & [ANS] \\\\ \\hline 14-21 & [ANS] & [ANS] \\\\ \\hline 22or more & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nNow, with that information, determine whether the length of stay is independent of retirement using $\\alpha=0.05$.\n(a) $\\chi^2=$ [ANS]\n(b) Find the degrees of freedom: [ANS]\n(c) Find the critical value: [ANS]\n(d) The final conclusion is [ANS] A. We can reject the null hypothesis that the length of stay is independent of retirement and accept the alternative hypothesis that the two are dependent. B. There is not sufficient evidence to reject the null hypothesis that the length of stay is independent of retirement.", "answer_v2": [ "213.933", "192.067", "79.0393", "70.9607", "43.7351", "39.2649", "25.2926", "22.7074", "20.5209", "3", "7.81473", "A" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [ "A", "B" ] ], "problem_v3": "It has been suggusted that the highest priority of retirees is travel. Thus, a study was conducted to investigate the differences in the length of stay of a trip for pre-and post-retirees. A sample of $694$ travelers were asked how long they stayed on a typical trip. The observed results of the study are found below.\n$\\begin{array}{cccc}\\hline Number of Nights & Pre-retirement & Post-retirement & Total \\\\ \\hline 4-7 & 236 & 163 & 399 \\\\ \\hline 8-13 & 81 & 65 & 146 \\\\ \\hline 14-21 & 33 & 58 & 91 \\\\ \\hline 22or more & 18 & 40 & 58 \\\\ \\hline Total & 368 & 326 & 694 \\\\ \\hline \\end{array}$\nWith this information, construct a table of estimated expected values.\n$\\begin{array}{ccc}\\hline Number of Nights & Pre-retirement & Post-retirement \\\\ \\hline 4-7 & [ANS] & [ANS] \\\\ \\hline 8-13 & [ANS] & [ANS] \\\\ \\hline 14-21 & [ANS] & [ANS] \\\\ \\hline 22or more & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nNow, with that information, determine whether the length of stay is independent of retirement using $\\alpha=0.05$.\n(a) $\\chi^2=$ [ANS]\n(b) Find the degrees of freedom: [ANS]\n(c) Find the critical value: [ANS]\n(d) The final conclusion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that the length of stay is independent of retirement. B. We can reject the null hypothesis that the length of stay is independent of retirement and accept the alternative hypothesis that the two are dependent.", "answer_v3": [ "211.573", "187.427", "77.4179", "68.5821", "48.2536", "42.7464", "30.755", "27.245", "27.8826", "3", "7.81473", "B" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [ "A", "B" ] ] }, { "id": "Statistics_0316", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Chi-squared test for independence", "level": "3", "keywords": [ "statistics", "chi-square", "contingency tables" ], "problem_v1": "For each problem, select the best response.\n(a) A study was performed to examine the personal goals of children in grades 4, 5 and 6. A random sample of students was selected from grades 4, 5 and 6 from schools in Idaho. The students received a questionnaire regarding achieving personal goals. They were asked what they would most like to do at school: make good grades, be good at sports, or be popular. Each student's gender was also recorded. Which hypotheses are being tested by the chi-square test? [ANS] A. The distribution of gender is different for the three personal goals. B. The null hypothesis is that there is no relationship between personal goals and gender, and the alternative is that there is a positive relationship. C. The null hypothesis is that the mean personal goal is the same for boys and girls, and the alternative is that the means differ. D. The null hypothesis is that there is no relationship between personal goals and gender, and the alternative is that there is some relationship. E. None of the above.\n(b) A $\\chi^2$ statistic provides persuasive evidence against the null hypothesis if its value is [ANS] A. a large positive number. B. close to 0. C. close to 1. D. a large negative number. E. 1.96\n(c) Chi-square tests are most appropriate for data that [ANS] A. can be averaged. B. are categorical. C. are normally distributed. D. have small standard deviations. E. have rounding errors.", "answer_v1": [ "D", "A", "B" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "For each problem, select the best response.\n(a) Chi-square tests are most appropriate for data that [ANS] A. are normally distributed. B. can be averaged. C. are categorical. D. have rounding errors. E. have small standard deviations.\n(b) A study was performed to examine the personal goals of children in grades 4, 5 and 6. A random sample of students was selected from grades 4, 5 and 6 from schools in Idaho. The students received a questionnaire regarding achieving personal goals. They were asked what they would most like to do at school: make good grades, be good at sports, or be popular. Each student's gender was also recorded. Which hypotheses are being tested by the chi-square test? [ANS] A. The distribution of gender is different for the three personal goals. B. The null hypothesis is that there is no relationship between personal goals and gender, and the alternative is that there is some relationship. C. The null hypothesis is that the mean personal goal is the same for boys and girls, and the alternative is that the means differ. D. The null hypothesis is that there is no relationship between personal goals and gender, and the alternative is that there is a positive relationship. E. None of the above.\n(c) A $\\chi^2$ statistic provides persuasive evidence against the null hypothesis if its value is [ANS] A. close to 0. B. close to 1. C. a large positive number. D. 1.96 E. a large negative number.", "answer_v2": [ "C", "B", "C" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "For each problem, select the best response.\n(a) Chi-square tests are most appropriate for data that [ANS] A. can be averaged. B. are normally distributed. C. have rounding errors. D. are categorical. E. have small standard deviations.\n(b) A study was performed to examine the personal goals of children in grades 4, 5 and 6. A random sample of students was selected from grades 4, 5 and 6 from schools in Idaho. The students received a questionnaire regarding achieving personal goals. They were asked what they would most like to do at school: make good grades, be good at sports, or be popular. Each student's gender was also recorded. Which hypotheses are being tested by the chi-square test? [ANS] A. The null hypothesis is that there is no relationship between personal goals and gender, and the alternative is that there is a positive relationship. B. The null hypothesis is that there is no relationship between personal goals and gender, and the alternative is that there is some relationship. C. The null hypothesis is that the mean personal goal is the same for boys and girls, and the alternative is that the means differ. D. The distribution of gender is different for the three personal goals. E. None of the above.\n(c) A $\\chi^2$ statistic provides persuasive evidence against the null hypothesis if its value is [ANS] A. a large negative number. B. close to 0. C. close to 1. D. a large positive number. E. 1.96", "answer_v3": [ "D", "B", "D" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Statistics_0317", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Chi-squared test for independence", "level": "3", "keywords": [ "statistics", "chi-square", "contingency tables" ], "problem_v1": "A certain health maintenance organization (HMO) wishes to study why patients leave the HMO. A SRS of 444 patients was taken. Data was collected on whether a patient had filed a complaint and, if so, whether the complaint was medical or nonmedical in nature. After a year, a tally from these patients was collected to count number who left the HMO voluntarily. Here are the data on the total number in each group and the number who voluntarily left the HMO:\n$\\begin{array}{cccc}\\hline & No complaint & Medical complaint & Nonmedical complaint \\\\ \\hline Total & 162 & 138 & 144 \\\\ \\hline Left & 59 & 42 & 47 \\\\ \\hline \\end{array}$\nIf the null hypothesis is $H_0: p_1=p_2=p_3$ and using $\\alpha=0.01$, then do the following:\n(a) Find the expected number of people with no complaint who leave the HMO: [ANS]\n(b) Find the expected number of people with a medical complaint who leave the HMO: [ANS]\n(c) Find the expected number of people with a nonmedical complaint who leave the HMO: [ANS]\n(d) Find the test statistic: [ANS]\n(e) Find the degrees of freedom: [ANS]\n(f) Find the critical value: [ANS]\n(g) The final conclusion is [ANS] A. We can reject the null hypothesis that the proportions are equal. B. There is not sufficient evidence to reject the null hypothesis.", "answer_v1": [ "54", "46", "48", "1.24743", "2", "9.21034", "B" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [], [], [], [ "A", "B" ] ], "problem_v2": "A certain health maintenance organization (HMO) wishes to study why patients leave the HMO. A SRS of 330 patients was taken. Data was collected on whether a patient had filed a complaint and, if so, whether the complaint was medical or nonmedical in nature. After a year, a tally from these patients was collected to count number who left the HMO voluntarily. Here are the data on the total number in each group and the number who voluntarily left the HMO:\n$\\begin{array}{cccc}\\hline & No complaint & Medical complaint & Nonmedical complaint \\\\ \\hline Total & 66 & 186 & 78 \\\\ \\hline Left & 19 & 71 & 20 \\\\ \\hline \\end{array}$\nIf the null hypothesis is $H_0: p_1=p_2=p_3$ and using $\\alpha=0.01$, then do the following:\n(a) Find the expected number of people with no complaint who leave the HMO: [ANS]\n(b) Find the expected number of people with a medical complaint who leave the HMO: [ANS]\n(c) Find the expected number of people with a nonmedical complaint who leave the HMO: [ANS]\n(d) Find the test statistic: [ANS]\n(e) Find the degrees of freedom: [ANS]\n(f) Find the critical value: [ANS]\n(g) The final conclusion is [ANS] A. We can reject the null hypothesis that the proportions are equal. B. There is not sufficient evidence to reject the null hypothesis.", "answer_v2": [ "22", "62", "26", "4.65024", "2", "9.21034", "B" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [], [], [], [ "A", "B" ] ], "problem_v3": "A certain health maintenance organization (HMO) wishes to study why patients leave the HMO. A SRS of 336 patients was taken. Data was collected on whether a patient had filed a complaint and, if so, whether the complaint was medical or nonmedical in nature. After a year, a tally from these patients was collected to count number who left the HMO voluntarily. Here are the data on the total number in each group and the number who voluntarily left the HMO:\n$\\begin{array}{cccc}\\hline & No complaint & Medical complaint & Nonmedical complaint \\\\ \\hline Total & 99 & 141 & 96 \\\\ \\hline Left & 34 & 41 & 37 \\\\ \\hline \\end{array}$\nIf the null hypothesis is $H_0: p_1=p_2=p_3$ and using $\\alpha=0.01$, then do the following:\n(a) Find the expected number of people with no complaint who leave the HMO: [ANS]\n(b) Find the expected number of people with a medical complaint who leave the HMO: [ANS]\n(c) Find the expected number of people with a nonmedical complaint who leave the HMO: [ANS]\n(d) Find the test statistic: [ANS]\n(e) Find the degrees of freedom: [ANS]\n(f) Find the critical value: [ANS]\n(g) The final conclusion is [ANS] A. We can reject the null hypothesis that the proportions are equal. B. There is not sufficient evidence to reject the null hypothesis.", "answer_v3": [ "33", "47", "32", "2.36627", "2", "9.21034", "B" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [], [], [], [ "A", "B" ] ] }, { "id": "Statistics_0318", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Chi-squared test for independence", "level": "3", "keywords": [ "statistics", "chi-square", "contingency tables", "Hypothesis Testing", "Multinomial", "Contingency", "Hypothesis", "Test Statistic", "Critical Value", "Chi Squared", "Rejection", "Region" ], "problem_v1": "Use software to test the null hypothesis of whether there is a relationship between the two classifications, A and B, of the $3 \\times 3$ contingency table shown below. Test using $\\alpha=0.01$. NOTE: You may do this by hand, but it will take a bit of time.\n$\\begin{array}{ccccc}\\hline & B_1 & B_2 & B_3 & Total \\\\ \\hline A_1 & 70 & 63 & 65 & 198 \\\\ \\hline A_2 & 69 & 52 & 53 & 174 \\\\ \\hline A_3 & 63 & 63 & 55 & 181 \\\\ \\hline Total & 202 & 178 & 173 & 553 \\\\ \\hline \\end{array}$\n(a) $\\chi^2=$ [ANS]\n(b) Find the degrees of freedom. [ANS]\n(c) Find the critical value. $\\chi^2=$ [ANS]\n(d) The final conclusion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that there is no relationship between A and B. B. We can reject the null hypothesis that A and B are not related and accept that there seems to be a relationship between A and B.", "answer_v1": [ "1.60343", "4", "13.2767", "A" ], "answer_type_v1": [ "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B" ] ], "problem_v2": "Use software to test the null hypothesis of whether there is a relationship between the two classifications, A and B, of the $3 \\times 3$ contingency table shown below. Test using $\\alpha=0.01$. NOTE: You may do this by hand, but it will take a bit of time.\n$\\begin{array}{ccccc}\\hline & B_1 & B_2 & B_3 & Total \\\\ \\hline A_1 & 43 & 78 & 46 & 167 \\\\ \\hline A_2 & 53 & 78 & 52 & 183 \\\\ \\hline A_3 & 47 & 53 & 63 & 163 \\\\ \\hline Total & 143 & 209 & 161 & 513 \\\\ \\hline \\end{array}$\n(a) $\\chi^2=$ [ANS]\n(b) Find the degrees of freedom. [ANS]\n(c) Find the critical value. $\\chi^2=$ [ANS]\n(d) The final conclusion is [ANS] A. We can reject the null hypothesis that A and B are not related and accept that there seems to be a relationship between A and B. B. There is not sufficient evidence to reject the null hypothesis that there is no relationship between A and B.", "answer_v2": [ "8.76763", "4", "13.2767", "B" ], "answer_type_v2": [ "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B" ] ], "problem_v3": "Use software to test the null hypothesis of whether there is a relationship between the two classifications, A and B, of the $3 \\times 3$ contingency table shown below. Test using $\\alpha=0.01$. NOTE: You may do this by hand, but it will take a bit of time.\n$\\begin{array}{ccccc}\\hline & B_1 & B_2 & B_3 & Total \\\\ \\hline A_1 & 52 & 64 & 51 & 167 \\\\ \\hline A_2 & 62 & 48 & 54 & 164 \\\\ \\hline A_3 & 73 & 77 & 75 & 225 \\\\ \\hline Total & 187 & 189 & 180 & 556 \\\\ \\hline \\end{array}$\n(a) $\\chi^2=$ [ANS]\n(b) Find the degrees of freedom. [ANS]\n(c) Find the critical value. $\\chi^2=$ [ANS]\n(d) The final conclusion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that there is no relationship between A and B. B. We can reject the null hypothesis that A and B are not related and accept that there seems to be a relationship between A and B.", "answer_v3": [ "3.5072", "4", "13.2767", "A" ], "answer_type_v3": [ "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B" ] ] }, { "id": "Statistics_0319", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Chi-squared test for independence", "level": "1", "keywords": [ "Statistics", "Hypothesis Testing" ], "problem_v1": "A chi-square independence test is to be conducted to decide whether an association exists between two variables of a populations. One variable has 8 possible values and the other variable has 7. What are the degrees of freedom for the $\\chi^2$-statistic? answer: [ANS]", "answer_v1": [ "42" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A chi-square independence test is to be conducted to decide whether an association exists between two variables of a populations. One variable has 2 possible values and the other variable has 10. What are the degrees of freedom for the $\\chi^2$-statistic? answer: [ANS]", "answer_v2": [ "9" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A chi-square independence test is to be conducted to decide whether an association exists between two variables of a populations. One variable has 4 possible values and the other variable has 7. What are the degrees of freedom for the $\\chi^2$-statistic? answer: [ANS]", "answer_v3": [ "18" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0320", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Chi-squared test for independence", "level": "3", "keywords": [ "hypothesis testing", "chi squared" ], "problem_v1": "There is a popular story (among data miners) that there is a correlation between men buying diapers and buying beer while shopping. A student tests this theory by surveying 132 male shoppers as they left a grocery store. The results are summarized in the contingency table. Test for a dependent relationship between buying beer and buying diapers. Conduct this test at the 0.01 significance level.\n$\\begin{array}{cccc}\\hline & Bought Diapers & Did Not Buy Diapers & Totals \\\\ \\hline Beer & 7 & 49 & 56 \\\\ \\hline No Beer & 10 & 66 & 76 \\\\ \\hline Totals & 17 & 115 & 132 \\\\ \\hline \\end{array}$ 132\n(a) Find the expected frequencies.\n$\\begin{array}{ccc}\\hline & Bought Diapers & Did Not Buy Diapers \\\\ \\hline Beer & [ANS] & [ANS] \\\\ \\hline No Beer & [ANS] & [ANS] \\\\ \\hline \\end{array}$ (b) Find the test statistic. [ANS]\n(c) Find the critical value. [ANS]\n(d) Is there sufficient data to support the claim? [ANS] [ANS]", "answer_v1": [ "7.21212", "48.7879", "9.78788", "66.2121", "0.0124377", "6.6349", "No" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "TF" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "There is a popular story (among data miners) that there is a correlation between men buying diapers and buying beer while shopping. A student tests this theory by surveying 131 male shoppers as they left a grocery store. The results are summarized in the contingency table. Test for a dependent relationship between buying beer and buying diapers. Conduct this test at the 0.10 significance level.\n$\\begin{array}{cccc}\\hline & Bought Diapers & Did Not Buy Diapers & Totals \\\\ \\hline Beer & 6 & 54 & 60 \\\\ \\hline No Beer & 7 & 64 & 71 \\\\ \\hline Totals & 13 & 118 & 131 \\\\ \\hline \\end{array}$ 131\n(a) Find the expected frequencies.\n$\\begin{array}{ccc}\\hline & Bought Diapers & Did Not Buy Diapers \\\\ \\hline Beer & [ANS] & [ANS] \\\\ \\hline No Beer & [ANS] & [ANS] \\\\ \\hline \\end{array}$ (b) Find the test statistic. [ANS]\n(c) Find the critical value. [ANS]\n(d) Is there sufficient data to support the claim? [ANS] [ANS]", "answer_v2": [ "5.9542", "54.0458", "7.0458", "63.9542", "0.00072167", "2.70554", "No" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "TF" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "There is a popular story (among data miners) that there is a correlation between men buying diapers and buying beer while shopping. A student tests this theory by surveying 129 male shoppers as they left a grocery store. The results are summarized in the contingency table. Test for a dependent relationship between buying beer and buying diapers. Conduct this test at the 0.10 significance level.\n$\\begin{array}{cccc}\\hline & Bought Diapers & Did Not Buy Diapers & Totals \\\\ \\hline Beer & 7 & 49 & 56 \\\\ \\hline No Beer & 9 & 64 & 73 \\\\ \\hline Totals & 16 & 113 & 129 \\\\ \\hline \\end{array}$ 129\n(a) Find the expected frequencies.\n$\\begin{array}{ccc}\\hline & Bought Diapers & Did Not Buy Diapers \\\\ \\hline Beer & [ANS] & [ANS] \\\\ \\hline No Beer & [ANS] & [ANS] \\\\ \\hline \\end{array}$ (b) Find the test statistic. [ANS]\n(c) Find the critical value. [ANS]\n(d) Is there sufficient data to support the claim? [ANS] [ANS]", "answer_v3": [ "6.94574", "49.0543", "9.05426", "63.9457", "0.000855217", "2.70554", "No" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "TF" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Statistics_0321", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Chi-squared test for goodness of fit", "level": "3", "keywords": [ "Multinomial", "Contingency", "Hypothesis", "Test Statistic", "Critical Value", "Expected Value", "Rejection", "Region" ], "problem_v1": "A multinomial experiment with $k=3$ cells and $n=400$ produced the data shown below.\n$\\begin{array}{cccc}\\hline \\ & Cell 1 & Cell 2 & Cell 3 \\\\ \\hline n_i & 102 & 103 & 195 \\\\ \\hline \\end{array}$\nIf the null hypothesis is $H_0: p_1=.25, \\ p_2=.25, \\ p_3=.5$ and using $\\alpha=0.05$, then do the following:\n(a) Find the expected value of Cell 1. E(Cell 1) $=$ [ANS]\n(b) Find the expected value of Cell 2. E(Cell 2) $=$ [ANS]\n(c) Find the expected value of Cell 3. E(Cell 3) $=$ [ANS]\n(d) Find the test statistic. $\\chi^2=$ [ANS]\n(e) Find the rejection region. $\\chi^2 >$ [ANS]\nThe final conclustion is [ANS] A. We can reject the null hypothesis that $p_1=.25, \\ p_2=.25, \\ p_3=.5$ and accept that at least one of the multinomial probabilities does not equal its hypothesized value. B. There is not sufficient evidence to reject the null hypothesis that $p_1=.25, \\ p_2=.25, \\ p_3=.5$.", "answer_v1": [ "100", "100", "200", "0.255", "5.99146", "B" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [], [], [ "A", "B" ] ], "problem_v2": "A multinomial experiment with $k=3$ cells and $n=280$ produced the data shown below.\n$\\begin{array}{cccc}\\hline \\ & Cell 1 & Cell 2 & Cell 3 \\\\ \\hline n_i & 79 & 63 & 138 \\\\ \\hline \\end{array}$\nIf the null hypothesis is $H_0: p_1=.25, \\ p_2=.25, \\ p_3=.5$ and using $\\alpha=0.01$, then do the following:\n(a) Find the expected value of Cell 1. E(Cell 1) $=$ [ANS]\n(b) Find the expected value of Cell 2. E(Cell 2) $=$ [ANS]\n(c) Find the expected value of Cell 3. E(Cell 3) $=$ [ANS]\n(d) Find the test statistic. $\\chi^2=$ [ANS]\n(e) Find the rejection region. $\\chi^2 >$ [ANS]\nThe final conclustion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that $p_1=.25, \\ p_2=.25, \\ p_3=.5$. B. We can reject the null hypothesis that $p_1=.25, \\ p_2=.25, \\ p_3=.5$ and accept that at least one of the multinomial probabilities does not equal its hypothesized value.", "answer_v2": [ "70", "70", "140", "1.88571428571429", "9.21034", "A" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [], [], [ "A", "B" ] ], "problem_v3": "A multinomial experiment with $k=3$ cells and $n=320$ produced the data shown below.\n$\\begin{array}{cccc}\\hline \\ & Cell 1 & Cell 2 & Cell 3 \\\\ \\hline n_i & 82 & 75 & 163 \\\\ \\hline \\end{array}$\nIf the null hypothesis is $H_0: p_1=.25, \\ p_2=.25, \\ p_3=.5$ and using $\\alpha=0.05$, then do the following:\n(a) Find the expected value of Cell 1. E(Cell 1) $=$ [ANS]\n(b) Find the expected value of Cell 2. E(Cell 2) $=$ [ANS]\n(c) Find the expected value of Cell 3. E(Cell 3) $=$ [ANS]\n(d) Find the test statistic. $\\chi^2=$ [ANS]\n(e) Find the rejection region. $\\chi^2 >$ [ANS]\nThe final conclustion is [ANS] A. We can reject the null hypothesis that $p_1=.25, \\ p_2=.25, \\ p_3=.5$ and accept that at least one of the multinomial probabilities does not equal its hypothesized value. B. There is not sufficient evidence to reject the null hypothesis that $p_1=.25, \\ p_2=.25, \\ p_3=.5$.", "answer_v3": [ "80", "80", "160", "0.41875", "5.99146", "B" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [], [], [ "A", "B" ] ] }, { "id": "Statistics_0322", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Chi-squared test for goodness of fit", "level": "2", "keywords": [ "Multinomial", "Contingency", "Hypothesis", "Test Statistic", "Critical Value", "Chi Squared", "Expected Frequency" ], "problem_v1": "A computer random number generator was used to generate 900 random digits (0,1,...,9). The observed frequences of the digits are given in the table below.\n$\\begin{array}{cccccccccc}\\hline 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\\\ \\hline 88 & 89 & 92 & 80 & 81 & 87 & 88 & 82 & 85 & 128 \\\\ \\hline \\end{array}$\nTest the claim that all the outcomes are equally likely using the significance level $\\alpha=0.05$. The expected frequency of each outcome is $E=$ [ANS]\nThe test statistic is $\\chi^2=$ [ANS]\nThe critical value is $\\chi^2=$ [ANS]\nIs there sufficient evidence to warrant the rejection of the claim that all the outcomes are equally likely? [ANS] A. No B. Yes", "answer_v1": [ "90", "19.2888888888889", "16.9190", "B" ], "answer_type_v1": [ "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B" ] ], "problem_v2": "A computer random number generator was used to generate 500 random digits (0,1,...,9). The observed frequences of the digits are given in the table below.\n$\\begin{array}{cccccccccc}\\hline 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\\\ \\hline 54 & 42 & 45 & 55 & 45 & 42 & 45 & 49 & 41 & 82 \\\\ \\hline \\end{array}$\nTest the claim that all the outcomes are equally likely using the significance level $\\alpha=0.05$. The expected frequency of each outcome is $E=$ [ANS]\nThe test statistic is $\\chi^2=$ [ANS]\nThe critical value is $\\chi^2=$ [ANS]\nIs there sufficient evidence to warrant the rejection of the claim that all the outcomes are equally likely? [ANS] A. Yes B. No", "answer_v2": [ "50", "27", "16.9190", "A" ], "answer_type_v2": [ "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B" ] ], "problem_v3": "A computer random number generator was used to generate 650 random digits (0,1,...,9). The observed frequences of the digits are given in the table below.\n$\\begin{array}{cccccccccc}\\hline 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\\\ \\hline 64 & 57 & 62 & 56 & 58 & 68 & 70 & 69 & 56 & 90 \\\\ \\hline \\end{array}$\nTest the claim that all the outcomes are equally likely using the significance level $\\alpha=0.01$. The expected frequency of each outcome is $E=$ [ANS]\nThe test statistic is $\\chi^2=$ [ANS]\nThe critical value is $\\chi^2=$ [ANS]\nIs there sufficient evidence to warrant the rejection of the claim that all the outcomes are equally likely? [ANS] A. Yes B. No", "answer_v3": [ "65", "14.7692307692308", "21.6660", "B" ], "answer_type_v3": [ "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B" ] ] }, { "id": "Statistics_0323", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Chi-squared test for goodness of fit", "level": "2", "keywords": [ "Multinomial", "Contingency", "Hypothesis", "Test Statistic", "Critical Value" ], "problem_v1": "Among drivers who have had a car crash in the last year, 250 were randomly selected and categorized by age, with the results listed in the table below. $\\begin{array}{ccccc}\\hline Age & Under 25 & 25-44 & 45-64 & Over 64 \\\\ \\hline Drivers & 101 & 61 & 35 & 53 \\\\ \\hline \\end{array}$ If all ages have the same crash rate, we would expect (because of the age distribution of licensed drivers) the given categories to have 16\\%, 44\\%, 27\\%, 13\\% of the subjects, respectively. At the 0.025 significance level, test the claim that the distribution of crashes conforms to the distribution of ages. The test statistic is $\\chi^2=$ [ANS]\nThe critical value is $\\chi^2=$ [ANS]\nThe conclusion is [ANS] A. There is not sufficient evidence to warrant the rejection of the claim that the distribution of crashes conforms to the distibuion of ages. B. There is sufficient evidence to warrant the rejection of the claim that the distribution of crashes conforms to the distibuion of ages.", "answer_v1": [ "143.43119010619", "9.34840", "B" ], "answer_type_v1": [ "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "A", "B" ] ], "problem_v2": "Among drivers who have had a car crash in the last year, 110 were randomly selected and categorized by age, with the results listed in the table below. $\\begin{array}{ccccc}\\hline Age & Under 25 & 25-44 & 45-64 & Over 64 \\\\ \\hline Drivers & 46 & 23 & 16 & 25 \\\\ \\hline \\end{array}$ If all ages have the same crash rate, we would expect (because of the age distribution of licensed drivers) the given categories to have 16\\%, 44\\%, 27\\%, 13\\% of the subjects, respectively. At the 0.05 significance level, test the claim that the distribution of crashes conforms to the distribution of ages. The test statistic is $\\chi^2=$ [ANS]\nThe critical value is $\\chi^2=$ [ANS]\nThe conclusion is [ANS] A. There is sufficient evidence to warrant the rejection of the claim that the distribution of crashes conforms to the distibuion of ages. B. There is not sufficient evidence to warrant the rejection of the claim that the distribution of crashes conforms to the distibuion of ages.", "answer_v2": [ "73.4828471192107", "7.81473", "A" ], "answer_type_v2": [ "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "A", "B" ] ], "problem_v3": "Among drivers who have had a car crash in the last year, 160 were randomly selected and categorized by age, with the results listed in the table below. $\\begin{array}{ccccc}\\hline Age & Under 25 & 25-44 & 45-64 & Over 64 \\\\ \\hline Drivers & 64 & 36 & 24 & 36 \\\\ \\hline \\end{array}$ If all ages have the same crash rate, we would expect (because of the age distribution of licensed drivers) the given categories to have 16\\%, 44\\%, 27\\%, 13\\% of the subjects, respectively. At the 0.025 significance level, test the claim that the distribution of crashes conforms to the distribution of ages. The test statistic is $\\chi^2=$ [ANS]\nThe critical value is $\\chi^2=$ [ANS]\nThe conclusion is [ANS] A. There is not sufficient evidence to warrant the rejection of the claim that the distribution of crashes conforms to the distibuion of ages. B. There is sufficient evidence to warrant the rejection of the claim that the distribution of crashes conforms to the distibuion of ages.", "answer_v3": [ "94.0501165501165", "9.34840", "B" ], "answer_type_v3": [ "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "A", "B" ] ] }, { "id": "Statistics_0324", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Chi-squared test for goodness of fit", "level": "2", "keywords": [ "Statistics", "Hypothesis Testing" ], "problem_v1": "In each question below, we have given the relative frequencies for the null hypothesis of a chi-square goodness-of-fit test and the sample size. In each case, decide whether the two assumptions for using that test are satisfied. Assumption 1: All expected frequencies are 1 or greater Assumption 2: At most 20 percent of the expected frequencies are less than 5\nSample size: n=100, Relative frequencies: 0.65, 0.30, 0.05 [ANS] A. Only assumption 2 is satisfied B. Only assumption 1 is satisfied C. Both assumptions are satisfied D. None of the assumptions are satisfied\nSample size: n=50, Relative frequencies: 0.22, 0.22, 0.25, 0.30, 0.01 [ANS] A. Only assumption 2 is satisfied B. Both assumptions are satisfied C. Only assumption 1 is satisfied D. None of the assumptions are satisfied\nSample size: n=50, Relative frequencies: 0.22, 0.21, 0.25, 0.30, 0.02 [ANS] A. Both assumptions are satisfied B. Only assumption 2 is satisfied C. Only assumption 1 is satisfied D. None of the assumptions are satisfied\nSample size: n=100, Relative frequencies: 0.44, 0.25, 0.30, 0.01 [ANS] A. Only assumption 1 is satisfied B. Only assumption 2 is satisfied C. Both assumptions are satisfied D. None of the assumptions are satisfied", "answer_v1": [ "C", "A", "A", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "In each question below, we have given the relative frequencies for the null hypothesis of a chi-square goodness-of-fit test and the sample size. In each case, decide whether the two assumptions for using that test are satisfied. Assumption 1: All expected frequencies are 1 or greater Assumption 2: At most 20 percent of the expected frequencies are less than 5\nSample size: n=100, Relative frequencies: 0.65, 0.30, 0.05 [ANS] A. Both assumptions are satisfied B. Only assumption 1 is satisfied C. Only assumption 2 is satisfied D. None of the assumptions are satisfied\nSample size: n=50, Relative frequencies: 0.22, 0.22, 0.25, 0.30, 0.01 [ANS] A. Only assumption 2 is satisfied B. Only assumption 1 is satisfied C. Both assumptions are satisfied D. None of the assumptions are satisfied\nSample size: n=50, Relative frequencies: 0.22, 0.21, 0.25, 0.30, 0.02 [ANS] A. Only assumption 1 is satisfied B. Both assumptions are satisfied C. Only assumption 2 is satisfied D. None of the assumptions are satisfied\nSample size: n=100, Relative frequencies: 0.44, 0.25, 0.30, 0.01 [ANS] A. Both assumptions are satisfied B. Only assumption 2 is satisfied C. Only assumption 1 is satisfied D. None of the assumptions are satisfied", "answer_v2": [ "A", "A", "B", "C" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "In each question below, we have given the relative frequencies for the null hypothesis of a chi-square goodness-of-fit test and the sample size. In each case, decide whether the two assumptions for using that test are satisfied. Assumption 1: All expected frequencies are 1 or greater Assumption 2: At most 20 percent of the expected frequencies are less than 5\nSample size: n=100, Relative frequencies: 0.65, 0.30, 0.05 [ANS] A. Only assumption 2 is satisfied B. Both assumptions are satisfied C. Only assumption 1 is satisfied D. None of the assumptions are satisfied\nSample size: n=50, Relative frequencies: 0.22, 0.22, 0.25, 0.30, 0.01 [ANS] A. Both assumptions are satisfied B. Only assumption 2 is satisfied C. Only assumption 1 is satisfied D. None of the assumptions are satisfied\nSample size: n=50, Relative frequencies: 0.22, 0.21, 0.25, 0.30, 0.02 [ANS] A. Both assumptions are satisfied B. Only assumption 1 is satisfied C. Only assumption 2 is satisfied D. None of the assumptions are satisfied\nSample size: n=100, Relative frequencies: 0.44, 0.25, 0.30, 0.01 [ANS] A. Only assumption 1 is satisfied B. Only assumption 2 is satisfied C. Both assumptions are satisfied D. None of the assumptions are satisfied", "answer_v3": [ "B", "B", "A", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0325", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Chi-squared test for goodness of fit", "level": "3", "keywords": [ "hypothesis testing", "chi squared" ], "problem_v1": "In the Super-Mega lottery there are 50 numbers (1 to 50), a player chooses ten different numbers and hopes that these get drawn. If the player's numbers get drawn, he/she wins an obscene amount of money. The table below displays the frequency with which classes of numbers are chosen (not drawn). These numbers came from a sample of 174 chosen numbers.\n$\\begin{array}{cccccc}\\hline & Chosen Numbers (n=174) \\\\ \\hline & 1 to 10 & 11 to 20 & 21 to 30 & 31 to 40 & 41 to 50 \\\\ \\hline Count & 55 & 30 & 21 & 29 & 39 \\\\ \\hline \\end{array}$ 39 Test the claim that chosen numbers are not evenly distributed across the five classes. Test this claim at the 0.01 significance level.\n(a) Find the test statistic. [ANS]\n(b) Find the critical value. [ANS]\n(c) Is there sufficient data to support the claim? [ANS] [ANS]", "answer_v1": [ "19.3333", "13.2767", "Yes" ], "answer_type_v1": [ "NV", "NV", "TF" ], "options_v1": [ [], [], [] ], "problem_v2": "In the Super-Mega lottery there are 50 numbers (1 to 50), a player chooses ten different numbers and hopes that these get drawn. If the player's numbers get drawn, he/she wins an obscene amount of money. The table below displays the frequency with which classes of numbers are chosen (not drawn). These numbers came from a sample of 195 chosen numbers.\n$\\begin{array}{cccccc}\\hline & Chosen Numbers (n=195) \\\\ \\hline & 1 to 10 & 11 to 20 & 21 to 30 & 31 to 40 & 41 to 50 \\\\ \\hline Count & 53 & 25 & 32 & 54 & 31 \\\\ \\hline \\end{array}$ 31 Test the claim that chosen numbers are not evenly distributed across the five classes. Test this claim at the 0.10 significance level.\n(a) Find the test statistic. [ANS]\n(b) Find the critical value. [ANS]\n(c) Is there sufficient data to support the claim? [ANS] [ANS]", "answer_v2": [ "18.7179", "7.77944", "Yes" ], "answer_type_v2": [ "NV", "NV", "TF" ], "options_v2": [ [], [], [] ], "problem_v3": "In the Super-Mega lottery there are 50 numbers (1 to 50), a player chooses ten different numbers and hopes that these get drawn. If the player's numbers get drawn, he/she wins an obscene amount of money. The table below displays the frequency with which classes of numbers are chosen (not drawn). These numbers came from a sample of 209 chosen numbers.\n$\\begin{array}{cccccc}\\hline & Chosen Numbers (n=209) \\\\ \\hline & 1 to 10 & 11 to 20 & 21 to 30 & 31 to 40 & 41 to 50 \\\\ \\hline Count & 49 & 52 & 51 & 27 & 30 \\\\ \\hline \\end{array}$ 30 Test the claim that chosen numbers are not evenly distributed across the five classes. Test this claim at the 0.10 significance level.\n(a) Find the test statistic. [ANS]\n(b) Find the critical value. [ANS]\n(c) Is there sufficient data to support the claim? [ANS] [ANS]", "answer_v3": [ "14.3254", "7.77944", "Yes" ], "answer_type_v3": [ "NV", "NV", "TF" ], "options_v3": [ [], [], [] ] }, { "id": "Statistics_0326", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "Chi-squared test for goodness of fit", "level": "3", "keywords": [ "hypothesis testing", "chi squared" ], "problem_v1": "A student wants to see if the correct answers to multiple choice problems are evenly distributed. She heard a rumor that if you don't know the answer, you should always pick C. pick C. In a sample of 95 multiple-choice questions from prior tests and quizzes, the distribution of correct answers are given in the table below. In all of these questions, there were four options {A, B, C, D}.\n$\\begin{array}{ccccc}\\hline & Correct Answers (n=95) \\\\ \\hline & A & B & C & D \\\\ \\hline Count & 33 & 35 & 17 & 10 \\\\ \\hline \\end{array}$ 10 Test the claim that correct answers for all multiple-choice questions are not evenly distributed. Test this claim at the 0.01 significance level.\n(a) Find the test statistic. [ANS]\n(b) Find the critical value. [ANS]\n(c) Is there sufficient data to support the claim? [ANS] [ANS]", "answer_v1": [ "18.8105", "11.3449", "Yes" ], "answer_type_v1": [ "NV", "NV", "TF" ], "options_v1": [ [], [], [] ], "problem_v2": "A student wants to see if the correct answers to multiple choice problems are evenly distributed. She heard a rumor that if you don't know the answer, you should always pick C. pick C. In a sample of 99 multiple-choice questions from prior tests and quizzes, the distribution of correct answers are given in the table below. In all of these questions, there were four options {A, B, C, D}.\n$\\begin{array}{ccccc}\\hline & Correct Answers (n=99) \\\\ \\hline & A & B & C & D \\\\ \\hline Count & 34 & 13 & 18 & 34 \\\\ \\hline \\end{array}$ 34 Test the claim that correct answers for all multiple-choice questions are not evenly distributed. Test this claim at the 0.10 significance level.\n(a) Find the test statistic. [ANS]\n(b) Find the critical value. [ANS]\n(c) Is there sufficient data to support the claim? [ANS] [ANS]", "answer_v2": [ "14.3333", "6.25139", "Yes" ], "answer_type_v2": [ "NV", "NV", "TF" ], "options_v2": [ [], [], [] ], "problem_v3": "A student wants to see if the correct answers to multiple choice problems are evenly distributed. She heard a rumor that if you don't know the answer, you should always pick C. pick C. In a sample of 83 multiple-choice questions from prior tests and quizzes, the distribution of correct answers are given in the table below. In all of these questions, there were four options {A, B, C, D}.\n$\\begin{array}{ccccc}\\hline & Correct Answers (n=83) \\\\ \\hline & A & B & C & D \\\\ \\hline Count & 12 & 17 & 26 & 28 \\\\ \\hline \\end{array}$ 28 Test the claim that correct answers for all multiple-choice questions are not evenly distributed. Test this claim at the 0.10 significance level.\n(a) Find the test statistic. [ANS]\n(b) Find the critical value. [ANS]\n(c) Is there sufficient data to support the claim? [ANS] [ANS]", "answer_v3": [ "8.22892", "6.25139", "Yes" ], "answer_type_v3": [ "NV", "NV", "TF" ], "options_v3": [ [], [], [] ] }, { "id": "Statistics_0327", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One-way ANOVA", "level": "3", "keywords": [ "ANOVA", "Analysis", "Variance", "F-test" ], "problem_v1": "The table below lists the body temperatures of six randomly selected subjects from each of three different age groups. Use the $\\alpha=0.01$ significance level to test the claim that the three age-group populations have different mean body temperatures. $\\begin{array}{cccc}\\hline & 16-20 & 21-29 & 30 and older \\\\ \\hline subject 1 & 98.5 & 98.7 & 97.9 \\\\ \\hline subject 2 & 98.4 & 98.1 & 97.5 \\\\ \\hline subject 3 & 98.1 & 98.6 & 97.5 \\\\ \\hline subject 4 & 98 & 98.8 & 97.4 \\\\ \\hline subject 5 & 98 & 98.4 & 97.6 \\\\ \\hline subject 6 & 97.9 & 98.5 & 97.2 \\\\ \\hline mean & 98.15 & 98.517 & 97.517 \\\\ \\hline standard deviation & 0.243 & 0.248 & 0.232 \\\\ \\hline \\end{array}$\nThe variance between samples is $ns_{\\overline{x}}^2=$ [ANS] The variance within samples is $s_p^2=$ [ANS] The test statistic is $F=$ [ANS] The critical value is $F=$ [ANS] Is there sufficient evidence to warrant the rejection of the claim that the three age-group populations have the same mean body temperature? [ANS] A. Yes B. No", "answer_v1": [ "1.53537800000001", "0.0581111111111112", "26.4214187380498", "6.35886", "A" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [], [ "A", "B" ] ], "problem_v2": "The table below lists the body temperatures of six randomly selected subjects from each of three different age groups. Use the $\\alpha=0.01$ significance level to test the claim that the three age-group populations have different mean body temperatures. $\\begin{array}{cccc}\\hline & 16-20 & 21-29 & 30 and older \\\\ \\hline subject 1 & 97.2 & 99.3 & 97.2 \\\\ \\hline subject 2 & 97.7 & 99.3 & 97.5 \\\\ \\hline subject 3 & 97.4 & 98.2 & 97.9 \\\\ \\hline subject 4 & 97.2 & 98.8 & 97.7 \\\\ \\hline subject 5 & 98.6 & 97.9 & 97.3 \\\\ \\hline subject 6 & 97.6 & 98.6 & 97.3 \\\\ \\hline mean & 97.617 & 98.683 & 97.483 \\\\ \\hline standard deviation & 0.523 & 0.571 & 0.271 \\\\ \\hline \\end{array}$\nThe variance between samples is $ns_{\\overline{x}}^2=$ [ANS] The variance within samples is $s_p^2=$ [ANS] The test statistic is $F=$ [ANS] The critical value is $F=$ [ANS] Is there sufficient evidence to warrant the rejection of the claim that the three age-group populations have the same mean body temperature? [ANS] A. No B. Yes", "answer_v2": [ "2.59431200000001", "0.224333333333336", "11.5645408618127", "6.35886", "B" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [], [ "A", "B" ] ], "problem_v3": "The table below lists the body temperatures of six randomly selected subjects from each of three different age groups. Use the $\\alpha=0.01$ significance level to test the claim that the three age-group populations have different mean body temperatures. $\\begin{array}{cccc}\\hline & 16-20 & 21-29 & 30 and older \\\\ \\hline subject 1 & 97.6 & 98.7 & 97.4 \\\\ \\hline subject 2 & 98.1 & 97.9 & 97.5 \\\\ \\hline subject 3 & 98.6 & 99.3 & 98.3 \\\\ \\hline subject 4 & 97.4 & 98.1 & 97.4 \\\\ \\hline subject 5 & 97.1 & 98.6 & 98.5 \\\\ \\hline subject 6 & 98.6 & 98.7 & 97.1 \\\\ \\hline mean & 97.9 & 98.55 & 97.7 \\\\ \\hline standard deviation & 0.632 & 0.497 & 0.562 \\\\ \\hline \\end{array}$\nThe variance between samples is $ns_{\\overline{x}}^2=$ [ANS] The variance within samples is $s_p^2=$ [ANS] The test statistic is $F=$ [ANS] The critical value is $F=$ [ANS] Is there sufficient evidence to warrant the rejection of the claim that the three age-group populations have the same mean body temperature? [ANS] A. Yes B. No", "answer_v3": [ "1.18499999999998", "0.321", "3.69158878504667", "6.35886", "B" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [], [ "A", "B" ] ] }, { "id": "Statistics_0328", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One-way ANOVA", "level": "3", "keywords": [ "ANOVA", "Analysis", "Variance" ], "problem_v1": "A study was conducted to see how people reacted to certain facial expressions. A sample group of $n=36$ was randomly divided into six groups. Each group was assigned to view one picture of a person making a facial expression. Each group saw a different picture, and the different expressions were (1) Surprised (2) Nervous (3) Scared (4) Sad (5) Excited (6) Angry. After viewing the pictures, the subjects were asked to rank the degree of dominance they inferred from the facial expression they saw. (The scale ranged from-10 to 10) The data collected is summarized in the table below.\n$\\begin{array}{cccccc}\\hline Surprised & Nervous & Scared & Sad & Excited & Angry \\\\ \\hline 1 & 0.3 & 0.5 & 0.9 &-0.8 &-0.7 \\\\ \\hline 0.3 & 0.3 &-0.5 & 0 & 0.6 &-0.9 \\\\ \\hline 0 &-0.3 &-0.4 &-0.3 & 0.1 &-1.5 \\\\ \\hline-1.4 &-0.4 & 0 & 1.2 &-0.2 &-0.7 \\\\ \\hline-0.0999999999999999 & 1.6 & 2 &-0.9 &-1.9 &-1 \\\\ \\hline 0.2 &-1.3 &-0.6 & 1 & 1.2 &-0.0999999999999999 \\\\ \\hline \\end{array}$\nComplete the following ANOVA table\n$\\begin{array}{ccccc}\\hline Source & df & SS & MS & F \\\\ \\hline Expressions & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline Error & [ANS] & [ANS] & [ANS] & \\ \\\\ \\hline Total & [ANS] & [ANS] & [ANS] & \\ \\\\ \\hline \\end{array}$", "answer_v1": [ "5", "4.72555555555556", "0.945111111111111", "1.22441341586296", "30", "23.1566666666667", "0.771888888888889", "35", "27.8822222222222", "1.717" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "A study was conducted to see how people reacted to certain facial expressions. A sample group of $n=36$ was randomly divided into six groups. Each group was assigned to view one picture of a person making a facial expression. Each group saw a different picture, and the different expressions were (1) Surprised (2) Nervous (3) Scared (4) Sad (5) Excited (6) Angry. After viewing the pictures, the subjects were asked to rank the degree of dominance they inferred from the facial expression they saw. (The scale ranged from-10 to 10) The data collected is summarized in the table below.\n$\\begin{array}{cccccc}\\hline Surprised & Nervous & Scared & Sad & Excited & Angry \\\\ \\hline-1.7 & 1.8 &-1.4 &-0.7 & 1.8 &-0.8 \\\\ \\hline-1.3 &-0.7 & 0.3 &-1.8 & 0.6 &-0.2 \\\\ \\hline 1.3 &-1.3 &-1.3 &-0.9 & 0.3 &-1.3 \\\\ \\hline-0.7 & 0.2 &-1.6 &-1.5 &-1.7 &-0.8 \\\\ \\hline-0.0999999999999999 &-1.7 & 1.6 &-0.8 &-0.5 & 0.7 \\\\ \\hline 1.8 &-0.2 & 0.2 & 1.6 &-1.8 &-0.3 \\\\ \\hline \\end{array}$\nComplete the following ANOVA table\n$\\begin{array}{ccccc}\\hline Source & df & SS & MS & F \\\\ \\hline Expressions & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline Error & [ANS] & [ANS] & [ANS] & \\ \\\\ \\hline Total & [ANS] & [ANS] & [ANS] & \\ \\\\ \\hline \\end{array}$", "answer_v2": [ "5", "1.16583333333333", "0.233166666666667", "0.154887994980994", "30", "45.1616666666667", "1.50538888888889", "35", "46.3275", "1.73855555555556" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "A study was conducted to see how people reacted to certain facial expressions. A sample group of $n=36$ was randomly divided into six groups. Each group was assigned to view one picture of a person making a facial expression. Each group saw a different picture, and the different expressions were (1) Surprised (2) Nervous (3) Scared (4) Sad (5) Excited (6) Angry. After viewing the pictures, the subjects were asked to rank the degree of dominance they inferred from the facial expression they saw. (The scale ranged from-10 to 10) The data collected is summarized in the table below.\n$\\begin{array}{cccccc}\\hline Surprised & Nervous & Scared & Sad & Excited & Angry \\\\ \\hline-0.8 & 0.4 &-0.9 & 0.2 &-1.2 &-0.6 \\\\ \\hline 1.3 & 1.7 & 1.5 &-1.2 &-0.9 &-1 \\\\ \\hline-1.9 & 0.3 & 2 & 1.2 & 0.5 &-1.6 \\\\ \\hline-0.8 & 0.6 & 0.8 & 0.1 &-0.3 & 1.7 \\\\ \\hline 1.2 & 1.4 & 0.3 & 0.3 & 1.9 & 1.1 \\\\ \\hline-1.4 & 0.2 & 1.8 & 1 &-1.2 & 0.6 \\\\ \\hline \\end{array}$\nComplete the following ANOVA table\n$\\begin{array}{ccccc}\\hline Source & df & SS & MS & F \\\\ \\hline Expressions & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline Error & [ANS] & [ANS] & [ANS] & \\ \\\\ \\hline Total & [ANS] & [ANS] & [ANS] & \\ \\\\ \\hline \\end{array}$", "answer_v3": [ "5", "8.28805555555556", "1.65761111111111", "1.36810491081664", "30", "36.3483333333333", "1.21161111111111", "35", "44.6363888888889", "2.86922222222222" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Statistics_0329", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One-way ANOVA", "level": "2", "keywords": [ "ANOVA", "Analysis", "Variance" ], "problem_v1": "Which of the following changes the analysis of variance results? [ANS] A. the order of the samples is changed B. the same constant is added to every one of the sample values C. the same constant is added to each value in one of the samples D. each value in one of the samples is multiplied by the same constant E. each of the sample values is multiplied by the same constant F. each of the sample values is converted to a different scale", "answer_v1": [ "CD" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Which of the following changes the analysis of variance results? [ANS] A. the same constant is added to every one of the sample values B. each of the sample values is multiplied by the same constant C. the order of the samples is changed D. each value in one of the samples is multiplied by the same constant E. each of the sample values is converted to a different scale F. the same constant is added to each value in one of the samples", "answer_v2": [ "DF" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Which of the following changes the analysis of variance results? [ANS] A. each of the sample values is multiplied by the same constant B. the same constant is added to every one of the sample values C. each value in one of the samples is multiplied by the same constant D. the order of the samples is changed E. the same constant is added to each value in one of the samples F. each of the sample values is converted to a different scale", "answer_v3": [ "CE" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Statistics_0330", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One-way ANOVA", "level": "3", "keywords": [ "ANOVA", "Analysis", "Variance", "F-test", "p-value" ], "problem_v1": "Use the Minitab display to test the claims using the significance level of $\\alpha=0.05$. The sample data are the numbers of support beams manufactured by $5$ different operators using $4$ different machines. Assume that there is no interaction effect from operator and machine. \\begin{array}{l} \\begin{array}{l} \\mbox{Analysis of Variance for Beams} \\end{array} \\cr \\begin{array}{lrrrrr} \\mbox{Source} & \\mbox{DF} & \\mbox{SS} & \\mbox{MS} & \\mbox{F} & \\mbox{P} \\cr \\mbox{Operator} & 4 & 59.68 & 19.66 & 2.43 & 0.105 \\cr \\mbox{Machine} & 3 & 93.43 & 46.62 & 5.76 & 0.008 \\cr \\mbox{Error} & 12 & 47.97 & 8.09 & & \\cr \\mbox{Total} & 19 & 201.08 & & & \\end{array} \\end{array} Test the claim that the four operators have the same mean production output. The $F-$ test statistic is [ANS]\nThe P-value is [ANS]\nIs there sufficient evidence to warrant the rejection of the claim that the four machine operators have the same mean production output? [ANS] A. Yes B. No\nTest the claim that the choice of machine has no effect on the production output. The $F-$ test statistic is [ANS]\nThe P-value is [ANS]\nIs there sufficient evidence to warrant the rejection of the claim that the choice of machine has no effect on the production output? [ANS] A. No B. Yes", "answer_v1": [ "2.43", "0.105", "B", "5.76", "0.008", "B" ], "answer_type_v1": [ "NV", "NV", "MCS", "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "A", "B" ], [], [], [ "A", "B" ] ], "problem_v2": "Use the Minitab display to test the claims using the significance level of $\\alpha=0.05$. The sample data are the numbers of support beams manufactured by $3$ different operators using $5$ different machines. Assume that there is no interaction effect from operator and machine. \\begin{array}{l} \\begin{array}{l} \\mbox{Analysis of Variance for Beams} \\end{array} \\cr \\begin{array}{lrrrrr} \\mbox{Source} & \\mbox{DF} & \\mbox{SS} & \\mbox{MS} & \\mbox{F} & \\mbox{P} \\cr \\mbox{Operator} & 2 & 59.18 & 19.66 & 2.5 & 0.143 \\cr \\mbox{Machine} & 4 & 93.05 & 46.22 & 5.89 & 0.027 \\cr \\mbox{Error} & 8 & 48.63 & 7.85 & & \\cr \\mbox{Total} & 14 & 200.86 & & & \\end{array} \\end{array} Test the claim that the four operators have the same mean production output. The $F-$ test statistic is [ANS]\nThe P-value is [ANS]\nIs there sufficient evidence to warrant the rejection of the claim that the four machine operators have the same mean production output? [ANS] A. No B. Yes\nTest the claim that the choice of machine has no effect on the production output. The $F-$ test statistic is [ANS]\nThe P-value is [ANS]\nIs there sufficient evidence to warrant the rejection of the claim that the choice of machine has no effect on the production output? [ANS] A. Yes B. No", "answer_v2": [ "2.5", "0.143", "A", "5.89", "0.027", "A" ], "answer_type_v2": [ "NV", "NV", "MCS", "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "A", "B" ], [], [], [ "A", "B" ] ], "problem_v3": "Use the Minitab display to test the claims using the significance level of $\\alpha=0.05$. The sample data are the numbers of support beams manufactured by $3$ different operators using $4$ different machines. Assume that there is no interaction effect from operator and machine. \\begin{array}{l} \\begin{array}{l} \\mbox{Analysis of Variance for Beams} \\end{array} \\cr \\begin{array}{lrrrrr} \\mbox{Source} & \\mbox{DF} & \\mbox{SS} & \\mbox{MS} & \\mbox{F} & \\mbox{P} \\cr \\mbox{Operator} & 2 & 59.38 & 19.66 & 2.33 & 0.178 \\cr \\mbox{Machine} & 3 & 93.25 & 46.82 & 5.55 & 0.043 \\cr \\mbox{Error} & 6 & 47.87 & 8.43 & & \\cr \\mbox{Total} & 11 & 200.5 & & & \\end{array} \\end{array} Test the claim that the four operators have the same mean production output. The $F-$ test statistic is [ANS]\nThe P-value is [ANS]\nIs there sufficient evidence to warrant the rejection of the claim that the four machine operators have the same mean production output? [ANS] A. No B. Yes\nTest the claim that the choice of machine has no effect on the production output. The $F-$ test statistic is [ANS]\nThe P-value is [ANS]\nIs there sufficient evidence to warrant the rejection of the claim that the choice of machine has no effect on the production output? [ANS] A. No B. Yes", "answer_v3": [ "2.33", "0.178", "A", "5.55", "0.043", "B" ], "answer_type_v3": [ "NV", "NV", "MCS", "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "A", "B" ], [], [], [ "A", "B" ] ] }, { "id": "Statistics_0331", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One-way ANOVA", "level": "3", "keywords": [ "ANOVA", "Analysis", "Variance" ], "problem_v1": "Complete the ANOVA table for a completely randomized design below.\n$\\begin{array}{ccccc}\\hline Source & df & SS & MS & F \\\\ \\hline Treatments & 18 & 18.1 & [ANS] & [ANS] \\\\ \\hline Error & [ANS] & [ANS] & [ANS] & \\ \\\\ \\hline Total & 46 & 47.2 & \\ & \\ \\\\ \\hline \\end{array}$", "answer_v1": [ "1.00555555555556", "0.967544864452081", "28", "29.1", "1.03928571428571" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Complete the ANOVA table for a completely randomized design below.\n$\\begin{array}{ccccc}\\hline Source & df & SS & MS & F \\\\ \\hline Treatments & 10 & 15.7 & [ANS] & [ANS] \\\\ \\hline Error & [ANS] & [ANS] & [ANS] & \\ \\\\ \\hline Total & 50 & 43.3 & \\ & \\ \\\\ \\hline \\end{array}$", "answer_v2": [ "1.57", "2.27536231884058", "40", "27.6", "0.69" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Complete the ANOVA table for a completely randomized design below.\n$\\begin{array}{ccccc}\\hline Source & df & SS & MS & F \\\\ \\hline Treatments & 13 & 16.4 & [ANS] & [ANS] \\\\ \\hline Error & [ANS] & [ANS] & [ANS] & \\ \\\\ \\hline Total & 46 & 45.5 & \\ & \\ \\\\ \\hline \\end{array}$", "answer_v3": [ "1.26153846153846", "1.43061062648692", "33", "29.1", "0.881818181818182" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Statistics_0332", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One-way ANOVA", "level": "2", "keywords": [ "ANOVA", "Analysis", "Variance", "SS", "Sum of Squares", "Rejection", "Region" ], "problem_v1": "138 322 Suppose the Total Sum of Squares for a completely randomzied design with $p=6$ treatments and $n=24$ total measurements (SS(Total))is equal to $460$. In each of the following cases, conduct an $F$-test of the null hypothesis that the mean responses for the $6$ treatments are the same. Use $\\alpha=0.05$.\n(a) Sum of Squares for Treatment (SST) is $30$ \\% of SS(Total) $F=$ [ANS]\nRejection region $F >$ [ANS]\nThe final conclustion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that the mean responses for the treatments are the same. B. We can reject the null hypothesis that the mean responses for the treatments are the same and accept the alternative hypothesis that at least two treatment means differ.\n(b) Sum of Squares for Treatment (SST) is $40$ \\% of SS(Total) $F=$ [ANS]\nRejection region $F >$ [ANS]\nThe final conclustion is [ANS] A. We can reject the null hypothesis that the mean responses for the treatments are the same and accept the alternative hypothesis that at least two treatment means differ. B. There is not sufficient evidence to reject the null hypothesis that the mean responses for the treatments are the same.\n(c) Sum of Squares for Treatment (SST) is $60$ \\% of SS(Total) $F=$ [ANS]\nRejection region $F >$ [ANS]\nThe final conclustion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that the mean responses for the treatments are the same. B. We can reject the null hypothesis that the mean responses for the treatments are the same and accept the alternative hypothesis that at least two treatment means differ.", "answer_v1": [ "1.54285714285714", "2.77285", "A", "2.4", "2.77285", "B", "5.4", "2.77285", "B" ], "answer_type_v1": [ "NV", "NV", "MCS", "NV", "NV", "MCS", "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "A", "B" ], [], [], [ "A", "B" ], [], [], [ "A", "B" ] ], "problem_v2": "369 41 Suppose the Total Sum of Squares for a completely randomzied design with $p=4$ treatments and $n=20$ total measurements (SS(Total))is equal to $410$. In each of the following cases, conduct an $F$-test of the null hypothesis that the mean responses for the $4$ treatments are the same. Use $\\alpha=0.025$.\n(a) Sum of Squares for Treatment (SST) is $90$ \\% of SS(Total) $F=$ [ANS]\nRejection region $F >$ [ANS]\nThe final conclustion is [ANS] A. We can reject the null hypothesis that the mean responses for the treatments are the same and accept the alternative hypothesis that at least two treatment means differ. B. There is not sufficient evidence to reject the null hypothesis that the mean responses for the treatments are the same.\n(b) Sum of Squares for Treatment (SST) is $30$ \\% of SS(Total) $F=$ [ANS]\nRejection region $F >$ [ANS]\nThe final conclustion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that the mean responses for the treatments are the same. B. We can reject the null hypothesis that the mean responses for the treatments are the same and accept the alternative hypothesis that at least two treatment means differ.\n(c) Sum of Squares for Treatment (SST) is $20$ \\% of SS(Total) $F=$ [ANS]\nRejection region $F >$ [ANS]\nThe final conclustion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that the mean responses for the treatments are the same. B. We can reject the null hypothesis that the mean responses for the treatments are the same and accept the alternative hypothesis that at least two treatment means differ.", "answer_v2": [ "48", "4.07682", "A", "2.28571428571429", "4.07682", "A", "1.33333333333333", "4.07682", "A" ], "answer_type_v2": [ "NV", "NV", "MCS", "NV", "NV", "MCS", "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "A", "B" ], [], [], [ "A", "B" ], [], [], [ "A", "B" ] ], "problem_v3": "86 344 Suppose the Total Sum of Squares for a completely randomzied design with $p=4$ treatments and $n=16$ total measurements (SS(Total))is equal to $430$. In each of the following cases, conduct an $F$-test of the null hypothesis that the mean responses for the $4$ treatments are the same. Use $\\alpha=0.05$.\n(a) Sum of Squares for Treatment (SST) is $20$ \\% of SS(Total) $F=$ [ANS]\nRejection region $F >$ [ANS]\nThe final conclustion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that the mean responses for the treatments are the same. B. We can reject the null hypothesis that the mean responses for the treatments are the same and accept the alternative hypothesis that at least two treatment means differ.\n(b) Sum of Squares for Treatment (SST) is $40$ \\% of SS(Total) $F=$ [ANS]\nRejection region $F >$ [ANS]\nThe final conclustion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that the mean responses for the treatments are the same. B. We can reject the null hypothesis that the mean responses for the treatments are the same and accept the alternative hypothesis that at least two treatment means differ.\n(c) Sum of Squares for Treatment (SST) is $80$ \\% of SS(Total) $F=$ [ANS]\nRejection region $F >$ [ANS]\nThe final conclustion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that the mean responses for the treatments are the same. B. We can reject the null hypothesis that the mean responses for the treatments are the same and accept the alternative hypothesis that at least two treatment means differ.", "answer_v3": [ "1", "3.49029", "A", "2.66666666666667", "3.49029", "A", "16", "3.49029", "B" ], "answer_type_v3": [ "NV", "NV", "MCS", "NV", "NV", "MCS", "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "A", "B" ], [], [], [ "A", "B" ], [], [], [ "A", "B" ] ] }, { "id": "Statistics_0333", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One-way ANOVA", "level": "3", "keywords": [ "statistics", "analysis of variance", "ANOVA" ], "problem_v1": "For each problem, select the best response.\n(a) The alternative hypothesis for an ANOVA F test is that [ANS] A. at least two of the population means are not equal. B. none of the population variances are equal. C. all of the population variances are not equal. D. all of the population means are not equal. E. none of the population means are equal.\n(b) The null hypothesis for an ANOVA F test is that [ANS] A. population means are equal. B. sample means are equal. C. sample standard deviations are not equal. D. population standard deviations are not equal. E. sample sizes are equal.\n(c) In contrast to a chi-square test, an ANOVA F test is most appropriate when the [ANS] A. standard deviations must be estimated. B. data are quantitative. C. number of samples is two. D. samples sizes are equal. E. standard deviations are not equal.", "answer_v1": [ "A", "A", "B" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "For each problem, select the best response.\n(a) In contrast to a chi-square test, an ANOVA F test is most appropriate when the [ANS] A. number of samples is two. B. standard deviations must be estimated. C. data are quantitative. D. standard deviations are not equal. E. samples sizes are equal.\n(b) The alternative hypothesis for an ANOVA F test is that [ANS] A. at least two of the population means are not equal. B. none of the population means are equal. C. all of the population means are not equal. D. none of the population variances are equal. E. all of the population variances are not equal.\n(c) The null hypothesis for an ANOVA F test is that [ANS] A. sample means are equal. B. sample standard deviations are not equal. C. population means are equal. D. sample sizes are equal. E. population standard deviations are not equal.", "answer_v2": [ "C", "A", "C" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "For each problem, select the best response.\n(a) In contrast to a chi-square test, an ANOVA F test is most appropriate when the [ANS] A. standard deviations must be estimated. B. number of samples is two. C. standard deviations are not equal. D. data are quantitative. E. samples sizes are equal.\n(b) The alternative hypothesis for an ANOVA F test is that [ANS] A. at least two of the population means are not equal. B. all of the population variances are not equal. C. none of the population means are equal. D. all of the population means are not equal. E. none of the population variances are equal.\n(c) The null hypothesis for an ANOVA F test is that [ANS] A. population standard deviations are not equal. B. sample means are equal. C. sample standard deviations are not equal. D. population means are equal. E. sample sizes are equal.", "answer_v3": [ "D", "A", "D" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Statistics_0334", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One-way ANOVA", "level": "3", "keywords": [ "statistics", "analysis of variance", "ANOVA" ], "problem_v1": "Use the Minitab display to test the claims. Use the $\\alpha=0.05$ significance level. The sample data are SAT scores on the verbal and math portions of SAT-I. \\begin{array}{l} \\begin{array}{l} \\mbox{Analysis of Variance for SAT} \\end{array} \\cr \\begin{array}{lrrrrr} \\mbox{Source} & \\mbox{DF} & \\mbox{SS} & \\mbox{MS} & \\mbox{F} & \\mbox{P} \\cr \\mbox{Gender} & 1 & 52668 & 52668 & 5.05 & 0.031 \\cr \\mbox{Ver/Math} & 1 & 6039 & 6039 & 0.58 & 0.452 \\cr \\mbox{Interaction} & 1 & 31617 & 31617 & 3.03 & 0.09 \\cr \\mbox{Error} & 37 & 376351 & 10431 & & \\cr \\mbox{Total} & 40 & 466675 & & & \\end{array} \\end{array} 1. Test the claim that SAT scores are not affected by an interaction between gender and test (verbal/math).\n(a) The $F-$ test statistic is [ANS]. (b) The P-value is [ANS]. (c) Does there appear to be a significant effect from the interaction between gender and test? [ANS] A. Yes B. No\n2. Test the claim that gender has an effect on SAT scores.\n(a) The $F-$ test statistic is [ANS]. (b) The P-value is [ANS]. (c) Is there sufficient evidence to support the claim that gender has an effect on SAT scores? [ANS] A. Yes B. No\n3. Test the claim that the type of test (math/verbal) has an effect on SAT scores.\n(a) The $F-$ test statistic is [ANS]. (b) The P-value is [ANS]. (c) Is there sufficient evidence to support the claim that the type of test has an effect on SAT scores? [ANS] A. Yes B. No", "answer_v1": [ "3.03", "0.09", "B", "5.05", "0.031", "A", "0.58", "0.452", "B" ], "answer_type_v1": [ "NV", "NV", "MCS", "NV", "NV", "MCS", "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "A", "B" ], [], [], [ "A", "B" ], [], [], [ "A", "B" ] ], "problem_v2": "Use the Minitab display to test the claims. Use the $\\alpha=0.05$ significance level. The sample data are SAT scores on the verbal and math portions of SAT-I. \\begin{array}{l} \\begin{array}{l} \\mbox{Analysis of Variance for SAT} \\end{array} \\cr \\begin{array}{lrrrrr} \\mbox{Source} & \\mbox{DF} & \\mbox{SS} & \\mbox{MS} & \\mbox{F} & \\mbox{P} \\cr \\mbox{Gender} & 1 & 52808 & 52808 & 5.06 & 0.031 \\cr \\mbox{Ver/Math} & 1 & 5992 & 5992 & 0.57 & 0.453 \\cr \\mbox{Interaction} & 1 & 31461 & 31461 & 3.02 & 0.091 \\cr \\mbox{Error} & 34 & 377642 & 10428 & & \\cr \\mbox{Total} & 37 & 467903 & & & \\end{array} \\end{array} 1. Test the claim that SAT scores are not affected by an interaction between gender and test (verbal/math).\n(a) The $F-$ test statistic is [ANS]. (b) The P-value is [ANS]. (c) Does there appear to be a significant effect from the interaction between gender and test? [ANS] A. Yes B. No\n2. Test the claim that gender has an effect on SAT scores.\n(a) The $F-$ test statistic is [ANS]. (b) The P-value is [ANS]. (c) Is there sufficient evidence to support the claim that gender has an effect on SAT scores? [ANS] A. Yes B. No\n3. Test the claim that the type of test (math/verbal) has an effect on SAT scores.\n(a) The $F-$ test statistic is [ANS]. (b) The P-value is [ANS]. (c) Is there sufficient evidence to support the claim that the type of test has an effect on SAT scores? [ANS] A. No B. Yes", "answer_v2": [ "3.02", "0.091", "B", "5.06", "0.031", "A", "0.57", "0.453", "A" ], "answer_type_v2": [ "NV", "NV", "MCS", "NV", "NV", "MCS", "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "A", "B" ], [], [], [ "A", "B" ], [], [], [ "A", "B" ] ], "problem_v3": "Use the Minitab display to test the claims. Use the $\\alpha=0.05$ significance level. The sample data are SAT scores on the verbal and math portions of SAT-I. \\begin{array}{l} \\begin{array}{l} \\mbox{Analysis of Variance for SAT} \\end{array} \\cr \\begin{array}{lrrrrr} \\mbox{Source} & \\mbox{DF} & \\mbox{SS} & \\mbox{MS} & \\mbox{F} & \\mbox{P} \\cr \\mbox{Gender} & 1 & 52677 & 52677 & 5.05 & 0.031 \\cr \\mbox{Ver/Math} & 1 & 6005 & 6005 & 0.58 & 0.453 \\cr \\mbox{Interaction} & 1 & 31547 & 31547 & 3.02 & 0.091 \\cr \\mbox{Error} & 35 & 376160 & 10434 & & \\cr \\mbox{Total} & 38 & 466389 & & & \\end{array} \\end{array} 1. Test the claim that SAT scores are not affected by an interaction between gender and test (verbal/math).\n(a) The $F-$ test statistic is [ANS]. (b) The P-value is [ANS]. (c) Does there appear to be a significant effect from the interaction between gender and test? [ANS] A. Yes B. No\n2. Test the claim that gender has an effect on SAT scores.\n(a) The $F-$ test statistic is [ANS]. (b) The P-value is [ANS]. (c) Is there sufficient evidence to support the claim that gender has an effect on SAT scores? [ANS] A. Yes B. No\n3. Test the claim that the type of test (math/verbal) has an effect on SAT scores.\n(a) The $F-$ test statistic is [ANS]. (b) The P-value is [ANS]. (c) Is there sufficient evidence to support the claim that the type of test has an effect on SAT scores? [ANS] A. No B. Yes", "answer_v3": [ "3.02", "0.091", "B", "5.05", "0.031", "A", "0.58", "0.453", "A" ], "answer_type_v3": [ "NV", "NV", "MCS", "NV", "NV", "MCS", "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "A", "B" ], [], [], [ "A", "B" ], [], [], [ "A", "B" ] ] }, { "id": "Statistics_0335", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One-way ANOVA", "level": "2", "keywords": [ "statistics", "analysis of variance", "ANOVA" ], "problem_v1": "For each problem, select the best response.\n(a) We anticipate a small P value for an ANOVA F statistic if the box plots for the samples are [ANS] A. narrow and located differently. B. identical. C. wide and have similar medians. D. wide and similarly located. E. symmetrical.\n(b) The P value for an ANOVA F statistic is large if the sample [ANS] A. means are about equal. B. sizes are small. C. standard deviations are small. D. standard deviations are equal. E. sizes are equal.\n(c) ANOVA stands for [ANS] A. add now or value after. B. analysis of variance. C. average number of votes. D. addition of values. E. average of values.", "answer_v1": [ "A", "A", "B" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "For each problem, select the best response.\n(a) ANOVA stands for [ANS] A. average number of votes. B. add now or value after. C. analysis of variance. D. average of values. E. addition of values.\n(b) We anticipate a small P value for an ANOVA F statistic if the box plots for the samples are [ANS] A. narrow and located differently. B. symmetrical. C. wide and similarly located. D. identical. E. wide and have similar medians.\n(c) The P value for an ANOVA F statistic is large if the sample [ANS] A. sizes are small. B. standard deviations are small. C. means are about equal. D. sizes are equal. E. standard deviations are equal.", "answer_v2": [ "C", "A", "C" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "For each problem, select the best response.\n(a) ANOVA stands for [ANS] A. add now or value after. B. average number of votes. C. average of values. D. analysis of variance. E. addition of values.\n(b) We anticipate a small P value for an ANOVA F statistic if the box plots for the samples are [ANS] A. narrow and located differently. B. wide and have similar medians. C. symmetrical. D. wide and similarly located. E. identical.\n(c) The P value for an ANOVA F statistic is large if the sample [ANS] A. standard deviations are equal. B. sizes are small. C. standard deviations are small. D. means are about equal. E. sizes are equal.", "answer_v3": [ "D", "A", "D" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Statistics_0336", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One-way ANOVA", "level": "1", "keywords": [ "statistics", "analysis of variance", "ANOVA", "percent", "Hypothesis Testing" ], "problem_v1": "One-way ANOVA is a procedure for comparing the means of several populations. It is the generalization of what procedure for comparing the means of two populations? [ANS] A. nonpooled t-procedure B. paired t-test C. pooled t-procedure D. None of the above\nIn One-way ANOVA, identify the statistics (SSTR, MSTR, SSE, MSE or F) used\n(a) as a measure of variation among the sample means [ANS]\n(b) as a measure of variation within the samples [ANS]\n(c) to compare the variation among the sample means to the variation within the samples [ANS]", "answer_v1": [ "C", "MSTR", "MSE", "F" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "TF" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "SSTR", "MSTR", "SSE", "MSE", "F" ], [ "SSTR", "MSTR", "SSE", "MSE", "F" ], [] ], "problem_v2": "One-way ANOVA is a procedure for comparing the means of several populations. It is the generalization of what procedure for comparing the means of two populations? [ANS] A. pooled t-procedure B. paired t-test C. nonpooled t-procedure D. None of the above\nIn One-way ANOVA, identify the statistics (SSTR, MSTR, SSE, MSE or F) used\n(a) as a measure of variation among the sample means [ANS]\n(b) as a measure of variation within the samples [ANS]\n(c) to compare the variation among the sample means to the variation within the samples [ANS]", "answer_v2": [ "A", "MSTR", "MSE", "F" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "TF" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "SSTR", "MSTR", "SSE", "MSE", "F" ], [ "SSTR", "MSTR", "SSE", "MSE", "F" ], [] ], "problem_v3": "One-way ANOVA is a procedure for comparing the means of several populations. It is the generalization of what procedure for comparing the means of two populations? [ANS] A. nonpooled t-procedure B. pooled t-procedure C. paired t-test D. None of the above\nIn One-way ANOVA, identify the statistics (SSTR, MSTR, SSE, MSE or F) used\n(a) as a measure of variation among the sample means [ANS]\n(b) as a measure of variation within the samples [ANS]\n(c) to compare the variation among the sample means to the variation within the samples [ANS]", "answer_v3": [ "B", "MSTR", "MSE", "F" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "TF" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "SSTR", "MSTR", "SSE", "MSE", "F" ], [ "SSTR", "MSTR", "SSE", "MSE", "F" ], [] ] }, { "id": "Statistics_0337", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One-way ANOVA", "level": "1", "keywords": [ "Statistics", "Hypothesis Testing" ], "problem_v1": "For an F-curve with degrees of freedom df=(12,5), find the F-value that has area 0.01 to its right. [ANS] A. 5.06 B. 3.27 C. 9.89 D. None of the above\nFor an F-curve with degrees of freedom df=(4,10), find the F-value that has area 0.005 to its right. [ANS] A. 7.34 B. 20.97 C. 2.61 D. None of the above", "answer_v1": [ "C", "A" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "For an F-curve with degrees of freedom df=(12,5), find the F-value that has area 0.01 to its right. [ANS] A. 9.89 B. 3.27 C. 5.06 D. None of the above\nFor an F-curve with degrees of freedom df=(4,10), find the F-value that has area 0.005 to its right. [ANS] A. 7.34 B. 2.61 C. 20.97 D. None of the above", "answer_v2": [ "A", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "For an F-curve with degrees of freedom df=(12,5), find the F-value that has area 0.01 to its right. [ANS] A. 5.06 B. 9.89 C. 3.27 D. None of the above\nFor an F-curve with degrees of freedom df=(4,10), find the F-value that has area 0.005 to its right. [ANS] A. 20.97 B. 7.34 C. 2.61 D. None of the above", "answer_v3": [ "B", "B" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0338", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One-way ANOVA", "level": "3", "keywords": [ "percent" ], "problem_v1": "What are the three conditions required for 0ne-way ANOVA? [ANS] A. normal populations, no outliers and equal sample standard deviations B. large sample sizes, paired samples and normal differences C. independent samples, normal populations and approximately equal population standard deviations D. None of the above", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "What are the three conditions required for 0ne-way ANOVA? [ANS] A. independent samples, normal populations and approximately equal population standard deviations B. large sample sizes, paired samples and normal differences C. normal populations, no outliers and equal sample standard deviations D. None of the above", "answer_v2": [ "A" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "What are the three conditions required for 0ne-way ANOVA? [ANS] A. normal populations, no outliers and equal sample standard deviations B. independent samples, normal populations and approximately equal population standard deviations C. large sample sizes, paired samples and normal differences D. None of the above", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0339", "subject": "Statistics", "topic": "Hypothesis tests", "subtopic": "One-way ANOVA", "level": "3", "keywords": [ "Statistics", "Hypothesis Testing", "ANOVA", "Analysis", "Variance", "F-test", "analysis of variance", "ANOVA" ], "problem_v1": "An experiment is conducted to determine whether there is a differnce among the mean increases in growth produced by five strains (A, B, C, D and E) of growth hormones for plants. The experimental material consists of 20 cuttings of a shrub (all of equal weight), with four cuttings randomly assigned to each of the five different strains. The increases in weight for each cutting along with the sample mean and sample standard deviation of each group are given in the table below.\n$\\begin{array}{cccccc}\\hline & A & B & C & D & E \\\\ \\hline Plant 1 & 17 & 24 & 23 & 17 & 9 \\\\ \\hline Plant 2 & 11 & 24 & 23 & 13 & 10 \\\\ \\hline Plant 3 & 15 & 19 & 23 & 14 & 9 \\\\ \\hline Plant 4 & 13 & 23 & 20 & 11 & 9 \\\\ \\hline Mean & 14 & 22.5 & 22.25 & 13.75 & 9.25 \\\\ \\hline Standard Dev. & 2.5820 & 2.3805 & 1.5000 & 2.5000 & 0.5000 \\\\ \\hline \\end{array}$\nIt is also given that the overall mean=16.35. Compute the following:\n(a) SSTR $=$ [ANS]\n(b) SSE $=$ [ANS]\n(c) MSTR $=$ [ANS]\n(d) MSE $=$ [ANS]\n(e) F $=$ [ANS]", "answer_v1": [ "541.3", "63.25", "135.325", "4.21666666666667", "32.0928853754941" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "An experiment is conducted to determine whether there is a differnce among the mean increases in growth produced by five strains (A, B, C, D and E) of growth hormones for plants. The experimental material consists of 20 cuttings of a shrub (all of equal weight), with four cuttings randomly assigned to each of the five different strains. The increases in weight for each cutting along with the sample mean and sample standard deviation of each group are given in the table below.\n$\\begin{array}{cccccc}\\hline & A & B & C & D & E \\\\ \\hline Plant 1 & 8 & 29 & 20 & 13 & 15 \\\\ \\hline Plant 2 & 11 & 17 & 21 & 15 & 6 \\\\ \\hline Plant 3 & 15 & 22 & 24 & 11 & 7 \\\\ \\hline Plant 4 & 11 & 24 & 21 & 13 & 11 \\\\ \\hline Mean & 11.25 & 23 & 21.5 & 13 & 9.75 \\\\ \\hline Standard Dev. & 2.8723 & 4.9666 & 1.7321 & 1.6330 & 4.1130 \\\\ \\hline \\end{array}$\nIt is also given that the overall mean=15.7. Compute the following:\n(a) SSTR $=$ [ANS]\n(b) SSE $=$ [ANS]\n(c) MSTR $=$ [ANS]\n(d) MSE $=$ [ANS]\n(e) F $=$ [ANS]", "answer_v2": [ "597.7", "166.5", "149.425", "11.1", "13.4617117117117" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "An experiment is conducted to determine whether there is a differnce among the mean increases in growth produced by five strains (A, B, C, D and E) of growth hormones for plants. The experimental material consists of 20 cuttings of a shrub (all of equal weight), with four cuttings randomly assigned to each of the five different strains. The increases in weight for each cutting along with the sample mean and sample standard deviation of each group are given in the table below.\n$\\begin{array}{cccccc}\\hline & A & B & C & D & E \\\\ \\hline Plant 1 & 11 & 24 & 21 & 15 & 8 \\\\ \\hline Plant 2 & 12 & 27 & 25 & 18 & 8 \\\\ \\hline Plant 3 & 11 & 19 & 20 & 15 & 15 \\\\ \\hline Plant 4 & 17 & 24 & 20 & 12 & 12 \\\\ \\hline Mean & 12.75 & 23.5 & 21.5 & 15 & 10.75 \\\\ \\hline Standard Dev. & 2.8723 & 3.3166 & 2.3805 & 2.4495 & 3.4034 \\\\ \\hline \\end{array}$\nIt is also given that the overall mean=16.7. Compute the following:\n(a) SSTR $=$ [ANS]\n(b) SSE $=$ [ANS]\n(c) MSTR $=$ [ANS]\n(d) MSE $=$ [ANS]\n(e) F $=$ [ANS]", "answer_v3": [ "492.7", "127.5", "123.175", "8.5", "14.4911764705882" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Statistics_0340", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Correlation", "level": "3", "keywords": [ "Correlation", "Coefficient" ], "problem_v1": "Given the following data set, $\\begin{array}{cccccccc}\\hline x & 3 & 2 & 0 & 1 & 0 & 2 & 1 \\\\ \\hline y & 4 & 5 & 2 & 4 & 3 & 1 & 2 \\\\ \\hline \\end{array}$\nCompute the coefficient of correlation $r$ $r=$ [ANS]", "answer_v1": [ "0.317744454651121" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Given the following data set, $\\begin{array}{cccccccc}\\hline x &-2 &-1 & 4 &-1 & 1 & 2 & 3 \\\\ \\hline y & 6 & 2 & 2 & 2 & 0 & 3 & 1 \\\\ \\hline \\end{array}$\nCompute the coefficient of correlation $r$ $r=$ [ANS]", "answer_v2": [ "-0.494444444444444" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Given the following data set, $\\begin{array}{cccccccc}\\hline x & 0 &-1 &-1 & 3 & 4 & 0 &-2 \\\\ \\hline y & 4 & 3 & 2 & 6 & 1 & 1 & 4 \\\\ \\hline \\end{array}$\nCompute the coefficient of correlation $r$ $r=$ [ANS]", "answer_v3": [ "0" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0341", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Correlation", "level": "2", "keywords": [ "statistic", "correlation" ], "problem_v1": "An outbreak of the deadly Ebola virus in 2002 and 2003 killed 91 of the 95 gorillas in 7 home ranges in the Congo. To study the spread of the virus, measure \" distance\" by the number of home ranges separating a group of gorillas from the first group infected. Here are data on distance and number of days until deaths began in each later group:\n$\\begin{array}{ccccccc}\\hline Distance x & 4 & 4 & 6 & 6 & 4 & 6 \\\\ \\hline Days y & 4 & 21 & 31 & 40 & 45 & 46 \\\\ \\hline \\end{array}$ Use a calculator or software to find the following values.\n(a) Find the mean and standard deviation of $x$. MEAN: [ANS]\nSTANDARD DEVIATION: [ANS]\n(b) Find the mean and standard deviation of $y$. MEAN: [ANS]\nSTANDARD DEVIATION: [ANS]\n(c) Find the correlation $r$ between $x$ and $y$. r: [ANS]", "answer_v1": [ "5", "1.09545", "31.1667", "16.3146", "0.525969" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "An outbreak of the deadly Ebola virus in 2002 and 2003 killed 91 of the 95 gorillas in 7 home ranges in the Congo. To study the spread of the virus, measure \" distance\" by the number of home ranges separating a group of gorillas from the first group infected. Here are data on distance and number of days until deaths began in each later group:\n$\\begin{array}{ccccccc}\\hline Distance x & 1 & 5 & 3 & 4 & 7 & 6 \\\\ \\hline Days y & 2 & 19 & 33 & 35 & 45 & 47 \\\\ \\hline \\end{array}$ Use a calculator or software to find the following values.\n(a) Find the mean and standard deviation of $x$. MEAN: [ANS]\nSTANDARD DEVIATION: [ANS]\n(b) Find the mean and standard deviation of $y$. MEAN: [ANS]\nSTANDARD DEVIATION: [ANS]\n(c) Find the correlation $r$ between $x$ and $y$. r: [ANS]", "answer_v2": [ "4.33333", "2.16025", "30.1667", "17.046", "0.807453" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "An outbreak of the deadly Ebola virus in 2002 and 2003 killed 91 of the 95 gorillas in 7 home ranges in the Congo. To study the spread of the virus, measure \" distance\" by the number of home ranges separating a group of gorillas from the first group infected. Here are data on distance and number of days until deaths began in each later group:\n$\\begin{array}{ccccccc}\\hline Distance x & 2 & 4 & 4 & 5 & 4 & 6 \\\\ \\hline Days y & 6 & 23 & 35 & 37 & 42 & 46 \\\\ \\hline \\end{array}$ Use a calculator or software to find the following values.\n(a) Find the mean and standard deviation of $x$. MEAN: [ANS]\nSTANDARD DEVIATION: [ANS]\n(b) Find the mean and standard deviation of $y$. MEAN: [ANS]\nSTANDARD DEVIATION: [ANS]\n(c) Find the correlation $r$ between $x$ and $y$. r: [ANS]", "answer_v3": [ "4.16667", "1.32916", "31.5", "14.7343", "0.87315" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Statistics_0342", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Correlation", "level": "1", "keywords": [ "statistic", "correlation" ], "problem_v1": "For each problem, select the best response.\n(a) Can one predict a student's score on the midterm exam in a statistics course from the number of hours the student spent studying for the exam? To explore this, the teacher of the course asks students how many hours they spent studying for the exam and then makes a scatterplot of the time students spent studying and their scores on the exam. In making the scatterplot, the teacher should [ANS] A. plot the score on the exam on the horizontal axis. B. first determine if the scores on the exam approximately follow a normal distribution. C. use a plotting scale that makes the overall trend roughly linear. D. plot time spent studying for the exam on the horizontal axis. E. None of the above.\n(b) A gambler conducts a study to determine whether the time it took a horse to run its last race can be used to predict the time it takes the horse to run its next race. In this study, the explanatory variable is [ANS] A. all horses used in the study. B. the time it takes a horse to run its next race. C. the gambler's winnings. D. the time it took a horse to run its last race. E. None of the above.\n(c) Does mandatory gun ownership prevent crime? To study this, the number of burglaries committed each month in a small town were recorded for 75 months prior to passage of a bill requiring citizens to own guns and for 56 months after passage of the bill. The goal was to see if the number of burglaries committed was affected by requiring citizens to own guns. The response variable here is [ANS] A. the number of burglaries committed. B. whether or not a burglary was committed by a gun owner. C. the number of guns owned. D. whether or not gun ownership is required by law. E. None of the above.", "answer_v1": [ "D", "D", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "For each problem, select the best response.\n(a) Does mandatory gun ownership prevent crime? To study this, the number of burglaries committed each month in a small town were recorded for 75 months prior to passage of a bill requiring citizens to own guns and for 56 months after passage of the bill. The goal was to see if the number of burglaries committed was affected by requiring citizens to own guns. The response variable here is [ANS] A. the number of guns owned. B. whether or not a burglary was committed by a gun owner. C. whether or not gun ownership is required by law. D. the number of burglaries committed. E. None of the above.\n(b) Can one predict a student's score on the midterm exam in a statistics course from the number of hours the student spent studying for the exam? To explore this, the teacher of the course asks students how many hours they spent studying for the exam and then makes a scatterplot of the time students spent studying and their scores on the exam. In making the scatterplot, the teacher should [ANS] A. plot the score on the exam on the horizontal axis. B. plot time spent studying for the exam on the horizontal axis. C. use a plotting scale that makes the overall trend roughly linear. D. first determine if the scores on the exam approximately follow a normal distribution. E. None of the above.\n(c) A gambler conducts a study to determine whether the time it took a horse to run its last race can be used to predict the time it takes the horse to run its next race. In this study, the explanatory variable is [ANS] A. the time it took a horse to run its last race. B. the gambler's winnings. C. all horses used in the study. D. the time it takes a horse to run its next race. E. None of the above.", "answer_v2": [ "D", "B", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "For each problem, select the best response.\n(a) Does mandatory gun ownership prevent crime? To study this, the number of burglaries committed each month in a small town were recorded for 75 months prior to passage of a bill requiring citizens to own guns and for 56 months after passage of the bill. The goal was to see if the number of burglaries committed was affected by requiring citizens to own guns. The response variable here is [ANS] A. the number of burglaries committed. B. whether or not gun ownership is required by law. C. the number of guns owned. D. whether or not a burglary was committed by a gun owner. E. None of the above.\n(b) Can one predict a student's score on the midterm exam in a statistics course from the number of hours the student spent studying for the exam? To explore this, the teacher of the course asks students how many hours they spent studying for the exam and then makes a scatterplot of the time students spent studying and their scores on the exam. In making the scatterplot, the teacher should [ANS] A. first determine if the scores on the exam approximately follow a normal distribution. B. plot time spent studying for the exam on the horizontal axis. C. use a plotting scale that makes the overall trend roughly linear. D. plot the score on the exam on the horizontal axis. E. None of the above.\n(c) A gambler conducts a study to determine whether the time it took a horse to run its last race can be used to predict the time it takes the horse to run its next race. In this study, the explanatory variable is [ANS] A. the time it took a horse to run its last race. B. the gambler's winnings. C. the time it takes a horse to run its next race. D. all horses used in the study. E. None of the above.", "answer_v3": [ "A", "B", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Statistics_0343", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Correlation", "level": "1", "keywords": [ "statistic", "correlation" ], "problem_v1": "For each problem, select the best response.\n(a) A study found a correlation of r=-0.61 between the gender of a worker and his or her income. You may correctly conclude [ANS] A. women earn less than men on average. B. women earn more than men on average. C. an arithmetic mistake was made. Correlation must be positive. D. this is incorrect because r makes no sense here. E. None of the above.\n(b) For a biology project, you measure the weight in grams and the tail length in millimeters of a group of mice. The correlation is r=0.6. If you had measured tail length in centimeters instead of millimeters, what would be the correlation? (There are 10 millimeters in a centimeter.) [ANS] A. 0.6 B. 0.6/10=0.06 C. (0.6)(10)=6 D. None of the above.", "answer_v1": [ "D", "A" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D" ] ], "problem_v2": "For each problem, select the best response.\n(a) For a biology project, you measure the weight in grams and the tail length in millimeters of a group of mice. The correlation is r=0.2. If you had measured tail length in centimeters instead of millimeters, what would be the correlation? (There are 10 millimeters in a centimeter.) [ANS] A. (0.2)(10)=2 B. 0.2/10=0.02 C. 0.2 D. None of the above.\n(b) A study found a correlation of r=-0.61 between the gender of a worker and his or her income. You may correctly conclude [ANS] A. this is incorrect because r makes no sense here. B. an arithmetic mistake was made. Correlation must be positive. C. women earn less than men on average. D. women earn more than men on average. E. None of the above.", "answer_v2": [ "C", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "For each problem, select the best response.\n(a) For a biology project, you measure the weight in grams and the tail length in millimeters of a group of mice. The correlation is r=0.3. If you had measured tail length in centimeters instead of millimeters, what would be the correlation? (There are 10 millimeters in a centimeter.) [ANS] A. 0.3 B. 0.3/10=0.03 C. (0.3)(10)=3 D. None of the above.\n(b) A study found a correlation of r=-0.61 between the gender of a worker and his or her income. You may correctly conclude [ANS] A. this is incorrect because r makes no sense here. B. an arithmetic mistake was made. Correlation must be positive. C. women earn more than men on average. D. women earn less than men on average. E. None of the above.", "answer_v3": [ "A", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Statistics_0344", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Correlation", "level": "3", "keywords": [ "statistic", "correlation", "scatterplots", "Correlation", "Coefficient" ], "problem_v1": "Use a scatterplot and the linear correlation coefficient $r$ to determine whether there is a correlation between the two variables. (Note: Use software, and don't forget to look at the scatterplot!)\n\\begin{array}{c|ccccccccccccccc} x & 0.5 & 1.6 & 2.2 & 3.5 & 4.4 & 5.2 & 6.4 & 7.4 & 8.1 & 9.3 & 10.7 & 11.3 & 12.8 & 13.2 & 14.2\\cr\ny & 9.3 & 4.5 & 8.5 & 5.6 & 9.8 & 7.6 & 5.9 & 7.7 & 2.3 & 7.7 & 6.9 & 7 & 15 & 0.4 & 8.2\\cr \\end{array}\n(a) $r=$ [ANS]\n(b) There is [ANS] A. no correlation between $x$ and $y$ B. a perfect positive correlation between $x$ and $y$ C. a positive correlation between $x$ and $y$ D. a negative correlation between $x$ and $y$ E. a perfect negative correlation between $x$ and $y$ F. a nonlinear correlation between $x$ and $y$", "answer_v1": [ "-0.00728503", "A" ], "answer_type_v1": [ "NV", "MCS" ], "options_v1": [ [], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Use a scatterplot and the linear correlation coefficient $r$ to determine whether there is a correlation between the two variables. (Note: Use software, and don't forget to look at the scatterplot!)\n\\begin{array}{c|ccccccccccccccc} x & 0.9 & 1.1 & 2.3 & 3.8 & 4.3 & 5.1 & 6.3 & 7.5 & 8 & 9.6 & 10.4 & 11.8 & 12.1 & 13.1 & 14.2\\cr\ny &-0.5 & 1.1 & 4.1 & 4.9 & 2.5 & 7.1 & 7 & 6.1 & 8.4 & 8.9 & 11.6 & 11.1 & 11.9 & 11.2 & 16.1\\cr \\end{array}\n(a) $r=$ [ANS]\n(b) There is [ANS] A. a nonlinear correlation between $x$ and $y$ B. no correlation between $x$ and $y$ C. a negative correlation between $x$ and $y$ D. a perfect positive correlation between $x$ and $y$ E. a perfect negative correlation between $x$ and $y$ F. a positive correlation between $x$ and $y$", "answer_v2": [ "0.954455", "F" ], "answer_type_v2": [ "NV", "MCS" ], "options_v2": [ [], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Use a scatterplot and the linear correlation coefficient $r$ to determine whether there is a correlation between the two variables. (Note: Use software, and don't forget to look at the scatterplot!)\n\\begin{array}{c|ccccccccccccccc} x & 0.6 & 1.2 & 2.4 & 3.1 & 4.3 & 5.8 & 6.9 & 7.8 & 8.2 & 9.2 & 10.2 & 11 & 12.5 & 13.9 & 14.7\\cr\ny & 0.6 & 1.2 & 2.4 & 3.1 & 4.3 & 5.8 & 6.9 & 7.8 & 8.2 & 9.2 & 10.2 & 11 & 12.5 & 13.9 & 14.7\\cr \\end{array}\n(a) $r=$ [ANS]\n(b) There is [ANS] A. a negative correlation between $x$ and $y$ B. a nonlinear correlation between $x$ and $y$ C. a perfect positive correlation between $x$ and $y$ D. no correlation between $x$ and $y$ E. a perfect negative correlation between $x$ and $y$ F. a positive correlation between $x$ and $y$", "answer_v3": [ "1", "C" ], "answer_type_v3": [ "NV", "MCS" ], "options_v3": [ [], [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Statistics_0345", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Correlation", "level": "1", "keywords": [ "statistic", "correlation" ], "problem_v1": "For each problem, select the best response.\n(a) If the correlation between two variables is close to 0, you can conclude that a scatterplot would show [ANS] A. a strong straight-line pattern. B. a cloud of points with no visible pattern. C. no straight-line pattern, but there might be a strong pattern of another form. D. None of the above.\n(b) In a scatterplot of the average price of a barrel of oil and the average retail price of a gallon of gasoline, you expect to see [ANS] A. a positive association. B. very little association. C. a negative association. D. None of the above.\n(c) What are all the values that a correlation r can possibly take? [ANS] A. 0 $\\le$ r $\\le$ 1 B.-1 $\\le$ r $\\le$ 1 C. r $\\ge$ 0 D. None of the above.", "answer_v1": [ "C", "A", "B" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "For each problem, select the best response.\n(a) You have data for many years on the average price of a barrel of oil and the average retail price of a gallon of unleaded regular gasoline. When you make a scatterplot, the explanatory variable on the x-axis [ANS] A. can be either oil price or gasoline price. B. is the price of gasoline. C. is the price of oil. D. None of the above.\n(b) If the correlation between two variables is close to 0, you can conclude that a scatterplot would show [ANS] A. a cloud of points with no visible pattern. B. a strong straight-line pattern. C. no straight-line pattern, but there might be a strong pattern of another form. D. None of the above.\n(c) In a scatterplot of the average price of a barrel of oil and the average retail price of a gallon of gasoline, you expect to see [ANS] A. a negative association. B. very little association. C. a positive association. D. None of the above.", "answer_v2": [ "C", "C", "C" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "For each problem, select the best response.\n(a) In a scatterplot of the average price of a barrel of oil and the average retail price of a gallon of gasoline, you expect to see [ANS] A. a negative association. B. a positive association. C. very little association. D. None of the above.\n(b) What are all the values that a correlation r can possibly take? [ANS] A. r $\\ge$ 0 B. 0 $\\le$ r $\\le$ 1 C.-1 $\\le$ r $\\le$ 1 D. None of the above.\n(c) You have data for many years on the average price of a barrel of oil and the average retail price of a gallon of unleaded regular gasoline. When you make a scatterplot, the explanatory variable on the x-axis [ANS] A. is the price of oil. B. is the price of gasoline. C. can be either oil price or gasoline price. D. None of the above.", "answer_v3": [ "B", "C", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0346", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Correlation", "level": "3", "keywords": [ "statistic", "correlation", "scatterplots", "Correlation", "Coefficient" ], "problem_v1": "Match the following sample correlation coefficients with the explanation of what that correlation coefficient means. Type the correct letter in each box. [ANS] 1. $r=.92$ [ANS] 2. $r=-.97$ [ANS] 3. $r=.1$ [ANS] 4. $r=-.15$\nA. a strong positive relationship between $x$ and $y$ B. a weak positive relationship between $x$ and $y$ C. a strong negative relationship between $x$ and $y$ D. a weak negative relationship between $x$ and $y$", "answer_v1": [ "A", "C", "B", "D" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Match the following sample correlation coefficients with the explanation of what that correlation coefficient means. Type the correct letter in each box. [ANS] 1. $r=0$ [ANS] 2. $r=-.15$ [ANS] 3. $r=.92$ [ANS] 4. $r=1$\nA. a strong positive relationship between $x$ and $y$ B. a perfect positive relationship between $x$ and $y$ C. a weak negative relationship between $x$ and $y$ D. no relationship between $x$ and $y$", "answer_v2": [ "D", "C", "A", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Match the following sample correlation coefficients with the explanation of what that correlation coefficient means. Type the correct letter in each box. [ANS] 1. $r=-1$ [ANS] 2. $r=1$ [ANS] 3. $r=0$ [ANS] 4. $r=-.97$\nA. a perfect negative relationship between $x$ and $y$ B. a perfect positive relationship between $x$ and $y$ C. a strong negative relationship between $x$ and $y$ D. no relationship between $x$ and $y$", "answer_v3": [ "A", "B", "D", "C" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0347", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Correlation", "level": "1", "keywords": [ "statistic", "correlation" ], "problem_v1": "Keeping water supplies clean requires regular measurement of levels of pollutants. The measurements are indirect-a typical analysis involves forming a dye by a chemical reaction with the dissolved pollutant, then passing light through the solution and measuring its \" absorbence.\" To calibrate such measurements, the laboratory measures known standard solutions and uses regression to relate absorbence and pollutant concentration. This is usually done every day. Here is one series of data on the absorbence for different levels of nitrates. Nitrates are measured in milligrams per liter of water.\n$\\begin{array}{ccccccccccc}\\hline Nitrates & 100 & 75 & 125 & 250 & 200 & 400 & 800 & 1200 & 1500 & 3000 \\\\ \\hline Absorbance & 5.8 & 7.3 & 12.1 & 24 & 44.3 & 95.1 & 132.7 & 183.1 & 215.8 & 230.2 \\\\ \\hline \\end{array}$\nChemical theory says that these data should lie on a straight line. If the correlation is not at least 0.997, something went wrong and the calibration procedure is repeated.\n(a) Find the correlation $r$. $r$=[ANS]\n(b) Must the calibration be done again? (Answer YES or NO). ANSWER: [ANS]", "answer_v1": [ "0.893", "yes" ], "answer_type_v1": [ "NV", "TF" ], "options_v1": [ [], [] ], "problem_v2": "Keeping water supplies clean requires regular measurement of levels of pollutants. The measurements are indirect-a typical analysis involves forming a dye by a chemical reaction with the dissolved pollutant, then passing light through the solution and measuring its \" absorbence.\" To calibrate such measurements, the laboratory measures known standard solutions and uses regression to relate absorbence and pollutant concentration. This is usually done every day. Here is one series of data on the absorbence for different levels of nitrates. Nitrates are measured in milligrams per liter of water.\n$\\begin{array}{ccccccccccc}\\hline Nitrates & 50 & 100 & 75 & 200 & 300 & 400 & 600 & 1100 & 1600 & 2100 \\\\ \\hline Absorbance & 6.3 & 7.4 & 10.9 & 21.9 & 42.7 & 95.7 & 133.7 & 186.4 & 222.3 & 222.2 \\\\ \\hline \\end{array}$\nChemical theory says that these data should lie on a straight line. If the correlation is not at least 0.997, something went wrong and the calibration procedure is repeated.\n(a) Find the correlation $r$. $r$=[ANS]\n(b) Must the calibration be done again? (Answer YES or NO). ANSWER: [ANS]", "answer_v2": [ "0.943", "yes" ], "answer_type_v2": [ "NV", "TF" ], "options_v2": [ [], [] ], "problem_v3": "Keeping water supplies clean requires regular measurement of levels of pollutants. The measurements are indirect-a typical analysis involves forming a dye by a chemical reaction with the dissolved pollutant, then passing light through the solution and measuring its \" absorbence.\" To calibrate such measurements, the laboratory measures known standard solutions and uses regression to relate absorbence and pollutant concentration. This is usually done every day. Here is one series of data on the absorbence for different levels of nitrates. Nitrates are measured in milligrams per liter of water.\n$\\begin{array}{ccccccccccc}\\hline Nitrates & 50 & 75 & 100 & 250 & 200 & 400 & 900 & 1400 & 1800 & 2400 \\\\ \\hline Absorbance & 5.7 & 7 & 12.9 & 29.8 & 47.9 & 96.2 & 132.1 & 185.8 & 225.7 & 234.1 \\\\ \\hline \\end{array}$\nChemical theory says that these data should lie on a straight line. If the correlation is not at least 0.997, something went wrong and the calibration procedure is repeated.\n(a) Find the correlation $r$. $r$=[ANS]\n(b) Must the calibration be done again? (Answer YES or NO). ANSWER: [ANS]", "answer_v3": [ "0.966", "yes" ], "answer_type_v3": [ "NV", "TF" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0348", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Correlation", "level": "2", "keywords": [ "statistic", "regression", "correlation" ], "problem_v1": "Given the following data set, let $x$ be the explanatory variable and $y$ be the response variable. $\\begin{array}{cccccccc}\\hline x & 7 & 5 & 6 & 6 & 2 & 3 & 5 \\\\ \\hline y & 4 & 5 & 6 & 4 & 8 & 8 & 6 \\\\ \\hline \\end{array}$\n(a) If a least squares line was fitted to this data, what percentage of the variation in the $y$ would be explained by the regression line? (Enter your answer as a percent.) ANSWER: [ANS] \\% (b) Compute the correlation coefficient: $r=$ [ANS]", "answer_v1": [ "81.9787", "-0.905421" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Given the following data set, let $x$ be the explanatory variable and $y$ be the response variable. $\\begin{array}{cccccccc}\\hline x & 1 & 8 & 2 & 5 & 8 & 3 & 2 \\\\ \\hline y & 10 & 2 & 9 & 7 & 3 & 7 & 8 \\\\ \\hline \\end{array}$\n(a) If a least squares line was fitted to this data, what percentage of the variation in the $y$ would be explained by the regression line? (Enter your answer as a percent.) ANSWER: [ANS] \\% (b) Compute the correlation coefficient: $r=$ [ANS]", "answer_v2": [ "93.62", "-0.967574" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Given the following data set, let $x$ be the explanatory variable and $y$ be the response variable. $\\begin{array}{cccccccc}\\hline x & 3 & 6 & 2 & 5 & 2 & 5 & 8 \\\\ \\hline y & 8 & 6 & 8 & 5 & 9 & 7 & 3 \\\\ \\hline \\end{array}$\n(a) If a least squares line was fitted to this data, what percentage of the variation in the $y$ would be explained by the regression line? (Enter your answer as a percent.) ANSWER: [ANS] \\% (b) Compute the correlation coefficient: $r=$ [ANS]", "answer_v3": [ "86.5385", "-0.930261" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0349", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Correlation", "level": "1", "keywords": [ "statistic", "regression", "correlation" ], "problem_v1": "For each problem, select the best response.\n(a) When possible, the best way to establish that an observed association is the result of a cause-and-effect relation is by means of [ANS] A. the least squares regression line. B. the correlation coefficient. C. the square of the correlation coefficient. D. a well designed experiment. E. None of the above.\n(b) A researcher observes that, on average, the number of divorces in cities with major league baseball teams is larger than in cities without major league baseball teams. The most plausible explanation for this observed association is [ANS] A. the presence of a major league baseball team causes the number of divorces to rise (perhaps husbands are spending too much time at the ballpark). B. the observed association is purely coincidental. It is implausible to believe the observed association could be anything other than accidental. C. the high number of divorces is responsible for the presence of a major league baseball teams (more single men means potentially more fans at the ballpark, making it attractive for an owner to relocate to such cities). D. the association is due to the presence of a lurking variable (major league teams tend to be in large cities with more people, hence a greater number of divorces). E. None of the above.\n(c) The owner of a chain of supermarkets notices that there is a positive correlation between the sales of beer and the sales of ice cream over the course of the previous year. Seasons when sales of beer were above average, sales of ice cream also tended to be above average. Likewise, during seasons when sales of beer were below average, sales of ice cream also tended to be below average. Which of the following would be a valid conclusion from these facts? [ANS] A. The sale of beer and ice cream may both be affected by another variable such as the outside temperature. B. A scatterplot of monthly ice cream sales versus monthly beer sales would show that a straight line describes the pattern in the plot, but it would have to be a horizontal line. C. Evidently, for a significant proportion of customers of these supermarkets, drinking beer causes a desire for ice cream or eating ice cream causes a thirst for beer. D. Sales records must be in error. There should be no association between beer and ice cream sales. E. None of the above.", "answer_v1": [ "D", "D", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "For each problem, select the best response.\n(a) The owner of a chain of supermarkets notices that there is a positive correlation between the sales of beer and the sales of ice cream over the course of the previous year. Seasons when sales of beer were above average, sales of ice cream also tended to be above average. Likewise, during seasons when sales of beer were below average, sales of ice cream also tended to be below average. Which of the following would be a valid conclusion from these facts? [ANS] A. Evidently, for a significant proportion of customers of these supermarkets, drinking beer causes a desire for ice cream or eating ice cream causes a thirst for beer. B. A scatterplot of monthly ice cream sales versus monthly beer sales would show that a straight line describes the pattern in the plot, but it would have to be a horizontal line. C. Sales records must be in error. There should be no association between beer and ice cream sales. D. The sale of beer and ice cream may both be affected by another variable such as the outside temperature. E. None of the above.\n(b) When possible, the best way to establish that an observed association is the result of a cause-and-effect relation is by means of [ANS] A. the least squares regression line. B. a well designed experiment. C. the square of the correlation coefficient. D. the correlation coefficient. E. None of the above.\n(c) A researcher observes that, on average, the number of divorces in cities with major league baseball teams is larger than in cities without major league baseball teams. The most plausible explanation for this observed association is [ANS] A. the association is due to the presence of a lurking variable (major league teams tend to be in large cities with more people, hence a greater number of divorces). B. the high number of divorces is responsible for the presence of a major league baseball teams (more single men means potentially more fans at the ballpark, making it attractive for an owner to relocate to such cities). C. the presence of a major league baseball team causes the number of divorces to rise (perhaps husbands are spending too much time at the ballpark). D. the observed association is purely coincidental. It is implausible to believe the observed association could be anything other than accidental. E. None of the above.", "answer_v2": [ "D", "B", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "For each problem, select the best response.\n(a) The owner of a chain of supermarkets notices that there is a positive correlation between the sales of beer and the sales of ice cream over the course of the previous year. Seasons when sales of beer were above average, sales of ice cream also tended to be above average. Likewise, during seasons when sales of beer were below average, sales of ice cream also tended to be below average. Which of the following would be a valid conclusion from these facts? [ANS] A. The sale of beer and ice cream may both be affected by another variable such as the outside temperature. B. Sales records must be in error. There should be no association between beer and ice cream sales. C. Evidently, for a significant proportion of customers of these supermarkets, drinking beer causes a desire for ice cream or eating ice cream causes a thirst for beer. D. A scatterplot of monthly ice cream sales versus monthly beer sales would show that a straight line describes the pattern in the plot, but it would have to be a horizontal line. E. None of the above.\n(b) When possible, the best way to establish that an observed association is the result of a cause-and-effect relation is by means of [ANS] A. the correlation coefficient. B. a well designed experiment. C. the square of the correlation coefficient. D. the least squares regression line. E. None of the above.\n(c) A researcher observes that, on average, the number of divorces in cities with major league baseball teams is larger than in cities without major league baseball teams. The most plausible explanation for this observed association is [ANS] A. the association is due to the presence of a lurking variable (major league teams tend to be in large cities with more people, hence a greater number of divorces). B. the high number of divorces is responsible for the presence of a major league baseball teams (more single men means potentially more fans at the ballpark, making it attractive for an owner to relocate to such cities). C. the observed association is purely coincidental. It is implausible to believe the observed association could be anything other than accidental. D. the presence of a major league baseball team causes the number of divorces to rise (perhaps husbands are spending too much time at the ballpark). E. None of the above.", "answer_v3": [ "A", "B", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Statistics_0350", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Correlation", "level": "2", "keywords": [ "Statistics" ], "problem_v1": "The linear correlation coefficient of a set of data points is-0.825. a) Is the slope of the regression line positive or negative? answer: [ANS]\nb) Determine the coefficient of determination. answer: [ANS]", "answer_v1": [ "NEGATIVE", "0.680625" ], "answer_type_v1": [ "MCS", "NV" ], "options_v1": [ [ "NEGATIVE", "POSITIVE" ], [] ], "problem_v2": "The linear correlation coefficient of a set of data points is-0.975. a) Is the slope of the regression line positive or negative? answer: [ANS]\nb) Determine the coefficient of determination. answer: [ANS]", "answer_v2": [ "NEGATIVE", "0.950625" ], "answer_type_v2": [ "MCS", "NV" ], "options_v2": [ [ "NEGATIVE", "POSITIVE" ], [] ], "problem_v3": "The linear correlation coefficient of a set of data points is-0.925. a) Is the slope of the regression line positive or negative? answer: [ANS]\nb) Determine the coefficient of determination. answer: [ANS]", "answer_v3": [ "NEGATIVE", "0.855625" ], "answer_type_v3": [ "MCS", "NV" ], "options_v3": [ [ "NEGATIVE", "POSITIVE" ], [] ] }, { "id": "Statistics_0351", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Correlation", "level": "2", "keywords": [ "Statistics" ], "problem_v1": "A study was conducted to detemine whether a the final grade of a student in an introductory psychology course is linearly related to his or her performance on the verbal ability test administered before college entrance. The verbal scores and final grades for $10$ students are shown in the table below.\n$\\begin{array}{ccc}\\hline Student & Verbal Score x & Final Grade y \\\\ \\hline 1 & 67 & 81 \\\\ \\hline 2 & 59 & 72 \\\\ \\hline 3 & 41 & 47 \\\\ \\hline 4 & 56 & 68 \\\\ \\hline 5 & 45 & 54 \\\\ \\hline 6 & 61 & 71 \\\\ \\hline 7 & 53 & 62 \\\\ \\hline 8 & 47 & 55 \\\\ \\hline 9 & 53 & 60 \\\\ \\hline 10 & 33 & 38 \\\\ \\hline \\end{array}$\nFind the following: a) Compute the value of SST (Total Sum of Squares) answer: [ANS]\n(b) Compute the value of SSR (Regression Sum of Squares) answer: [ANS]\n(c) Compute the value of SSE (Error Sum of Squares) answer: [ANS]\n(d) The coefficient of determination is answer: [ANS]\n(e) What percent of variation in the observed values of the response variable is explained by the regression? answer: [ANS]", "answer_v1": [ "1481.6", "1462.38100377766", "19.2189962223406", "0.987028215292697", "98.7028215292697" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "A study was conducted to detemine whether a the final grade of a student in an introductory psychology course is linearly related to his or her performance on the verbal ability test administered before college entrance. The verbal scores and final grades for $10$ students are shown in the table below.\n$\\begin{array}{ccc}\\hline Student & Verbal Score x & Final Grade y \\\\ \\hline 1 & 29 & 38 \\\\ \\hline 2 & 33 & 38 \\\\ \\hline 3 & 78 & 91 \\\\ \\hline 4 & 35 & 40 \\\\ \\hline 5 & 56 & 63 \\\\ \\hline 6 & 60 & 71 \\\\ \\hline 7 & 70 & 81 \\\\ \\hline 8 & 35 & 40 \\\\ \\hline 9 & 57 & 65 \\\\ \\hline 10 & 43 & 52 \\\\ \\hline \\end{array}$\nFind the following: a) Compute the value of SST (Total Sum of Squares) answer: [ANS]\n(b) Compute the value of SSR (Regression Sum of Squares) answer: [ANS]\n(c) Compute the value of SSE (Error Sum of Squares) answer: [ANS]\n(d) The coefficient of determination is answer: [ANS]\n(e) What percent of variation in the observed values of the response variable is explained by the regression? answer: [ANS]", "answer_v2": [ "3344.9", "3316.85301186257", "28.0469881374279", "0.99161499951047", "99.161499951047" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "A study was conducted to detemine whether a the final grade of a student in an introductory psychology course is linearly related to his or her performance on the verbal ability test administered before college entrance. The verbal scores and final grades for $10$ students are shown in the table below.\n$\\begin{array}{ccc}\\hline Student & Verbal Score x & Final Grade y \\\\ \\hline 1 & 42 & 51 \\\\ \\hline 2 & 40 & 48 \\\\ \\hline 3 & 36 & 42 \\\\ \\hline 4 & 70 & 88 \\\\ \\hline 5 & 73 & 84 \\\\ \\hline 6 & 41 & 47 \\\\ \\hline 7 & 27 & 33 \\\\ \\hline 8 & 79 & 97 \\\\ \\hline 9 & 59 & 66 \\\\ \\hline 10 & 41 & 50 \\\\ \\hline \\end{array}$\nFind the following: a) Compute the value of SST (Total Sum of Squares) answer: [ANS]\n(b) Compute the value of SSR (Regression Sum of Squares) answer: [ANS]\n(c) Compute the value of SSE (Error Sum of Squares) answer: [ANS]\n(d) The coefficient of determination is answer: [ANS]\n(e) What percent of variation in the observed values of the response variable is explained by the regression? answer: [ANS]", "answer_v3": [ "4308.4", "4245.55824008841", "62.8417599115874", "0.985414130556219", "98.5414130556219" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Statistics_0352", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Correlation", "level": "2", "keywords": [ "Statistics" ], "problem_v1": "Is the number of games won by a major league baseball team in a season related to the team batting average? The table below shows the number of games won and the batting average of 8 teams.\n$\\begin{array}{ccc}\\hline Team & Games Won & Batting Average \\\\ \\hline 1 & 105 & 0.277 \\\\ \\hline 2 & 97 & 0.281 \\\\ \\hline 3 & 78 & 0.269 \\\\ \\hline 4 & 94 & 0.276 \\\\ \\hline 5 & 82 & 0.274 \\\\ \\hline 6 & 99 & 0.267 \\\\ \\hline 7 & 90 & 0.272 \\\\ \\hline 8 & 84 & 0.272 \\\\ \\hline \\end{array}$\nUsing games won as the independent variable $x$, do the following:\n(a) Compute the value of SST (Total Sum of Squares) answer: [ANS]\n(b) Compute the value of SSR (Regression Sum of Squares) answer: [ANS]\n(c) Compute the value of SSE (Error Sum of Squares) answer: [ANS]\n(d) The coefficient of determination is answer: [ANS]\n(e) What percent of variation in the observed values of the response variable is explained by the regression? answer: [ANS]", "answer_v1": [ "0.000142000000000087", "2.20545567265954E-05", "0.000119945443273491", "0.155313779764661", "15.5313779764661" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Is the number of games won by a major league baseball team in a season related to the team batting average? The table below shows the number of games won and the batting average of 8 teams.\n$\\begin{array}{ccc}\\hline Team & Games Won & Batting Average \\\\ \\hline 1 & 65 & 0.287 \\\\ \\hline 2 & 69 & 0.269 \\\\ \\hline 3 & 117 & 0.268 \\\\ \\hline 4 & 71 & 0.269 \\\\ \\hline 5 & 94 & 0.261 \\\\ \\hline 6 & 99 & 0.272 \\\\ \\hline 7 & 109 & 0.264 \\\\ \\hline 8 & 71 & 0.267 \\\\ \\hline \\end{array}$\nUsing games won as the independent variable $x$, do the following:\n(a) Compute the value of SST (Total Sum of Squares) answer: [ANS]\n(b) Compute the value of SSR (Regression Sum of Squares) answer: [ANS]\n(c) Compute the value of SSE (Error Sum of Squares) answer: [ANS]\n(d) The coefficient of determination is answer: [ANS]\n(e) What percent of variation in the observed values of the response variable is explained by the regression? answer: [ANS]", "answer_v2": [ "0.000423875000000073", "9.45575458468131E-05", "0.00032931745415326", "0.223078845996571", "22.3078845996571" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Is the number of games won by a major league baseball team in a season related to the team batting average? The table below shows the number of games won and the batting average of 8 teams.\n$\\begin{array}{ccc}\\hline Team & Games Won & Batting Average \\\\ \\hline 1 & 79 & 0.277 \\\\ \\hline 2 & 76 & 0.276 \\\\ \\hline 3 & 72 & 0.269 \\\\ \\hline 4 & 109 & 0.287 \\\\ \\hline 5 & 113 & 0.265 \\\\ \\hline 6 & 77 & 0.266 \\\\ \\hline 7 & 62 & 0.276 \\\\ \\hline 8 & 119 & 0.283 \\\\ \\hline \\end{array}$\nUsing games won as the independent variable $x$, do the following:\n(a) Compute the value of SST (Total Sum of Squares) answer: [ANS]\n(b) Compute the value of SSR (Regression Sum of Squares) answer: [ANS]\n(c) Compute the value of SSE (Error Sum of Squares) answer: [ANS]\n(d) The coefficient of determination is answer: [ANS]\n(e) What percent of variation in the observed values of the response variable is explained by the regression? answer: [ANS]", "answer_v3": [ "0.000430875000000053", "4.58894148915981E-05", "0.000384985585108455", "0.106502848602478", "10.6502848602478" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Statistics_0353", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Correlation", "level": "4", "keywords": [ "statistics", "variance" ], "problem_v1": "Consider $n=5$ pairs $(x_1,y_1),\\ldots,(x_n,y_n)$. Let ${\\overline x}=n^{-1}\\sum_{j=1}^n x_j$ and ${\\overline y}=n^{-1}\\sum_{j=1}^n y_j$ be the sample means of the $x$ and $y$ variables. Let $s_x$ and $s_y$ be the corresponding standard deviations. Let $s_{xy}$ and $r_{xy}$ be the sample covariance and sample correlation respectively. Suppose ${\\overline x}=5.8$, ${\\overline y}=7.8$, $s_x=0.837$, $s_y=1.924$, $s_{xy}=1.45$. Part a) What is the sample correlation of the $(x_i,y_i)$? [ANS]\nFor parts (b) to (e), consider a linearly transformed variable $x^*_i=3+3\\cdot x_i$ for $i=1,\\ldots,n$. Part b) What is ${\\overline x}^*$? [ANS]\nPart c) What is $s_{x^*}$? [ANS]\nPart d) What is the sample covariance of the $(x_i^*,y_i)$? [ANS]\nPart e) What is the sample correlation of the $(x_i^*,y_i)$? [ANS]\nPart f) Consider a quadratic transformed variable $x^{**}_i=3+3\\cdot x_i+x_i^2$ for $i=1,\\ldots,n$. As above $n=5$, ${\\overline x}=5.8$ and $s_x=0.837$. What is ${\\overline x}^{**}$, the sample mean of the transformed variable? [ANS]", "answer_v1": [ "0.900404126210578", "20.4", "2.511", "4.35", "0.900404126210578", "54.6004552" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Consider $n=5$ pairs $(x_1,y_1),\\ldots,(x_n,y_n)$. Let ${\\overline x}=n^{-1}\\sum_{j=1}^n x_j$ and ${\\overline y}=n^{-1}\\sum_{j=1}^n y_j$ be the sample means of the $x$ and $y$ variables. Let $s_x$ and $s_y$ be the corresponding standard deviations. Let $s_{xy}$ and $r_{xy}$ be the sample covariance and sample correlation respectively. Suppose ${\\overline x}=5$, ${\\overline y}=6.6$, $s_x=3.162$, $s_y=4.278$, $s_{xy}=13.5$. Part a) What is the sample correlation of the $(x_i,y_i)$? [ANS]\nFor parts (b) to (e), consider a linearly transformed variable $x^*_i=2+3\\cdot x_i$ for $i=1,\\ldots,n$. Part b) What is ${\\overline x}^*$? [ANS]\nPart c) What is $s_{x^*}$? [ANS]\nPart d) What is the sample covariance of the $(x_i^*,y_i)$? [ANS]\nPart e) What is the sample correlation of the $(x_i^*,y_i)$? [ANS]\nPart f) Consider a quadratic transformed variable $x^{**}_i=2+3\\cdot x_i+x_i^2$ for $i=1,\\ldots,n$. As above $n=5$, ${\\overline x}=5$ and $s_x=3.162$. What is ${\\overline x}^{**}$, the sample mean of the transformed variable? [ANS]", "answer_v2": [ "0.998001335991122", "17", "9.486", "40.5", "0.998001335991122", "49.9985952" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Consider $n=5$ pairs $(x_1,y_1),\\ldots,(x_n,y_n)$. Let ${\\overline x}=n^{-1}\\sum_{j=1}^n x_j$ and ${\\overline y}=n^{-1}\\sum_{j=1}^n y_j$ be the sample means of the $x$ and $y$ variables. Let $s_x$ and $s_y$ be the corresponding standard deviations. Let $s_{xy}$ and $r_{xy}$ be the sample covariance and sample correlation respectively. Suppose ${\\overline x}=4.6$, ${\\overline y}=7.4$, $s_x=1.673$, $s_y=3.286$, $s_{xy}=5.2$. Part a) What is the sample correlation of the $(x_i,y_i)$? [ANS]\nFor parts (b) to (e), consider a linearly transformed variable $x^*_i=2+2\\cdot x_i$ for $i=1,\\ldots,n$. Part b) What is ${\\overline x}^*$? [ANS]\nPart c) What is $s_{x^*}$? [ANS]\nPart d) What is the sample covariance of the $(x_i^*,y_i)$? [ANS]\nPart e) What is the sample correlation of the $(x_i^*,y_i)$? [ANS]\nPart f) Consider a quadratic transformed variable $x^{**}_i=2+2\\cdot x_i+x_i^2$ for $i=1,\\ldots,n$. As above $n=5$, ${\\overline x}=4.6$ and $s_x=1.673$. What is ${\\overline x}^{**}$, the sample mean of the transformed variable? [ANS]", "answer_v3": [ "0.945888278225033", "11.2", "3.346", "10.4", "0.945888278225033", "34.5991432" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Statistics_0354", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Correlation", "level": "3", "keywords": [ "statistics", "Two Populations", "Inference" ], "problem_v1": "If the coefficient of correlation is 0.90, the percentage of the variation in the dependent variable y that is explained by the variation in the independent variable x is: [ANS] A. 81\\% B. 0.81\\% C. 0.90\\% D. 90\\%\nIf the coefficient of determination is 0.975, then the slope of the regression line: [ANS] A. could be either positive or negative B. must be negative C. must be positive D. none of the above answers is correct", "answer_v1": [ "A", "A" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "If the coefficient of correlation is 0.90, the percentage of the variation in the dependent variable y that is explained by the variation in the independent variable x is: [ANS] A. 0.90\\% B. 0.81\\% C. 90\\% D. 81\\%\nWhich value of the coefficient of correlation r indicates a stronger correlation than 0.65? [ANS] A.-0.75 B.-0.45 C. 0.55 D. 0.60", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "If the coefficient of correlation is 0.90, the percentage of the variation in the dependent variable y that is explained by the variation in the independent variable x is: [ANS] A. 0.90\\% B. 81\\% C. 0.81\\% D. 90\\%\nCorrelation analysis is used to determine: [ANS] A. the strength of the relationship between x and y B. the predicted value of y for a given value of x C. the least squares estimates of the regression parameters D. the coefficient of determination", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0355", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Correlation", "level": "2", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "Suppose that for a given least-squares regression, the sum of squares for error is 70 and the sum of squares for regression is 95. Find the coefficient of determination. Coefficient of Determination=[ANS]", "answer_v1": [ "0.575757575757576" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that for a given least-squares regression, the sum of squares for error is 120 and the sum of squares for regression is 95. Find the coefficient of determination. Coefficient of Determination=[ANS]", "answer_v2": [ "0.441860465116279" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that for a given least-squares regression, the sum of squares for error is 60 and the sum of squares for regression is 105. Find the coefficient of determination. Coefficient of Determination=[ANS]", "answer_v3": [ "0.636363636363636" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0356", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Correlation", "level": "2", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "Suppose that the line $\\hat{y}=3+2x$ is fitted to the data points (-1,2), (2,7), and (5,13). Determine the sum of the squared residuals. Sum of the Squared Residuals=[ANS]", "answer_v1": [ "1" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that the line $\\hat{y}=5+2x$ is fitted to the data points (-3,1), (3,9), and (4,13). Determine the sum of the squared residuals. Sum of the Squared Residuals=[ANS]", "answer_v2": [ "8" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that the line $\\hat{y}=5+2x$ is fitted to the data points (-3,1), (2,6), and (4,14). Determine the sum of the squared residuals. Sum of the Squared Residuals=[ANS]", "answer_v3": [ "14" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0357", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Correlation", "level": "2", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "Suppose that a regression yields the following sum of squares: $\\Sigma(y_i-\\bar{y})^2=360, \\quad \\Sigma(y_i-\\hat{y}_i)^2=90, \\quad \\Sigma(\\hat{y}_i-\\bar{y})^2=270$ Then the percentage of the variation in y that is explained by the variation in x is: Answer=[ANS] \\%", "answer_v1": [ "75" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that a regression yields the following sum of squares: $\\Sigma(y_i-\\bar{y})^2=200, \\quad \\Sigma(y_i-\\hat{y}_i)^2=110, \\quad \\Sigma(\\hat{y}_i-\\bar{y})^2=90$ Then the percentage of the variation in y that is explained by the variation in x is: Answer=[ANS] \\%", "answer_v2": [ "45" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that a regression yields the following sum of squares: $\\Sigma(y_i-\\bar{y})^2=260, \\quad \\Sigma(y_i-\\hat{y}_i)^2=90, \\quad \\Sigma(\\hat{y}_i-\\bar{y})^2=170$ Then the percentage of the variation in y that is explained by the variation in x is: Answer=[ANS] \\%", "answer_v3": [ "65.3846153846154" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0358", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Correlation", "level": "2", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "Find the standard error of the estimate and the coefficient of determination for the least-squares regression line $\\hat{y}=b_0+b_1x$ through the points (-1,0), (2,8), (5,14), (9,19), (10,25). Standard Error of the Estimate=[ANS]\nCoefficient of Determination=[ANS]", "answer_v1": [ "1.8716572269499", "0.971960239259388" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the standard error of the estimate and the coefficient of determination for the least-squares regression line $\\hat{y}=b_0+b_1x$ through the points (-3,0), (3,6), (4,13), (8,19), (12,23). Standard Error of the Estimate=[ANS]\nCoefficient of Determination=[ANS]", "answer_v2": [ "2.49836960506069", "0.946620433151445" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the standard error of the estimate and the coefficient of determination for the least-squares regression line $\\hat{y}=b_0+b_1x$ through the points (-3,1), (2,9), (4,15), (8,20), (9,24). Standard Error of the Estimate=[ANS]\nCoefficient of Determination=[ANS]", "answer_v3": [ "1.29126909919074", "0.984878695104067" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0359", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Correlation", "level": "2", "keywords": [ "statistics", "numerical", "descriptive statistics", "covariance", "coefficient of correlation" ], "problem_v1": "A sample of four 35-year-old males is asked about the average number of hours per week that he exercises, and is also given a blood cholesterol test. The data is recorded in the order pairs given below, in the form (Hours Exercising, Cholesterol Level):\n(2.8,218), \\quad (3.6,205), \\quad (4.6,198), \\quad (6.7,187) Find the covariance and coefficient of correlation for this sample. Covariance=[ANS]\nCoefficient of Correlation=[ANS]", "answer_v1": [ "-21.0999999999999", "-0.963646143179892" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A sample of four 35-year-old males is asked about the average number of hours per week that he exercises, and is also given a blood cholesterol test. The data is recorded in the order pairs given below, in the form (Hours Exercising, Cholesterol Level):\n(2,232), \\quad (4,205), \\quad (4.1,191), \\quad (6.3,181) Find the covariance and coefficient of correlation for this sample. Covariance=[ANS]\nCoefficient of Correlation=[ANS]", "answer_v2": [ "-36.5", "-0.938294557633103" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A sample of four 35-year-old males is asked about the average number of hours per week that he exercises, and is also given a blood cholesterol test. The data is recorded in the order pairs given below, in the form (Hours Exercising, Cholesterol Level):\n(2.3,216), \\quad (3.6,205), \\quad (4.3,202), \\quad (6.6,194) Find the covariance and coefficient of correlation for this sample. Covariance=[ANS]\nCoefficient of Correlation=[ANS]", "answer_v3": [ "-15.8666666666668", "-0.967043320558631" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0360", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Regression", "level": "4", "keywords": [ "Correlation", "Test Statistic", "Critical Value", "Confidence", "Interval" ], "problem_v1": "The amounts of 6 restaurant bills and the corresponding amounts of the tips are given in the below. \\begin{array}{c|cccccc} \\mbox{Bill} & 52.44 & 64.30 & 106.27 & 70.29 & 49.72 & 88.01 \\cr \\hline \\mbox{Tip} & 7.00 & 7.70 & 16.00 & 10.00 & 5.28 & 10.00 \\cr \\end{array} Use a 0.05 confidence level to find the following: The test statistic $r=$ [ANS]\nIs there a significant correlation? [ANS] A. No B. Yes\nThe regression equation is $\\hat{y}=$ [ANS] $+$ [ANS] $x.$ If the amount of the bill is \\$ $75,$ the best prediction for the amount of the tip is [ANS]", "answer_v1": [ "0.949453564146192", "B", "-2.37716249001277", "0.162965396701103", "9.84524226256998" ], "answer_type_v1": [ "NV", "MCS", "NV", "NV", "NV" ], "options_v1": [ [], [ "A", "B" ], [], [], [] ], "problem_v2": "The amounts of 6 restaurant bills and the corresponding amounts of the tips are given in the below. \\begin{array}{c|cccccc} \\mbox{Bill} & 32.98 & 43.58 & 88.01 & 64.30 & 70.29 & 97.34 \\cr \\hline \\mbox{Tip} & 4.50 & 5.50 & 10.00 & 7.70 & 10.00 & 16.00 \\cr \\end{array} Use a 0.05 confidence level to find the following: The test statistic $r=$ [ANS]\nIs there a significant correlation? [ANS] A. No B. Yes\nThe regression equation is $\\hat{y}=$ [ANS] $+$ [ANS] $x.$ If the amount of the bill is \\$ $60,$ the best prediction for the amount of the tip is [ANS]", "answer_v2": [ "0.927210520744121", "B", "-1.23802574251118", "0.154169368108618", "8.0121363440059" ], "answer_type_v2": [ "NV", "MCS", "NV", "NV", "NV" ], "options_v2": [ [], [ "A", "B" ], [], [], [] ], "problem_v3": "The amounts of 6 restaurant bills and the corresponding amounts of the tips are given in the below. \\begin{array}{c|cccccc} \\mbox{Bill} & 88.01 & 106.27 & 49.72 & 52.44 & 97.34 & 64.30 \\cr \\hline \\mbox{Tip} & 10.00 & 16.00 & 5.28 & 7.00 & 16.00 & 7.70 \\cr \\end{array} Use a 0.05 confidence level to find the following: The test statistic $r=$ [ANS]\nIs there a significant correlation? [ANS] A. No B. Yes\nThe regression equation is $\\hat{y}=$ [ANS] $+$ [ANS] $x.$ If the amount of the bill is \\$ $95,$ the best prediction for the amount of the tip is [ANS]", "answer_v3": [ "0.948849153660483", "B", "-3.64959992263529", "0.183106879880833", "13.7455536660438" ], "answer_type_v3": [ "NV", "MCS", "NV", "NV", "NV" ], "options_v3": [ [], [ "A", "B" ], [], [], [] ] }, { "id": "Statistics_0361", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Regression", "level": "2", "keywords": [ "statistic", "regression" ], "problem_v1": "For each problem, select the best response.\n(a) Measurements on young children in Mumbai, India, found this least-squares line for predicting height $y$ from armspan $x$:\n$\\hat{y}=6.4+0.93x$ All measurements are in centimeters (cm). How much on the average does height increase for each additional centimeter of armspan? [ANS] A. 6.4 cm B. 0.93 cm C. 7.33 cm D. 0.64 cm E. None of the above.\n(b) Smokers don't live as long (on the average) as nonsmokers, and heavy smokers don't live as long as light smokers. You regress the age at death of a group of male smokers on the number of packs per day they smoked. The slope of your regression line [ANS] A. must be between-1 and 1. B. will be less than zero. C. will be greater than zero. D. can't tell without seeing the data.\n(c) Fred keeps his savings in his mattress. He began with \\$500 from his mother and adds \\$125 each year. His total savings $y$ after $x$ years are given by the equation [ANS] A. $y=500-125x$ B. $y=500+125x$ C. $y=125+500x$ D. $y=125+x$ E. None of the above.", "answer_v1": [ "B", "B", "B" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "For each problem, select the best response.\n(a) The points on a scatterplot lie close to the line whose equation is $y=4x-9$. The slope of the line is [ANS] A. 4 B. 9 C. 13 D.-4 E. None of the above.\n(b) Measurements on young children in Mumbai, India, found this least-squares line for predicting height $y$ from armspan $x$:\n$\\hat{y}=6.4+0.93x$ All measurements are in centimeters (cm). How much on the average does height increase for each additional centimeter of armspan? [ANS] A. 0.64 cm B. 7.33 cm C. 0.93 cm D. 6.4 cm E. None of the above.\n(c) Smokers don't live as long (on the average) as nonsmokers, and heavy smokers don't live as long as light smokers. You regress the age at death of a group of male smokers on the number of packs per day they smoked. The slope of your regression line [ANS] A. will be less than zero. B. must be between-1 and 1. C. will be greater than zero. D. can't tell without seeing the data.", "answer_v2": [ "A", "C", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D" ] ], "problem_v3": "For each problem, select the best response.\n(a) Smokers don't live as long (on the average) as nonsmokers, and heavy smokers don't live as long as light smokers. You regress the age at death of a group of male smokers on the number of packs per day they smoked. The slope of your regression line [ANS] A. must be between-1 and 1. B. will be greater than zero. C. will be less than zero. D. can't tell without seeing the data.\n(b) Fred keeps his savings in his mattress. He began with \\$500 from his mother and adds \\$175 each year. His total savings $y$ after $x$ years are given by the equation [ANS] A. $y=500+175x$ B. $y=175+x$ C. $y=175+500x$ D. $y=500-175x$ E. None of the above.\n(c) The points on a scatterplot lie close to the line whose equation is $y=5x-2$. The slope of the line is [ANS] A. 5 B.-5 C. 7 D. 2 E. None of the above.", "answer_v3": [ "C", "A", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Statistics_0362", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Regression", "level": "2", "keywords": [ "statistic", "regression" ], "problem_v1": "We have data on the lean body mass and resting metabolic rate for 12 women who are subjects in a study of dieting. Lean body mass, given in kilograms, is a person's weight leaving out all fat. Metabolic rate, in calories burned per 24 hours, is the rate at which the body consumes energy.\n$\\begin{array}{ccccccccccccc}\\hline Mass & 50.3 & 47.8 & 41.7 & 46.8 & 43 & 48.4 & 45.6 & 43.5 & 45.8 & 39 & 45.7 & 44.7 \\\\ \\hline Rate & 1310 & 1410 & 1130 & 1300 & 1250 & 1090 & 1200 & 1200 & 990 & 1170 & 1450 & 1120 \\\\ \\hline \\end{array}$\nFind the least-squares regression line for predicting metabolic rate from body mass. ANSWER: $\\hat{y}=$ [ANS]", "answer_v1": [ "668.836+12.1593*x" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "We have data on the lean body mass and resting metabolic rate for 12 women who are subjects in a study of dieting. Lean body mass, given in kilograms, is a person's weight leaving out all fat. Metabolic rate, in calories burned per 24 hours, is the rate at which the body consumes energy.\n$\\begin{array}{ccccccccccccc}\\hline Mass & 37.5 & 38.8 & 54 & 39.4 & 46.7 & 48.2 & 51.4 & 39.6 & 46.9 & 42.1 & 38.1 & 37.8 \\\\ \\hline Rate & 1560 & 1130 & 1120 & 1130 & 940 & 1210 & 1030 & 1090 & 1030 & 1290 & 980 & 1120 \\\\ \\hline \\end{array}$\nFind the least-squares regression line for predicting metabolic rate from body mass. ANSWER: $\\hat{y}=$ [ANS]", "answer_v2": [ "1515.49+(-8.75278)*x" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "We have data on the lean body mass and resting metabolic rate for 12 women who are subjects in a study of dieting. Lean body mass, given in kilograms, is a person's weight leaving out all fat. Metabolic rate, in calories burned per 24 hours, is the rate at which the body consumes energy.\n$\\begin{array}{ccccccccccccc}\\hline Mass & 41.9 & 41.3 & 39.9 & 51.4 & 52.6 & 41.5 & 36.8 & 54.7 & 47.7 & 41.5 & 49.4 & 44 \\\\ \\hline Rate & 1320 & 1280 & 1140 & 1540 & 1040 & 1070 & 1300 & 1460 & 970 & 1350 & 1270 & 1550 \\\\ \\hline \\end{array}$\nFind the least-squares regression line for predicting metabolic rate from body mass. ANSWER: $\\hat{y}=$ [ANS]", "answer_v3": [ "1102.9+3.78703*x" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Statistics_0363", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Regression", "level": "2", "keywords": [ "statistic", "regression" ], "problem_v1": "Because elderly people may have difficulty standing to have their height measured, a study looked at the relationship between overall height and height to the knee. Here are data (in centimeters) for five elderly men:\n$\\begin{array}{cccccc}\\hline Knee Height x & 57.3 & 45.9 & 42.5 & 47.3 & 54.1 \\\\ \\hline Height y & 190.7 & 153.4 & 147 & 163.2 & 171.8 \\\\ \\hline \\end{array}$\nWhat is the equation of the least-squares regression line for predicting height from knee height? ANSWER: $\\hat{y}=$ [ANS]", "answer_v1": [ "31.1633+2.7126*x" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Because elderly people may have difficulty standing to have their height measured, a study looked at the relationship between overall height and height to the knee. Here are data (in centimeters) for five elderly men:\n$\\begin{array}{cccccc}\\hline Knee Height x & 56.2 & 47.5 & 40.8 & 45.1 & 58.4 \\\\ \\hline Height y & 190.6 & 151.4 & 145.3 & 164.4 & 168.5 \\\\ \\hline \\end{array}$\nWhat is the equation of the least-squares regression line for predicting height from knee height? ANSWER: $\\hat{y}=$ [ANS]", "answer_v2": [ "74.1375+1.81255*x" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Because elderly people may have difficulty standing to have their height measured, a study looked at the relationship between overall height and height to the knee. Here are data (in centimeters) for five elderly men:\n$\\begin{array}{cccccc}\\hline Knee Height x & 56.6 & 46 & 41.3 & 46.3 & 53.4 \\\\ \\hline Height y & 190.7 & 154.7 & 149.2 & 166.2 & 169.5 \\\\ \\hline \\end{array}$\nWhat is the equation of the least-squares regression line for predicting height from knee height? ANSWER: $\\hat{y}=$ [ANS]", "answer_v3": [ "49.0089+2.40253*x" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Statistics_0364", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Regression", "level": "2", "keywords": [ "linear equation", "slope", "intercept" ], "problem_v1": "For the equation $y=6.5x-6$, a. the $y$-intercept is [ANS], and the slope is [ANS]. b. the line [ANS] A. slopes upward B. is horizontal C. slopes downward D. none of the above\nc. use two points to graph the equation.", "answer_v1": [ "-6", "6.5", "A" ], "answer_type_v1": [ "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "A", "B", "C", "D" ] ], "problem_v2": "For the equation $y=0.5x-9$, a. the $y$-intercept is [ANS], and the slope is [ANS]. b. the line [ANS] A. slopes downward B. slopes upward C. is horizontal D. none of the above\nc. use two points to graph the equation.", "answer_v2": [ "-9", "0.5", "B" ], "answer_type_v2": [ "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "A", "B", "C", "D" ] ], "problem_v3": "For the equation $y=2.5x-6$, a. the $y$-intercept is [ANS], and the slope is [ANS]. b. the line [ANS] A. slopes upward B. slopes downward C. is horizontal D. none of the above\nc. use two points to graph the equation.", "answer_v3": [ "-6", "2.5", "A" ], "answer_type_v3": [ "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0365", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Regression", "level": "1", "keywords": [ "Statistics" ], "problem_v1": "The regression line is the straight line that bests fits a set of data points according to what? [ANS] A. Most accurate regression criterion B. Greatest-squares criterion C. Least-squares criterion. D. None of the above", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "The regression line is the straight line that bests fits a set of data points according to what? [ANS] A. Least-squares criterion. B. Greatest-squares criterion C. Most accurate regression criterion D. None of the above", "answer_v2": [ "A" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "The regression line is the straight line that bests fits a set of data points according to what? [ANS] A. Most accurate regression criterion B. Least-squares criterion. C. Greatest-squares criterion D. None of the above", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0366", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Regression", "level": "2", "keywords": [ "Statistics" ], "problem_v1": "Before determining a regression line, it is important to do what? [ANS] A. Plot the data to make sure it does not appear linear. B. Make sure that every x value has exactly one corresponding y value C. Plot the data to make sure it appears somewhat linear. D. None of the above", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "Before determining a regression line, it is important to do what? [ANS] A. Plot the data to make sure it appears somewhat linear. B. Make sure that every x value has exactly one corresponding y value C. Plot the data to make sure it does not appear linear. D. None of the above", "answer_v2": [ "A" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "Before determining a regression line, it is important to do what? [ANS] A. Plot the data to make sure it does not appear linear. B. Plot the data to make sure it appears somewhat linear. C. Make sure that every x value has exactly one corresponding y value D. None of the above", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0367", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Regression", "level": "2", "keywords": [ "linear equation", "slope", "intercept" ], "problem_v1": "Consider the linear equation $y=b_0+b_1x$. a. In the equation, $b_0$ is [ANS] A. the $y$-intercept B. the dependent variable C. the slope D. the independent variable\nb. In the equation, $b_1$ is [ANS] A. the slope B. the dependent variable C. the independent variable D. the $y$-intercept\nc. Give the geometric interpretation of $b_0$. It indicates [ANS] A. how much the $x$-value on the straight line changes when the $y$-value increases by unit B. the $x$-value where the straight-line graph of the linear equation intersects the $x$-axis C. the $y$-value where the straight-line graph of the linear equation intersects the $y$-axis D. how much the $y$-value on the straight line changes when the $x$-value increases by unit\nd. Give the geometric interpretation of $b_1$. It indicates [ANS] A. how much the $y$-value on the straight line changes when the $x$-value increases by unit B. how much the $x$-value on the straight line changes when the $y$-value increases by unit C. the $y$-value where the straight-line graph of the linear equation intersects the $y$-axis D. the $x$-value where the straight-line graph of the linear equation intersects the $x$-axis", "answer_v1": [ "A", "A", "C", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Consider the linear equation $y=b_0+b_1x$. a. In the equation, $b_0$ is [ANS] A. the $y$-intercept B. the dependent variable C. the slope D. the independent variable\nb. In the equation, $b_1$ is [ANS] A. the independent variable B. the dependent variable C. the $y$-intercept D. the slope\nc. Give the geometric interpretation of $b_0$. It indicates [ANS] A. how much the $x$-value on the straight line changes when the $y$-value increases by unit B. the $x$-value where the straight-line graph of the linear equation intersects the $x$-axis C. how much the $y$-value on the straight line changes when the $x$-value increases by unit D. the $y$-value where the straight-line graph of the linear equation intersects the $y$-axis\nd. Give the geometric interpretation of $b_1$. It indicates [ANS] A. the $y$-value where the straight-line graph of the linear equation intersects the $y$-axis B. how much the $x$-value on the straight line changes when the $y$-value increases by unit C. the $x$-value where the straight-line graph of the linear equation intersects the $x$-axis D. how much the $y$-value on the straight line changes when the $x$-value increases by unit", "answer_v2": [ "A", "D", "D", "D" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Consider the linear equation $y=b_0+b_1x$. a. In the equation, $b_0$ is [ANS] A. the $y$-intercept B. the slope C. the dependent variable D. the independent variable\nb. In the equation, $b_1$ is [ANS] A. the independent variable B. the slope C. the dependent variable D. the $y$-intercept\nc. Give the geometric interpretation of $b_0$. It indicates [ANS] A. how much the $x$-value on the straight line changes when the $y$-value increases by unit B. the $y$-value where the straight-line graph of the linear equation intersects the $y$-axis C. how much the $y$-value on the straight line changes when the $x$-value increases by unit D. the $x$-value where the straight-line graph of the linear equation intersects the $x$-axis\nd. Give the geometric interpretation of $b_1$. It indicates [ANS] A. how much the $y$-value on the straight line changes when the $x$-value increases by unit B. the $x$-value where the straight-line graph of the linear equation intersects the $x$-axis C. the $y$-value where the straight-line graph of the linear equation intersects the $y$-axis D. how much the $x$-value on the straight line changes when the $y$-value increases by unit", "answer_v3": [ "A", "B", "B", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0368", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Regression", "level": "4", "keywords": [ "Confidence Intervals" ], "problem_v1": "Calculate the least-squares regression line for the following set of data points: (-1,3), (1,6), (1,6), (4,5), (4,7), (4,9), (6,7). Answer: $y=$ [ANS]", "answer_v1": [ "4.74193548387097 + 0.516129032258065 * x" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Calculate the least-squares regression line for the following set of data points: (-1,3), (0,0), (1,4), (0,7), (4,10). Answer: $y=$ [ANS]", "answer_v2": [ "3.62162162162162 + 1.47297297297297 * x" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Calculate the least-squares regression line for the following set of data points: (0,-1), (0,4), (5,6), (1,3), (2,2). Answer: $y=$ [ANS]", "answer_v3": [ "1.44186046511628 + 0.848837209302326 * x" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Statistics_0369", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Regression", "level": "4", "keywords": [ "statistics", "regression" ], "problem_v1": "For simple linear regression, with data $x_i,y_i$, $i=1,\\ldots,n$, and model $Y_i=\\beta_0+\\beta_1x_i+\\epsilon_i$, the multiple correlation coefficient is define as $R^2=1-{SS(Res)\\over SS(Total)}$ An identity for simple linear regression is that $R^2$ is the same as square of the sample correlation of the $x$ and $y$ variables.\nPart a) Which of the following is a correct statement used in the proof of the claim.? By choosing the items in a correct order, you will derive the claimed result. Note that an identity for the residual SD (previous homework question) is being used. Choose all appropriate items: [ANS] A. ${\\hat \\sigma}^2=(n-1)(1-r_{xy}^2)s_y^2/(n-2)$ B. $SS(Total)=s_y^2$ C. $SS(Total)=(n-1)s_y^2$ D. $SS(Total)=(n-2)s_y^2$ E. $R^2=1-(1-r_{xy}^2)$ F. $SS(Res)=(n-1)(1-r_{xy}^2)s_y^2$ G. $SS(Res)=(n-1){\\hat \\sigma}^2$ H. $SS(Res)=(n-2){\\hat \\sigma}^2$ I. None of the above", "answer_v1": [ "ACEFH" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ] ], "problem_v2": "For simple linear regression, with data $x_i,y_i$, $i=1,\\ldots,n$, and model $Y_i=\\beta_0+\\beta_1x_i+\\epsilon_i$, the multiple correlation coefficient is define as $R^2=1-{SS(Res)\\over SS(Total)}$ An identity for simple linear regression is that $R^2$ is the same as square of the sample correlation of the $x$ and $y$ variables.\nPart a) Which of the following is a correct statement used in the proof of the claim.? By choosing the items in a correct order, you will derive the claimed result. Note that an identity for the residual SD (previous homework question) is being used. Choose all appropriate items: [ANS] A. $SS(Res)=(n-1){\\hat \\sigma}^2$ B. $SS(Total)=s_y^2$ C. $R^2=1-(1-r_{xy}^2)$ D. $SS(Total)=(n-1)s_y^2$ E. $SS(Res)=(n-1)(1-r_{xy}^2)s_y^2$ F. $SS(Res)=(n-2){\\hat \\sigma}^2$ G. ${\\hat \\sigma}^2=(n-1)(1-r_{xy}^2)s_y^2/(n-2)$ H. $SS(Total)=(n-2)s_y^2$ I. None of the above", "answer_v2": [ "CDEFG" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ] ], "problem_v3": "For simple linear regression, with data $x_i,y_i$, $i=1,\\ldots,n$, and model $Y_i=\\beta_0+\\beta_1x_i+\\epsilon_i$, the multiple correlation coefficient is define as $R^2=1-{SS(Res)\\over SS(Total)}$ An identity for simple linear regression is that $R^2$ is the same as square of the sample correlation of the $x$ and $y$ variables.\nPart a) Which of the following is a correct statement used in the proof of the claim.? By choosing the items in a correct order, you will derive the claimed result. Note that an identity for the residual SD (previous homework question) is being used. Choose all appropriate items: [ANS] A. $R^2=1-(1-r_{xy}^2)$ B. ${\\hat \\sigma}^2=(n-1)(1-r_{xy}^2)s_y^2/(n-2)$ C. $SS(Total)=(n-2)s_y^2$ D. $SS(Res)=(n-1){\\hat \\sigma}^2$ E. $SS(Total)=s_y^2$ F. $SS(Res)=(n-1)(1-r_{xy}^2)s_y^2$ G. $SS(Total)=(n-1)s_y^2$ H. $SS(Res)=(n-2){\\hat \\sigma}^2$ I. None of the above", "answer_v3": [ "ABFGH" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ] ] }, { "id": "Statistics_0370", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Regression", "level": "4", "keywords": [ "statistics", "simple linear regression" ], "problem_v1": "The variables are x=height of father and y=height of corresponding son. The unit is centimetre (cm).\nFor input into R, the data vectors for heights of fathers and corresponding heights of their sons are x=c(170.2, 176.8, 183.5, 172.9, 163.8, 165, 171.2, 169.8, 158.5, 173.1, 182.3, 176, 173.5, 171.7, 181.9, 173.5, 163.9, 166.1, 173.9, 173.2, 164.9, 175.2, 164.4, 174.3, 176.1, 171.6, 183.1, 189.3, 169.4, 173.5, 171.1, 179.5, 164.3, 183.2, 172.5, 176.5, 174.9, 158.2, 178.8, 171.8, 167.1, 174.2, 161.4, 173.6, 159.9, 163.2, 183.2, 177, 174.6, 179.3, 171.4, 177.1, 181.9, 171, 168.6, 167.3, 177.3, 175.6, 184.1, 174.3, 172.8, 178.8, 186.1, 174, 171.7, 170.8, 171.9, 167.6, 162.3, 166.2, 161.1, 177.5, 177.8, 166.2, 177, 162.5, 166.6, 169.7, 178.4, 175.8, 170.4, 168.4, 173.9, 176.4, 172.8, 181.5, 180.1, 167.7, 174.1, 173.4, 166.7, 174, 163.3, 165.1, 170.3, 172.8, 181.2, 176.6, 181.3, 170.8, 169.3, 167, 175.1, 155.3, 185.6, 170.9, 169.6, 166.2, 164.4, 178, 169.8, 167, 180, 187.8, 172.5, 169.5, 170.8, 161.8, 170.2, 175.8, 185, 170.2, 163, 187.1, 170.7, 184.8, 176.5, 175, 179.7, 158.4, 172.7, 184.4, 173.6, 175, 179.6, 171.9, 173, 180.1, 184.1, 175.1, 185.4, 172.2, 166.6, 158.5, 168.1, 166.4, 156.5, 164.1, 169.6, 178.7, 163.9, 162.9, 166, 171.6, 169.5, 185.4, 181.4, 170, 176.6, 186.2, 168.6, 179.3, 173.3, 166.5, 183.2, 159, 170.7, 160.5, 178.6, 178.5, 157.8, 173.2, 165.6, 171.4, 171.2, 171, 181, 184.3, 171.6, 180) and y=c(181.8, 179.3, 166.8, 176.1, 169.1, 165.3, 170.8, 169, 179.8, 182.6, 184.8, 182.8, 162.8, 168.3, 176.1, 175.6, 167.1, 178.4, 176.6, 172.6, 169.2, 181, 172.5, 179.1, 183.6, 177.7, 186.7, 192.5, 174, 169.3, 180, 169.8, 168.6, 175.7, 172.1, 174, 178.8, 160.4, 179.9, 166.8, 180, 179.4, 163.3, 174.6, 172.1, 163.6, 177.6, 176.1, 170.1, 177.6, 171.6, 173.8, 169.6, 173.3, 176.7, 182.6, 177.9, 176, 183.2, 163.8, 181.8, 171.7, 177.1, 174.6, 176, 168.2, 171.6, 169.8, 166.3, 163.6, 164.9, 186.1, 176.1, 166.9, 167, 167.6, 174.6, 171.7, 187.1, 181.4, 174.7, 164.2, 178.8, 181.9, 172.6, 172.3, 178, 170.4, 174.3, 178.2, 168.1, 168.3, 172, 167.6, 181.1, 176.1, 178.8, 180.8, 185.2, 174.1, 182.9, 170.9, 176.4, 164.7, 185, 170.7, 171.2, 178.2, 167.8, 175.1, 178.8, 168.4, 172.2, 187.7, 167.1, 160.4, 177.8, 168.1, 176.7, 181.5, 177.5, 169.1, 171, 169.9, 168.6, 182.6, 169.9, 174.3, 186.2, 159.9, 156.8, 175, 178.2, 177.6, 174.4, 177, 169.6, 171.4, 179.6, 178.4, 186, 171.2, 172.9, 171.6, 175.8, 170, 159.5, 171.8, 174.3, 183.5, 164.7, 161.5, 164.4, 172.1, 178.7, 177.7, 181.6, 178.5, 174.9, 187.7, 173.1, 178.7, 169.3, 166.8, 177.7, 172.1, 163.9, 166.9, 182.5, 193, 165.4, 176.7, 172.9, 178, 172, 172, 170.3, 189.3, 172.9, 178.8) For the questions below, use 3 decimal places. Part a) The summary statistics (sample means of x and y, SDs of x and y, correlation coefficient): ${\\bar x}$=[ANS]\n${\\bar y}$=[ANS]\n$s_x$=[ANS]\n$s_y$=[ANS]\n${r_{xy}}$=[ANS]\nPart b) The coefficients of the least square regression line are ${\\hat \\beta}_0$=[ANS]\n${\\hat \\beta}_1$=[ANS]\nPart c) Suppose we want to get a confidence interval for the subpopulation mean of son's height $\\mu_{Y}(x)$ for fathers with a height of $x=182$. The estimated subpopulation mean is [ANS]. The standard error for this estimate is [ANS]. The t critical value for the 95\\% confidence interval is [ANS]. The lower endpoint of the 95\\% confidence interval for $\\mu_{Y}(x)$ is [ANS]. The upper endpoint of the 95\\% confidence interval for $\\mu_{Y}(x)$ is [ANS].\nPart d) For comparison, let's not use regression to get a confidence interval for the subpopulation mean. Consider instead the subset of fathers whose height is within 2.5 cm of $x=182$, including the endpoints. Based on the corresponding sons of these fathers obtain an approximate 95\\% confidence interval of the form ${\\overline y}(x) \\pm 2 \\times s_y(x)/\\sqrt{n(x)}$ The lower endpoint of the 95\\% confidence interval is [ANS]. The upper endpoint of the 95\\% confidence interval is [ANS].\nPart e) Why is the confidence interval in (c) shorter than the confidence interval in (d)? It is based on more [ANS] (put in one suitable word).", "answer_v1": [ "172.680555555555", "174.228888888889", "7.17799319321462", "6.7688366501569", "0.592589807558812", "77.7330455545177", "0.558811286099472", "179.436699624622", "0.669035334049935", "1.973381", "178.116438008079", "180.756961241164", "175.033660673377", "180.058005993289", "information" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "OE" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "The variables are x=height of father and y=height of corresponding son. The unit is centimetre (cm).\nFor input into R, the data vectors for heights of fathers and corresponding heights of their sons are x=c(179.9, 163.9, 179.1, 180.1, 157.4, 167.6, 168.6, 165, 172.6, 169.4, 170.8, 183.2, 168.2, 168.6, 163.5, 174.2, 162.5, 171.8, 162.5, 171.2, 166.9, 169.3, 173.9, 181.4, 178.5, 170.6, 169.9, 171.9, 178.5, 170.8, 172, 176.1, 174.9, 178, 168.9, 185.3, 170.8, 167, 168.4, 172.5, 161.8, 173.4, 177.4, 165.4, 174.5, 166.2, 167.9, 182.2, 183.2, 164.4, 177.6, 169.1, 170.6, 162.6, 169.1, 163.1, 174.9, 173.1, 181.2, 172.9, 167.6, 172.3, 168.4, 175.1, 170.7, 175.6, 175.4, 178.8, 175.4, 170.3, 168.5, 176.9, 162.9, 180.1, 178.9, 169.4, 169.7, 177.2, 174.1, 169, 178.1, 177.4, 172.9, 173.5, 165.7, 167.5, 158.5, 185.4, 171, 170.4, 180, 165.2, 173.5, 165.2, 173.3, 173.3, 176.4, 171.4, 172.1, 170, 172.8, 179, 171.6, 169.6, 163.5, 161.6, 160.6, 180.2, 169.6, 171.9, 173.9, 169.9, 174.5, 177, 172.8, 170.3, 171.6, 174.8, 171, 170.4, 166.5, 176.6, 166.1, 170.2, 187.9, 184.9, 169.9, 157.8, 165.7, 166.6, 182.6, 177.3, 168.1, 180.1, 171.2, 191.3, 167, 175.1, 178.4, 155.3, 176.9, 175, 170.5, 165.4, 171.2, 167.9, 174.6, 168.9, 172.2, 175.4, 178.6, 163.1, 169.4, 183.2, 168.8, 170.2, 166.4, 172.9, 162.8, 161.4, 176.4, 171.9, 173.4, 173.5, 168.4, 177.5, 168.4, 176.2, 178.8, 175.1, 160.7, 168.2, 183.4, 174, 163.4, 184.1, 170.6, 162.3, 174.9, 167.8) and y=c(186.2, 168.5, 183.2, 169.7, 171.8, 169.6, 175.9, 165.3, 169.5, 174, 174.1, 188.6, 163.7, 173.1, 163.6, 179.4, 171.7, 166.8, 167.6, 170.8, 172.7, 182.9, 166.2, 193.7, 183.5, 162.4, 176.6, 171.6, 193, 168.2, 175.2, 183.6, 174.3, 176.6, 175.2, 188.8, 173.2, 170.9, 181.9, 167.1, 168.7, 170.7, 175.9, 173.9, 177.1, 163.6, 167.9, 175.3, 175.7, 172.5, 162.4, 164.1, 176, 174.2, 177.4, 167.5, 168.9, 173.6, 178.8, 172.2, 169.8, 177.5, 167.4, 186.9, 163.9, 176, 184.1, 174.1, 166.7, 168.3, 181.4, 177.5, 161.5, 171.4, 176.9, 165.4, 171.7, 171.8, 172.8, 169.6, 173.2, 187.6, 169.2, 180.5, 169.3, 168.7, 171.6, 177.7, 178.9, 174.7, 172.2, 176.3, 181.3, 153.5, 173.6, 169.3, 181.9, 178, 175.7, 177.7, 172.4, 180, 168.9, 165.5, 163.1, 167.2, 164.3, 181.8, 171.2, 169.5, 176.6, 170.7, 168.3, 159.2, 181.8, 174.2, 186.8, 181.7, 178, 151.5, 166.8, 174.9, 178.4, 175.2, 184.1, 182.9, 171.8, 165.4, 177.3, 174.6, 181.6, 176.2, 170.6, 178, 174.3, 198, 168.4, 178.4, 187.1, 164.7, 175.2, 180.8, 177.5, 166.5, 171.9, 167.7, 168, 177.5, 171.2, 179.1, 174, 172.4, 168.1, 177.7, 179.9, 169.1, 170, 176.1, 179.7, 163.3, 165.7, 169.9, 178.2, 175.6, 164.2, 186.1, 166.9, 183, 179.9, 180.5, 171.8, 183.8, 178.7, 168.3, 170.1, 179.6, 175.9, 163.6, 178.8, 168.4) For the questions below, use 3 decimal places. Part a) The summary statistics (sample means of x and y, SDs of x and y, correlation coefficient): ${\\bar x}$=[ANS]\n${\\bar y}$=[ANS]\n$s_x$=[ANS]\n$s_y$=[ANS]\n${r_{xy}}$=[ANS]\nPart b) The coefficients of the least square regression line are ${\\hat \\beta}_0$=[ANS]\n${\\hat \\beta}_1$=[ANS]\nPart c) Suppose we want to get a confidence interval for the subpopulation mean of son's height $\\mu_{Y}(x)$ for fathers with a height of $x=181$. The estimated subpopulation mean is [ANS]. The standard error for this estimate is [ANS]. The t critical value for the 95\\% confidence interval is [ANS]. The lower endpoint of the 95\\% confidence interval for $\\mu_{Y}(x)$ is [ANS]. The upper endpoint of the 95\\% confidence interval for $\\mu_{Y}(x)$ is [ANS].\nPart d) For comparison, let's not use regression to get a confidence interval for the subpopulation mean. Consider instead the subset of fathers whose height is within 2.5 cm of $x=181$, including the endpoints. Based on the corresponding sons of these fathers obtain an approximate 95\\% confidence interval of the form ${\\overline y}(x) \\pm 2 \\times s_y(x)/\\sqrt{n(x)}$ The lower endpoint of the 95\\% confidence interval is [ANS]. The upper endpoint of the 95\\% confidence interval is [ANS].\nPart e) Why is the confidence interval in (c) shorter than the confidence interval in (d)? It is based on more [ANS] (put in one suitable word).", "answer_v2": [ "171.902222222222", "173.826666666667", "6.25695550893789", "7.19352284695943", "0.551397456174071", "64.8521856091192", "0.63393293960257", "179.594047677184", "0.793065444856907", "1.973381", "178.029027396547", "181.159067957821", "176.982638234912", "182.471907219633", "information" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "OE" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "The variables are x=height of father and y=height of corresponding son. The unit is centimetre (cm).\nFor input into R, the data vectors for heights of fathers and corresponding heights of their sons are x=c(180, 160.7, 159.9, 163.4, 168.6, 174.3, 164.2, 174.3, 170.3, 171.8, 162.8, 177.8, 179.2, 176.8, 171.3, 181, 156.5, 173.9, 172.9, 168.1, 173.8, 171.6, 171.2, 178.3, 172.1, 179, 187.4, 165.2, 155, 174.7, 177, 171.2, 165, 170.1, 175.9, 163.6, 186.1, 166.4, 173.3, 170.2, 168.5, 174.8, 170.8, 168.9, 179, 171.1, 166.2, 181.4, 166.2, 183.2, 173.9, 163.2, 169.8, 163.3, 172.1, 180.1, 172.2, 178.6, 170.8, 173.5, 174.1, 184.3, 174.5, 167.8, 177.6, 174.8, 173.9, 176.9, 169.1, 179, 171.6, 182.6, 167.9, 167.9, 172.8, 180, 167, 177.7, 173.5, 168.4, 167.7, 169, 176.2, 173.1, 184.5, 179.7, 169.3, 161.4, 175, 174.8, 164.6, 175.6, 182.4, 169.9, 172.6, 180.2, 185.6, 168.6, 183.6, 174.4, 184, 164.1, 177.4, 169.9, 158, 180.1, 178.7, 169.1, 167.6, 167.6, 177.3, 182.3, 160.5, 161.3, 171, 160.6, 172.7, 168.9, 175.1, 177.3, 181.5, 173.1, 175.4, 170.5, 175, 176.2, 186.8, 185.4, 168.9, 170.6, 167.8, 174.3, 178.4, 177.3, 168, 176.5, 176.5, 170.2, 168.6, 168.6, 165.8, 165.7, 182.1, 165.1, 164.1, 169.4, 155.3, 175.5, 159.5, 161.8, 157.8, 154.3, 179.5, 178.5, 171.9, 187.1, 170.3, 173.6, 185.4, 179.2, 183.5, 166.1, 158.5, 173.5, 166.4, 176.6, 187.8, 168.8, 167.1, 162.3, 175.2, 166.2, 163.5, 170.5, 184.2, 178.9, 171.6, 184.8, 163.4, 175.1) and y=c(178.8, 171.8, 172.1, 170.1, 173.1, 179.1, 175.4, 178.6, 181.1, 166.8, 179.7, 176.1, 167.9, 179.3, 165.8, 170.3, 159.5, 166.2, 169.2, 164.6, 170.6, 169.2, 174.3, 187, 175.7, 186.6, 182.1, 162, 162.8, 172.7, 176.1, 170.8, 169.6, 184.5, 183, 171, 177.1, 170, 173.6, 185.2, 181.4, 170.8, 174.7, 175.2, 180, 180, 174.1, 193.7, 178.2, 188.6, 167, 163.6, 178.8, 172, 176.4, 173.1, 175.2, 174, 173.2, 162.8, 174.3, 189.3, 184.4, 182.5, 181, 191, 176.6, 177.5, 164.1, 183.6, 186.8, 179.8, 167.7, 167.9, 181.8, 172.2, 170.9, 174.3, 169.3, 164.2, 171.9, 169.6, 181.4, 182.6, 179.5, 179.7, 182.9, 163.3, 177.6, 181.7, 172.8, 176, 176.8, 171.8, 179.6, 171.5, 185, 175.9, 176.2, 172.1, 184.3, 165.7, 187.6, 170.7, 163.2, 183.2, 183.5, 172.2, 170, 169.8, 176.2, 184.8, 166.9, 163.6, 178, 159, 156.8, 173, 178.4, 178.2, 172.3, 173.6, 179.1, 176.8, 180.8, 183, 183.3, 177.7, 177.5, 175.9, 168.4, 170.1, 187.1, 166.9, 181.4, 174, 169.9, 176.5, 163.3, 176.7, 169.3, 169.3, 190.5, 171.5, 165, 165.4, 164.7, 181.5, 178.4, 168.1, 165.4, 161.9, 169.8, 193, 168.1, 169.9, 174.2, 174.6, 186, 166.9, 166.8, 178.4, 171.6, 170, 179.8, 174.9, 187.7, 179.9, 180, 166.3, 181, 176.2, 163.1, 177.5, 181.1, 176.9, 177.7, 182.6, 156.4, 180.5) For the questions below, use 3 decimal places. Part a) The summary statistics (sample means of x and y, SDs of x and y, correlation coefficient): ${\\bar x}$=[ANS]\n${\\bar y}$=[ANS]\n$s_x$=[ANS]\n$s_y$=[ANS]\n${r_{xy}}$=[ANS]\nPart b) The coefficients of the least square regression line are ${\\hat \\beta}_0$=[ANS]\n${\\hat \\beta}_1$=[ANS]\nPart c) Suppose we want to get a confidence interval for the subpopulation mean of son's height $\\mu_{Y}(x)$ for fathers with a height of $x=177$. The estimated subpopulation mean is [ANS]. The standard error for this estimate is [ANS]. The t critical value for the 95\\% confidence interval is [ANS]. The lower endpoint of the 95\\% confidence interval for $\\mu_{Y}(x)$ is [ANS]. The upper endpoint of the 95\\% confidence interval for $\\mu_{Y}(x)$ is [ANS].\nPart d) For comparison, let's not use regression to get a confidence interval for the subpopulation mean. Consider instead the subset of fathers whose height is within 2.5 cm of $x=177$, including the endpoints. Based on the corresponding sons of these fathers obtain an approximate 95\\% confidence interval of the form ${\\overline y}(x) \\pm 2 \\times s_y(x)/\\sqrt{n(x)}$ The lower endpoint of the 95\\% confidence interval is [ANS]. The upper endpoint of the 95\\% confidence interval is [ANS].\nPart e) Why is the confidence interval in (c) shorter than the confidence interval in (d)? It is based on more [ANS] (put in one suitable word).", "answer_v3": [ "172.395", "174.836666666667", "7.25597634625738", "7.41856270619238", "0.550515420175211", "77.8039774829018", "0.562850948019171", "177.428595282295", "0.548704159066356", "1.973381", "176.345792920172", "178.511397644418", "176.859836713635", "180.759675481487", "information" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "OE" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Statistics_0371", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Regression", "level": "4", "keywords": [ "statistics", "variance" ], "problem_v1": "Consider $n$ numbers $x_1,\\ldots,x_n$. Let ${\\overline x}=n^{-1}\\sum_{j=1}^n x_j$ be the sample mean. For notation, also let $A_k=\\sum_{i=1}^n x_i^k$ for $k=1,2,3,\\ldots$. Hence, ${\\overline x}=A_1/n$ and $\\sum_{i=1}^n (x_i-{\\overline x})^2=\\sum_{i=1}^n x_i^2-n {\\overline x}^2=A_2-A_1^2/n$. Check that you can derive these equations before trying the questions below. Part a) What is the summation of deviations: $\\sum_{i=1}^n (x_i-{\\overline x})\\,\\,$? [ANS]\nPart b) Consider $\\sum_{i=1}^n (x_i-{\\overline x})^3$? Simplify this to an expression involving $A_1,A_2,A_3,n$. Now, suppose $n=5$, $A_1={20}$, $A_2={86}$, $A_3={386}$. Applying your formula, what is the numerical value of $\\sum_{i=1}^n (x_i-{\\overline x})^3$? [ANS]", "answer_v1": [ "0", "-6.000" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Consider $n$ numbers $x_1,\\ldots,x_n$. Let ${\\overline x}=n^{-1}\\sum_{j=1}^n x_j$ be the sample mean. For notation, also let $A_k=\\sum_{i=1}^n x_i^k$ for $k=1,2,3,\\ldots$. Hence, ${\\overline x}=A_1/n$ and $\\sum_{i=1}^n (x_i-{\\overline x})^2=\\sum_{i=1}^n x_i^2-n {\\overline x}^2=A_2-A_1^2/n$. Check that you can derive these equations before trying the questions below. Part a) What is the summation of deviations: $\\sum_{i=1}^n (x_i-{\\overline x})\\,\\,$? [ANS]\nPart b) Consider $\\sum_{i=1}^n (x_i-{\\overline x})^3$? Simplify this to an expression involving $A_1,A_2,A_3,n$. Now, suppose $n=5$, $A_1={17}$, $A_2={83}$, $A_3={461}$. Applying your formula, what is the numerical value of $\\sum_{i=1}^n (x_i-{\\overline x})^3$? [ANS]", "answer_v2": [ "0", "7.440" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Consider $n$ numbers $x_1,\\ldots,x_n$. Let ${\\overline x}=n^{-1}\\sum_{j=1}^n x_j$ be the sample mean. For notation, also let $A_k=\\sum_{i=1}^n x_i^k$ for $k=1,2,3,\\ldots$. Hence, ${\\overline x}=A_1/n$ and $\\sum_{i=1}^n (x_i-{\\overline x})^2=\\sum_{i=1}^n x_i^2-n {\\overline x}^2=A_2-A_1^2/n$. Check that you can derive these equations before trying the questions below. Part a) What is the summation of deviations: $\\sum_{i=1}^n (x_i-{\\overline x})\\,\\,$? [ANS]\nPart b) Consider $\\sum_{i=1}^n (x_i-{\\overline x})^3$? Simplify this to an expression involving $A_1,A_2,A_3,n$. Now, suppose $n=5$, $A_1={14}$, $A_2={44}$, $A_3={152}$. Applying your formula, what is the numerical value of $\\sum_{i=1}^n (x_i-{\\overline x})^3$? [ANS]", "answer_v3": [ "0", "1.920" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0372", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Regression", "level": "4", "keywords": [ "statistics", "simple linear regression" ], "problem_v1": "The variables are x=SP500 market monthly log return and y=monthly return of American Express for 48 months beginning in January 2009. For input into R, the data vectors for monthly market return and monthly stock return are x=c(-0.08955,-0.116457, 0.081953, 0.089772, 0.051721, 0.000196, 0.071522, 0.033009, 0.0351,-0.01996, 0.055779, 0.017615,-0.037675, 0.028115, 0.057133, 0.014651,-0.085532,-0.055388, 0.066516,-0.048612, 0.083928, 0.036193,-0.002293, 0.063257, 0.022393, 0.031457,-0.001048, 0.028097,-0.013593,-0.018426,-0.021708,-0.058467,-0.074467, 0.102307,-0.005071, 0.008497, 0.04266, 0.039787, 0.030852,-0.007526,-0.064699, 0.038793, 0.012519, 0.019571, 0.023947,-0.019988, 0.002843, 0.007043) and y=c(-0.094831,-0.327553, 0.122848, 0.628447,-0.01477,-0.059504, 0.197838, 0.177291, 0.007631, 0.027493, 0.182835,-0.031815,-0.069106, 0.014097, 0.081678, 0.111456,-0.145839, 0.000267, 0.117249,-0.112987, 0.052807,-0.008864, 0.041605,-0.007102, 0.01488, 0.004349, 0.036686, 0.086333, 0.050099, 0.00546,-0.032588,-0.006479,-0.101895, 0.124084,-0.052417,-0.018101, 0.064759, 0.053416, 0.089713, 0.043379,-0.075531, 0.041759,-0.005326, 0.010095,-0.024979,-0.012179,-0.001281, 0.02781)\nFor the questions below, use 3 decimal places. Part a) The coefficients of the least square regression line are ${\\hat \\beta}_0$=[ANS]\n${\\hat \\beta}_1$=[ANS]\nPart b) Next, the least squares equation in (a) is used for out-of-sample assessment. Suppose we want to get a prediction interval for each of the next 10 months (beginning January 2013) when the SP500 returns are values in the following R vector. xnext=c(0.049198, 0.011, 0.035355, 0.017924, 0.02055,-0.015113, 0.048278,-0.031798, 0.029316, 0.04363)\nThe t critical value for the 95\\% prediction interval is [ANS]. Using the fitted regression equation for January 2009 to December 2012, the lower endpoint of the 95\\% prediction interval for January 2013 (SP500 return 0.049198) is [ANS]. The upper endpoint of this 95\\% prediction interval is [ANS]. The lower endpoint of the 95\\% prediction interval for October 2013 (SP500 return 0.04363) is [ANS]. The upper endpoint of this 95\\% prediction interval is [ANS].\nPart c) Get the 10 out-of-sample prediction intervals for January to October 2013 from part (b) of which you were asked to enter two intervals. The actual values of the monthly stock returns for American Express are in the following vector ynext=c(0.026364, 0.055335, 0.081897, 0.016943, 0.101428,-0.012572,-0.010393,-0.025476, 0.048978, 0.08293) How many of these observed values (not used in the fitted regression equation) are contained in the corresponding prediction intervals. (The response here is an integer between 0 and 10; theoretically it is close to 9.) [ANS]", "answer_v1": [ "0.00831242415822129", "1.78257716293546", "2.012896", "-0.090534602029955", "0.282557912870595", "-0.10013933334609", "0.272311864900281", "10" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "The variables are x=SP500 market monthly log return and y=monthly return of Yahoo for 48 months beginning in January 2010. For input into R, the data vectors for monthly market return and monthly stock return are x=c(-0.037675, 0.028115, 0.057133, 0.014651,-0.085532,-0.055388, 0.066516,-0.048612, 0.083928, 0.036193,-0.002293, 0.063257, 0.022393, 0.031457,-0.001048, 0.028097,-0.013593,-0.018426,-0.021708,-0.058467,-0.074467, 0.102307,-0.005071, 0.008497, 0.04266, 0.039787, 0.030852,-0.007526,-0.064699, 0.038793, 0.012519, 0.019571, 0.023947,-0.019988, 0.002843, 0.007043, 0.049198, 0.011, 0.035355, 0.017924, 0.02055,-0.015113, 0.048278,-0.031798, 0.029316, 0.04363, 0.027663, 0.02329) and y=c(-0.111471, 0.01979, 0.076671, 0,-0.074713,-0.102901, 0.002886,-0.057074, 0.077752, 0.151627,-0.041479, 0.049933,-0.031148, 0.017221, 0.016929, 0.059354,-0.067179,-0.095673,-0.138101, 0.038193,-0.032863, 0.17189, 0.004466, 0.026383,-0.041778,-0.042251, 0.025958, 0.020807,-0.019494, 0.037983, 0.000632,-0.078098, 0.086898, 0.052419, 0.108503, 0.05846,-0.013661, 0.082117, 0.0991, 0.049741, 0.061552,-0.045507, 0.111351,-0.035142, 0.201374,-0.006958, 0.11569, 0.089442)\nFor the questions below, use 3 decimal places. Part a) The coefficients of the least square regression line are ${\\hat \\beta}_0$=[ANS]\n${\\hat \\beta}_1$=[ANS]\nPart b) Next, the least squares equation in (a) is used for out-of-sample assessment. Suppose we want to get a prediction interval for each of the next 10 months (beginning January 2014) when the SP500 returns are values in the following R vector. xnext=c(-0.036231, 0.042213, 0.006908, 0.006182, 0.020812, 0.018879,-0.015195, 0.036964,-0.015635, 0.022936)\nThe t critical value for the 95\\% prediction interval is [ANS]. Using the fitted regression equation for January 2010 to December 2013, the lower endpoint of the 95\\% prediction interval for January 2014 (SP500 return-0.036231) is [ANS]. The upper endpoint of this 95\\% prediction interval is [ANS]. The lower endpoint of the 95\\% prediction interval for October 2014 (SP500 return 0.022936) is [ANS]. The upper endpoint of this 95\\% prediction interval is [ANS].\nPart c) Get the 10 out-of-sample prediction intervals for January to October 2014 from part (b) of which you were asked to enter two intervals. The actual values of the monthly stock returns for Yahoo are in the following vector ynext=c(-0.116023, 0.071267,-0.074327, 0.001392,-0.036831, 0.013758, 0.019172, 0.072691, 0.056538, 0.122272) How many of these observed values (not used in the fitted regression equation) are contained in the corresponding prediction intervals. (The response here is an integer between 0 and 10; theoretically it is close to 9.) [ANS]", "answer_v2": [ "0.007395925966368", "1.03812646774736", "2.012896", "-0.15996383588938", "0.0995309677162071", "-0.0968891824283947", "0.159301971689637", "10" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "The variables are x=SP500 market monthly log return and y=monthly return of Microsoft for 48 months beginning in January 2009. For input into R, the data vectors for monthly market return and monthly stock return are x=c(-0.08955,-0.116457, 0.081953, 0.089772, 0.051721, 0.000196, 0.071522, 0.033009, 0.0351,-0.01996, 0.055779, 0.017615,-0.037675, 0.028115, 0.057133, 0.014651,-0.085532,-0.055388, 0.066516,-0.048612, 0.083928, 0.036193,-0.002293, 0.063257, 0.022393, 0.031457,-0.001048, 0.028097,-0.013593,-0.018426,-0.021708,-0.058467,-0.074467, 0.102307,-0.005071, 0.008497, 0.04266, 0.039787, 0.030852,-0.007526,-0.064699, 0.038793, 0.012519, 0.019571, 0.023947,-0.019988, 0.002843, 0.007043) and y=c(-0.12843,-0.05008, 0.128347, 0.09793, 0.037403, 0.128863,-0.010654, 0.052631, 0.042502, 0.075435, 0.063229, 0.035583,-0.078494, 0.021883, 0.021414, 0.041869,-0.164237,-0.114252, 0.114687,-0.089626, 0.042401, 0.085223,-0.048057, 0.099895,-0.006785,-0.036293,-0.045887, 0.020655,-0.02895, 0.038698, 0.052579,-0.023483,-0.066479, 0.067708,-0.032858, 0.014691, 0.128833, 0.078859, 0.016119,-0.007355,-0.086158, 0.047152,-0.037324, 0.051185,-0.035018,-0.041847,-0.061298, 0.003156)\nFor the questions below, use 3 decimal places. Part a) The coefficients of the least square regression line are ${\\hat \\beta}_0$=[ANS]\n${\\hat \\beta}_1$=[ANS]\nPart b) Next, the least squares equation in (a) is used for out-of-sample assessment. Suppose we want to get a prediction interval for each of the next 10 months (beginning January 2013) when the SP500 returns are values in the following R vector. xnext=c(0.049198, 0.011, 0.035355, 0.017924, 0.02055,-0.015113, 0.048278,-0.031798, 0.029316, 0.04363)\nThe t critical value for the 95\\% prediction interval is [ANS]. Using the fitted regression equation for January 2009 to December 2012, the lower endpoint of the 95\\% prediction interval for January 2013 (SP500 return 0.049198) is [ANS]. The upper endpoint of this 95\\% prediction interval is [ANS]. The lower endpoint of the 95\\% prediction interval for October 2013 (SP500 return 0.04363) is [ANS]. The upper endpoint of this 95\\% prediction interval is [ANS].\nPart c) Get the 10 out-of-sample prediction intervals for January to October 2013 from part (b) of which you were asked to enter two intervals. The actual values of the monthly stock returns for Microsoft are in the following vector ynext=c(0.02758, 0.020854, 0.028848, 0.145625, 0.059938,-0.010443,-0.081193, 0.054752,-0.003703, 0.062024) How many of these observed values (not used in the fitted regression equation) are contained in the corresponding prediction intervals. (The response here is an integer between 0 and 10; theoretically it is close to 9.) [ANS]", "answer_v3": [ "-0.00113845724092884", "1.02899722738685", "2.012896", "-0.0494702641505977", "0.148442560854697", "-0.0550296224318562", "0.142543006011775", "8" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Statistics_0373", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Regression", "level": "4", "keywords": [ "statistics", "regression" ], "problem_v1": "Consider simple linear regression with $n$ pairs of numbers $x_i,y_i$. Consider the $n\\times 2$ matrix ${\\bf X}$ with the first column consisting of 1s and the second column consisting of the $x_i$. Below are some questions on the resulting $({\\bf X}^T{\\bf X})$ and $({\\bf X}^T{\\bf X})^{-1}$. To do these questions, please review the results for the determinant and inverse of a 2x2 matrix.\nPart a) Which expressions match the determinant of ${\\bf X}^T{\\bf X}$? Possibly more than one item is correct. [ANS] A. $n\\sum_i (x_i-{\\bar x})^2$ B. $n(n-1)s_x^2$ C. $\\sum_i (x_i-{\\bar x})^2$ D. $n s_x^2$ E. None of the above\nPart b) Let det denote the determinant of $({\\bf X}^T{\\bf X})$. Which expression matches the (2,2) element of $({\\bf X}^T{\\bf X})^{-1}$? [ANS] A. $n$/det B. $-n$/det C. 1/det D. $-\\sum_i x_i^2$/det E. $\\sum_i x_i^2$/det F. None of the above\nPart c) Which expression matches the (1,2) element of $({\\bf X}^T{\\bf X})^{-1}$? [ANS] A. $n$/det B. 1/det C. $-\\sum_i x_i$/det D. $\\sum_i x_i$/det E. $-n$/det F. None of the above\nPart d) In simple linear regression, the standard error of an estimate of the subpopulation mean at $x^*$ is ${\\hat\\sigma}\\sqrt{{1\\over n}+{(x^*-{\\overline x})^2 \\over (n-1)s_x^2}} \\quad (A1)$ For multiple regression, the standard error of an estimate of the subpopulation mean at ${\\bf x}^*=(1,x_1^*,\\ldots,x^*_p)^T$ is ${\\hat\\sigma}\\sqrt{{{\\bf x}^*}^T({\\bf X}^T{\\bf X})^{-1}{\\bf x}^*} \\quad (A2)$ When $p=1$ and ${\\bf x}^*=(1,x^*)^T$, the two equations should be the same. Check this by simplifying det $\\times\\left[(1, x^*)({\\bf X}^T{\\bf X})^{-1}\\left(\\begin{array}{cc} 1 \\\\ x^*\\\\ \\end{array}\\right) \\right] \\quad (A3)$ (The above expression means the determinant multiplied by remainder). Which are following are equivalent to the equation (A3)? Possibly more than one item is correct. [ANS] A. $\\sum_i x_i^2+2x^*\\sum_i x_i+n(x^*)^2$ B. $(n-1)s_x^2+n(x^*+{\\bar x})^2$ C. $\\sum_i x_i^2+2nx^*{\\bar x}+n(x^*)^2$ D. $(n-1)s_x^2+n{\\bar x}^2-2nx^*{\\bar x}+n(x^*)^2$ E. $n(n-1)s_x^2$ F. $(n-1)s_x^2+n(x^*-{\\bar x})^2$ G. $(n-1)s_x^2+n{\\bar x}^2+2nx^*{\\bar x}+n(x^*)^2$ H. $\\sum_i x_i^2-2nx^*{\\bar x}+n(x^*)^2$ I. $\\sum_i x_i^2-2x^*\\sum_i x_i+n(x^*)^2$ J. None of the above\nPart e) After doing (d), you should be able to complete the derivation of the equivalence of (A1) and (A2) for $p=1$.", "answer_v1": [ "AB", "A", "C", "DFHI" ], "answer_type_v1": [ "MCM", "MCS", "MCS", "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J" ] ], "problem_v2": "Consider simple linear regression with $n$ pairs of numbers $x_i,y_i$. Consider the $n\\times 2$ matrix ${\\bf X}$ with the first column consisting of 1s and the second column consisting of the $x_i$. Below are some questions on the resulting $({\\bf X}^T{\\bf X})$ and $({\\bf X}^T{\\bf X})^{-1}$. To do these questions, please review the results for the determinant and inverse of a 2x2 matrix.\nPart a) Which expressions match the determinant of ${\\bf X}^T{\\bf X}$? Possibly more than one item is correct. [ANS] A. $n(n-1)s_x^2$ B. $n\\sum_i (x_i-{\\bar x})^2$ C. $\\sum_i (x_i-{\\bar x})^2$ D. $n s_x^2$ E. None of the above\nPart b) Let det denote the determinant of $({\\bf X}^T{\\bf X})$. Which expression matches the (2,2) element of $({\\bf X}^T{\\bf X})^{-1}$? [ANS] A. $-n$/det B. 1/det C. $-\\sum_i x_i^2$/det D. $n$/det E. $\\sum_i x_i^2$/det F. None of the above\nPart c) Which expression matches the (1,2) element of $({\\bf X}^T{\\bf X})^{-1}$? [ANS] A. 1/det B. $-n$/det C. $\\sum_i x_i$/det D. $-\\sum_i x_i$/det E. $n$/det F. None of the above\nPart d) In simple linear regression, the standard error of an estimate of the subpopulation mean at $x^*$ is ${\\hat\\sigma}\\sqrt{{1\\over n}+{(x^*-{\\overline x})^2 \\over (n-1)s_x^2}} \\quad (A1)$ For multiple regression, the standard error of an estimate of the subpopulation mean at ${\\bf x}^*=(1,x_1^*,\\ldots,x^*_p)^T$ is ${\\hat\\sigma}\\sqrt{{{\\bf x}^*}^T({\\bf X}^T{\\bf X})^{-1}{\\bf x}^*} \\quad (A2)$ When $p=1$ and ${\\bf x}^*=(1,x^*)^T$, the two equations should be the same. Check this by simplifying det $\\times\\left[(1, x^*)({\\bf X}^T{\\bf X})^{-1}\\left(\\begin{array}{cc} 1 \\\\ x^*\\\\ \\end{array}\\right) \\right] \\quad (A3)$ (The above expression means the determinant multiplied by remainder). Which are following are equivalent to the equation (A3)? Possibly more than one item is correct. [ANS] A. $(n-1)s_x^2+n{\\bar x}^2+2nx^*{\\bar x}+n(x^*)^2$ B. $(n-1)s_x^2+n(x^*+{\\bar x})^2$ C. $(n-1)s_x^2+n(x^*-{\\bar x})^2$ D. $\\sum_i x_i^2-2x^*\\sum_i x_i+n(x^*)^2$ E. $n(n-1)s_x^2$ F. $\\sum_i x_i^2-2nx^*{\\bar x}+n(x^*)^2$ G. $\\sum_i x_i^2+2x^*\\sum_i x_i+n(x^*)^2$ H. $(n-1)s_x^2+n{\\bar x}^2-2nx^*{\\bar x}+n(x^*)^2$ I. $\\sum_i x_i^2+2nx^*{\\bar x}+n(x^*)^2$ J. None of the above\nPart e) After doing (d), you should be able to complete the derivation of the equivalence of (A1) and (A2) for $p=1$.", "answer_v2": [ "AB", "D", "D", "CDFH" ], "answer_type_v2": [ "MCM", "MCS", "MCS", "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J" ] ], "problem_v3": "Consider simple linear regression with $n$ pairs of numbers $x_i,y_i$. Consider the $n\\times 2$ matrix ${\\bf X}$ with the first column consisting of 1s and the second column consisting of the $x_i$. Below are some questions on the resulting $({\\bf X}^T{\\bf X})$ and $({\\bf X}^T{\\bf X})^{-1}$. To do these questions, please review the results for the determinant and inverse of a 2x2 matrix.\nPart a) Which expressions match the determinant of ${\\bf X}^T{\\bf X}$? Possibly more than one item is correct. [ANS] A. $n(n-1)s_x^2$ B. $n\\sum_i (x_i-{\\bar x})^2$ C. $\\sum_i (x_i-{\\bar x})^2$ D. $n s_x^2$ E. None of the above\nPart b) Let det denote the determinant of $({\\bf X}^T{\\bf X})$. Which expression matches the (2,2) element of $({\\bf X}^T{\\bf X})^{-1}$? [ANS] A. $\\sum_i x_i^2$/det B. $-n$/det C. $-\\sum_i x_i^2$/det D. 1/det E. $n$/det F. None of the above\nPart c) Which expression matches the (1,2) element of $({\\bf X}^T{\\bf X})^{-1}$? [ANS] A. 1/det B. $-\\sum_i x_i$/det C. $n$/det D. $\\sum_i x_i$/det E. $-n$/det F. None of the above\nPart d) In simple linear regression, the standard error of an estimate of the subpopulation mean at $x^*$ is ${\\hat\\sigma}\\sqrt{{1\\over n}+{(x^*-{\\overline x})^2 \\over (n-1)s_x^2}} \\quad (A1)$ For multiple regression, the standard error of an estimate of the subpopulation mean at ${\\bf x}^*=(1,x_1^*,\\ldots,x^*_p)^T$ is ${\\hat\\sigma}\\sqrt{{{\\bf x}^*}^T({\\bf X}^T{\\bf X})^{-1}{\\bf x}^*} \\quad (A2)$ When $p=1$ and ${\\bf x}^*=(1,x^*)^T$, the two equations should be the same. Check this by simplifying det $\\times\\left[(1, x^*)({\\bf X}^T{\\bf X})^{-1}\\left(\\begin{array}{cc} 1 \\\\ x^*\\\\ \\end{array}\\right) \\right] \\quad (A3)$ (The above expression means the determinant multiplied by remainder). Which are following are equivalent to the equation (A3)? Possibly more than one item is correct. [ANS] A. $\\sum_i x_i^2-2x^*\\sum_i x_i+n(x^*)^2$ B. $\\sum_i x_i^2+2nx^*{\\bar x}+n(x^*)^2$ C. $n(n-1)s_x^2$ D. $(n-1)s_x^2+n{\\bar x}^2-2nx^*{\\bar x}+n(x^*)^2$ E. $\\sum_i x_i^2+2x^*\\sum_i x_i+n(x^*)^2$ F. $(n-1)s_x^2+n(x^*+{\\bar x})^2$ G. $(n-1)s_x^2+n(x^*-{\\bar x})^2$ H. $(n-1)s_x^2+n{\\bar x}^2+2nx^*{\\bar x}+n(x^*)^2$ I. $\\sum_i x_i^2-2nx^*{\\bar x}+n(x^*)^2$ J. None of the above\nPart e) After doing (d), you should be able to complete the derivation of the equivalence of (A1) and (A2) for $p=1$.", "answer_v3": [ "AB", "E", "B", "ADGI" ], "answer_type_v3": [ "MCM", "MCS", "MCS", "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J" ] ] }, { "id": "Statistics_0374", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Regression", "level": "", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "A regression line using 33 observations produced SSR=140 and SSE=57. Determine the standard error of the estimate. Standard Error of the Estimate=[ANS]", "answer_v1": [ "1.35599029399895" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A regression line using 23 observations produced SSR=113 and SSE=64. Determine the standard error of the estimate. Standard Error of the Estimate=[ANS]", "answer_v2": [ "1.74574312188794" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A regression line using 25 observations produced SSR=122 and SSE=57. Determine the standard error of the estimate. Standard Error of the Estimate=[ANS]", "answer_v3": [ "1.57424930349841" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0375", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Regression", "level": "2", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "Suppose that for a given data set, $s_x^2=550, \\quad s_y^2=820, \\quad \\mbox{cov}(X,Y)=140, \\quad n=9$ Then the standard error of estimate is: Answer=[ANS]", "answer_v1": [ "29.9402001398719" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that for a given data set, $s_x^2=410, \\quad s_y^2=890, \\quad \\mbox{cov}(X,Y)=70, \\quad n=7$ Then the standard error of estimate is: Answer=[ANS]", "answer_v2": [ "32.4601068480276" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that for a given data set, $s_x^2=460, \\quad s_y^2=820, \\quad \\mbox{cov}(X,Y)=90, \\quad n=8$ Then the standard error of estimate is: Answer=[ANS]", "answer_v3": [ "30.5961302848219" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0376", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Prediction", "level": "4", "keywords": [ "Confidence", "Interval" ], "problem_v1": "Construct both a $90$ \\% and a $95$ \\% confidence interval for $\\beta_1$. $\\hat{\\beta}_1=42, \\ s=7.3, \\ SS_{xx}=59, \\ n=21$ $90$ \\%: [ANS] $\\leq \\beta_1 \\leq$ [ANS]\n$95$ \\%: [ANS] $\\leq \\beta_1 \\leq$ [ANS]", "answer_v1": [ "40.3566710729968", "43.6433289270032", "40.0108376404341", "43.9891623595659" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Construct both a $99$ \\% and a $90$ \\% confidence interval for $\\beta_1$. $\\hat{\\beta}_1=49, \\ s=2.5, \\ SS_{xx}=44, \\ n=15$ $99$ \\%: [ANS] $\\leq \\beta_1 \\leq$ [ANS]\n$90$ \\%: [ANS] $\\leq \\beta_1 \\leq$ [ANS]", "answer_v2": [ "47.8647080275859", "50.1352919724141", "48.3325556431836", "49.6674443568164" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Construct both a $90$ \\% and a $95$ \\% confidence interval for $\\beta_1$. $\\hat{\\beta}_1=42, \\ s=4.2, \\ SS_{xx}=48, \\ n=18$ $90$ \\%: [ANS] $\\leq \\beta_1 \\leq$ [ANS]\n$95$ \\%: [ANS] $\\leq \\beta_1 \\leq$ [ANS]", "answer_v3": [ "40.9416164976286", "43.0583835023714", "40.7148728603843", "43.2851271396157" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Statistics_0377", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Prediction", "level": "1", "keywords": [ "statistic", "regression" ], "problem_v1": "A study of king penguins looked for a relationship between how deep the penguins dive to seek food and how long they stay underwater. For all but the shallowest dives, there is a linear relationship that is different for different penguins. The study report gives a scatterplot for one penguin titled \" The relation of dive duration (DD) to depth (D).\" Duration DD is measured in minutes and depth D is in meters. The report then says, \" The regression equation for this bird is: DD=2.76+0.0087\" D.\n(a) What is the slope of the regression line?. ANSWER [ANS] minutes per meter. (b) According to the regression line, how long does a typical dive to a depth of 300 meters last? ANSWER [ANS] minutes.", "answer_v1": [ "0.0087", "5.37" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A study of king penguins looked for a relationship between how deep the penguins dive to seek food and how long they stay underwater. For all but the shallowest dives, there is a linear relationship that is different for different penguins. The study report gives a scatterplot for one penguin titled \" The relation of dive duration (DD) to depth (D).\" Duration DD is measured in minutes and depth D is in meters. The report then says, \" The regression equation for this bird is: DD=2.08+0.014\" D.\n(a) What is the slope of the regression line?. ANSWER [ANS] minutes per meter. (b) According to the regression line, how long does a typical dive to a depth of 125 meters last? ANSWER [ANS] minutes.", "answer_v2": [ "0.014", "3.83" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A study of king penguins looked for a relationship between how deep the penguins dive to seek food and how long they stay underwater. For all but the shallowest dives, there is a linear relationship that is different for different penguins. The study report gives a scatterplot for one penguin titled \" The relation of dive duration (DD) to depth (D).\" Duration DD is measured in minutes and depth D is in meters. The report then says, \" The regression equation for this bird is: DD=2.31+0.0091\" D.\n(a) What is the slope of the regression line?. ANSWER [ANS] minutes per meter. (b) According to the regression line, how long does a typical dive to a depth of 175 meters last? ANSWER [ANS] minutes.", "answer_v3": [ "0.0091", "3.9025" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0378", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Prediction", "level": "1", "keywords": [ "Statistics", "Regression Equation", "Correlation", "Regression", "statistic" ], "problem_v1": "Heights (in centimeters) and weights (in kilograms) of $7$ supermodels are given below. Find the regression equation, letting the first variable be the independent $(x)$ variable, and predict the weight of a supermodel who is $173$ cm tall. \\begin{array}{c|ccccccc} \\mbox{Height} & 168 & 174 & 174 & 172 & 176 & 176 & 178 \\cr \\hline \\mbox{Weight} & 50 & 54 & 55 & 53 & 54 & 56 & 58 \\cr \\end{array} The regression equation is $\\hat{y}=$ [ANS] $+$ [ANS] $x.$ The best predicted weight of a supermodel who is $173$ cm tall is [ANS].", "answer_v1": [ "-70.7767857142857", "0.71875", "53.5669642857143" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Heights (in centimeters) and weights (in kilograms) of $7$ supermodels are given below. Find the regression equation, letting the first variable be the independent $(x)$ variable, and predict the weight of a supermodel who is $169$ cm tall. \\begin{array}{c|ccccccc} \\mbox{Height} & 176 & 172 & 176 & 178 & 166 & 178 & 176 \\cr \\hline \\mbox{Weight} & 55 & 52 & 54 & 58 & 47 & 57 & 56 \\cr \\end{array} The regression equation is $\\hat{y}=$ [ANS] $+$ [ANS] $x.$ The best predicted weight of a supermodel who is $169$ cm tall is [ANS].", "answer_v2": [ "-94.515625", "0.8515625", "49.3984375" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Heights (in centimeters) and weights (in kilograms) of $7$ supermodels are given below. Find the regression equation, letting the first variable be the independent $(x)$ variable, and predict the weight of a supermodel who is $177$ cm tall. \\begin{array}{c|ccccccc} \\mbox{Height} & 178 & 172 & 176 & 174 & 176 & 174 & 172 \\cr \\hline \\mbox{Weight} & 57 & 53 & 54 & 55 & 56 & 54 & 52 \\cr \\end{array} The regression equation is $\\hat{y}=$ [ANS] $+$ [ANS] $x.$ The best predicted weight of a supermodel who is $177$ cm tall is [ANS].", "answer_v3": [ "-64.75", "0.682692307692308", "56.0865384615385" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Statistics_0379", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Prediction", "level": "2", "keywords": [ "linear equation", "slope", "intercept" ], "problem_v1": "Measuring Temperature. The two most commonly used scales for measuring temperature are the Fahrenheit and Celsius scales. If you let $y$ denote the Fahrenheit temperature and $x$ denote Celsius temperature, you can express the relationship between those two scales with the linear equation $y=32+1.8x$. a. Determine the $y$-intercept $b_0$ and the slope $b_1$.\n$b_0$=[ANS]\n$b_1$=[ANS]\nb. Find the Fahrenheit temperature corresponding to the Celsius temperature $-40^{\\circ}$. [ANS] degrees c. Find the Fahrenheit temperature corresponding to the Celsius temperature $0^{\\circ}$. [ANS] degrees d. Find the Fahrenheit temperature corresponding to the Celsius temperature $33^{\\circ}$. [ANS] degrees e. Find the Fahrenheit temperature corresponding to the Celsius temperature $100^{\\circ}$. [ANS] degrees f. Graph the linear equation $y=32+1.8x$, using the four points found in (b), (c), (d) and (e). g. Apply the graph obtained in part (f) to estimate visually the Fahrenheit temperature corresponding to a Celsius temperature of $33^{\\circ}$. Then calculate that temperature exactly by using the linear equation $y=32+1.8x$. The exact Fahrenheit temperature corresponding to the Celsius temperature of $33^{\\circ}$ is [ANS] degrees", "answer_v1": [ "32", "1.8", "-40", "32", "91.4", "212", "91.4" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "Measuring Temperature. The two most commonly used scales for measuring temperature are the Fahrenheit and Celsius scales. If you let $y$ denote the Fahrenheit temperature and $x$ denote Celsius temperature, you can express the relationship between those two scales with the linear equation $y=32+1.8x$. a. Determine the $y$-intercept $b_0$ and the slope $b_1$.\n$b_0$=[ANS]\n$b_1$=[ANS]\nb. Find the Fahrenheit temperature corresponding to the Celsius temperature $-13^{\\circ}$. [ANS] degrees c. Find the Fahrenheit temperature corresponding to the Celsius temperature $0^{\\circ}$. [ANS] degrees d. Find the Fahrenheit temperature corresponding to the Celsius temperature $48^{\\circ}$. [ANS] degrees e. Find the Fahrenheit temperature corresponding to the Celsius temperature $100^{\\circ}$. [ANS] degrees f. Graph the linear equation $y=32+1.8x$, using the four points found in (b), (c), (d) and (e). g. Apply the graph obtained in part (f) to estimate visually the Fahrenheit temperature corresponding to a Celsius temperature of $23^{\\circ}$. Then calculate that temperature exactly by using the linear equation $y=32+1.8x$. The exact Fahrenheit temperature corresponding to the Celsius temperature of $23^{\\circ}$ is [ANS] degrees", "answer_v2": [ "32", "1.8", "8.6", "32", "118.4", "212", "73.4" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "Measuring Temperature. The two most commonly used scales for measuring temperature are the Fahrenheit and Celsius scales. If you let $y$ denote the Fahrenheit temperature and $x$ denote Celsius temperature, you can express the relationship between those two scales with the linear equation $y=32+1.8x$. a. Determine the $y$-intercept $b_0$ and the slope $b_1$.\n$b_0$=[ANS]\n$b_1$=[ANS]\nb. Find the Fahrenheit temperature corresponding to the Celsius temperature $-22^{\\circ}$. [ANS] degrees c. Find the Fahrenheit temperature corresponding to the Celsius temperature $0^{\\circ}$. [ANS] degrees d. Find the Fahrenheit temperature corresponding to the Celsius temperature $34^{\\circ}$. [ANS] degrees e. Find the Fahrenheit temperature corresponding to the Celsius temperature $100^{\\circ}$. [ANS] degrees f. Graph the linear equation $y=32+1.8x$, using the four points found in (b), (c), (d) and (e). g. Apply the graph obtained in part (f) to estimate visually the Fahrenheit temperature corresponding to a Celsius temperature of $25^{\\circ}$. Then calculate that temperature exactly by using the linear equation $y=32+1.8x$. The exact Fahrenheit temperature corresponding to the Celsius temperature of $25^{\\circ}$ is [ANS] degrees", "answer_v3": [ "32", "1.8", "-7.6", "32", "93.2", "212", "77" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Statistics_0380", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Prediction", "level": "2", "keywords": [ "Statistics", "Linear regression", "line" ], "problem_v1": "The table below shows (lifetime) peptic ulcer rates (per 100 population), $U$, for various family incomes, $x$, as reported by the 1989 National Health Interview Survey.\n$\\begin{array}{cccccccccc}\\hline Income & 4000 & 6000 & 8000 & 12000 & 16000 & 20000 & 30000 & 45000 & 60000 \\\\ \\hline Ulcer rate & 14.2 & 13.4 & 13.2 & 13.1 & 12.7 & 12.1 & 11.2 & 9.9 & 8.1 \\\\ \\hline \\end{array}$\n(a) Find the equation of the regression line. Ulcer rate, $U(x)=$ [ANS]. (b) Estimate the peptic ulcer rate for an income level of $x_0=$ 26000 according to the linear model in part (a). Ulcer rate, $U(x_0)=$ [ANS].", "answer_v1": [ "-0.000100632340* x+14.2363444745559", "11.6199036434809" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The table below shows (lifetime) peptic ulcer rates (per 100 population), $U$, for various family incomes, $x$, as reported by the 1989 National Health Interview Survey.\n$\\begin{array}{cccccccccc}\\hline Income & 4000 & 6000 & 8000 & 12000 & 16000 & 20000 & 30000 & 45000 & 60000 \\\\ \\hline Ulcer rate & 13 & 13.5 & 12.5 & 12 & 11.7 & 11.6 & 10.1 & 9.1 & 7.4 \\\\ \\hline \\end{array}$\n(a) Find the equation of the regression line. Ulcer rate, $U(x)=$ [ANS]. (b) Estimate the peptic ulcer rate for an income level of $x_0=$ 22000 according to the linear model in part (a). Ulcer rate, $U(x_0)=$ [ANS].", "answer_v2": [ "-0.000101874435* x+13.4863068352906", "11.2450692562481" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The table below shows (lifetime) peptic ulcer rates (per 100 population), $U$, for various family incomes, $x$, as reported by the 1989 National Health Interview Survey.\n$\\begin{array}{cccccccccc}\\hline Income & 4000 & 6000 & 8000 & 12000 & 16000 & 20000 & 30000 & 45000 & 60000 \\\\ \\hline Ulcer rate & 13.5 & 12.8 & 12.8 & 13 & 12.7 & 12.2 & 10.5 & 9.1 & 7.6 \\\\ \\hline \\end{array}$\n(a) Find the equation of the regression line. Ulcer rate, $U(x)=$ [ANS]. (b) Estimate the peptic ulcer rate for an income level of $x_0=$ 23000 according to the linear model in part (a). Ulcer rate, $U(x_0)=$ [ANS].", "answer_v3": [ "-0.000105092593* x+13.9248456790123", "11.5077160493827" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0381", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Prediction", "level": "2", "keywords": [ "Statistics", "Linear regression", "line" ], "problem_v1": "The average life expectancy in the United States, $L$, has been rising steadily over the past few decades, as shown in the table below. Let $t$ be the number of years since 1920, (so $t=0$ represents the year 1920).\n$\\begin{array}{ccccccccc}\\hline Year & 1920 & 1930 & 1940 & 1950 & 1960 & 1970 & 1980 & 1990 \\\\ \\hline Life expectancy & 57.5 & 59.4 & 62.4 & 65.9 & 68.8 & 71.1 & 74.3 & 77.7 \\\\ \\hline \\end{array}$\n(a) Find the equation of the regression line. Average life expectancy, $L(t)=$ [ANS]. Note: this is a function of $t$ (not $x$). (b) Estimate the average life expectancy in the United States in the year 1956 according to the linear model in part (a). Average life expectancy in 1956 $\\approx$ [ANS].", "answer_v1": [ "56.9333333333333+0.291547619047619*t", "67.4290476190476" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The average life expectancy in the United States, $L$, has been rising steadily over the past few decades, as shown in the table below. Let $t$ be the number of years since 1920, (so $t=0$ represents the year 1920).\n$\\begin{array}{ccccccccc}\\hline Year & 1920 & 1930 & 1940 & 1950 & 1960 & 1970 & 1980 & 1990 \\\\ \\hline Life expectancy & 54 & 58.2 & 59.7 & 62.2 & 65.4 & 69 & 70.4 & 75 \\\\ \\hline \\end{array}$\n(a) Find the equation of the regression line. Average life expectancy, $L(t)=$ [ANS]. Note: this is a function of $t$ (not $x$). (b) Estimate the average life expectancy in the United States in the year 1952 according to the linear model in part (a). Average life expectancy in 1952 $\\approx$ [ANS].", "answer_v2": [ "54.275+0.284642857142857*t", "63.3835714285714" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The average life expectancy in the United States, $L$, has been rising steadily over the past few decades, as shown in the table below. Let $t$ be the number of years since 1920, (so $t=0$ represents the year 1920).\n$\\begin{array}{ccccccccc}\\hline Year & 1920 & 1930 & 1940 & 1950 & 1960 & 1970 & 1980 & 1990 \\\\ \\hline Life expectancy & 55.4 & 57.5 & 60.7 & 64.7 & 67.9 & 70.7 & 71.7 & 74.8 \\\\ \\hline \\end{array}$\n(a) Find the equation of the regression line. Average life expectancy, $L(t)=$ [ANS]. Note: this is a function of $t$ (not $x$). (b) Estimate the average life expectancy in the United States in the year 1953 according to the linear model in part (a). Average life expectancy in 1953 $\\approx$ [ANS].", "answer_v3": [ "55.425+0.285714285714286*t", "64.8535714285714" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0382", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Prediction", "level": "2", "keywords": [ "Confidence Intervals" ], "problem_v1": "An train station has determined that the relationship between the number of passengers on a train and the total weight of luggage stored in the baggage compartment can be estimated by the least squares regression equation $y=193+25x$. Predict the weight of luggage for a flight with 98 passengers. Answer: [ANS] pounds", "answer_v1": [ "2643" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "An airline has determined that the relationship between the number of passengers on a flight and the total weight of luggage stored in the baggage compartment can be estimated by the least squares regression equation $y=122+17x$. Predict the weight of luggage for a flight with 142 passengers. Answer: [ANS] pounds", "answer_v2": [ "2536" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "An airline has determined that the relationship between the number of passengers on a flight and the total weight of luggage stored in the baggage compartment can be estimated by the least squares regression equation $y=142+21x$. Predict the weight of luggage for a flight with 101 passengers. Answer: [ANS] pounds", "answer_v3": [ "2263" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0383", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Prediction", "level": "4", "keywords": [ "statistics", "regression", "least squares" ], "problem_v1": "An analyst working for a telecommunications company has been asked to gauge the stress on its cellular networks due to the increasing use of smartphones. She decides to first look at the relationship between the number of minutes customers spent talking on their phones and the amount of cellular data they use. She collects data on 300 customers who have smartphones with data plans. The mean monthly call time was found to be 276 minutes, and the mean amount of data consumed was found to be 516 megabytes. Based on the least squares regression line fitted to the data, it is found that for every minute increase in calling time, the data usage is expected to increase by 3.8 megabytes. Predict the amount of data used by a customer who spends 422 minutes on the phone. Do not round in intermediate steps. The predicted the amount of data used by a customer who spends 422 minutes on the phone is: [ANS] (in megabytes, rounded to one decimal place).", "answer_v1": [ "1070.8" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "An analyst working for a telecommunications company has been asked to gauge the stress on its cellular networks due to the increasing use of smartphones. She decides to first look at the relationship between the number of minutes customers spent talking on their phones and the amount of cellular data they use. She collects data on 300 customers who have smartphones with data plans. The mean monthly call time was found to be 208 minutes, and the mean amount of data consumed was found to be 587 megabytes. Based on the least squares regression line fitted to the data, it is found that for every minute increase in calling time, the data usage is expected to increase by 2.4 megabytes. Predict the amount of data used by a customer who spends 383 minutes on the phone. Do not round in intermediate steps. The predicted the amount of data used by a customer who spends 383 minutes on the phone is: [ANS] (in megabytes, rounded to one decimal place).", "answer_v2": [ "1007" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "An analyst working for a telecommunications company has been asked to gauge the stress on its cellular networks due to the increasing use of smartphones. She decides to first look at the relationship between the number of minutes customers spent talking on their phones and the amount of cellular data they use. She collects data on 300 customers who have smartphones with data plans. The mean monthly call time was found to be 231 minutes, and the mean amount of data consumed was found to be 521 megabytes. Based on the least squares regression line fitted to the data, it is found that for every minute increase in calling time, the data usage is expected to increase by 2.8 megabytes. Predict the amount of data used by a customer who spends 405 minutes on the phone. Do not round in intermediate steps. The predicted the amount of data used by a customer who spends 405 minutes on the phone is: [ANS] (in megabytes, rounded to one decimal place).", "answer_v3": [ "1008.2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0384", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Prediction", "level": "4", "keywords": [ "statistics", "regression", "least squares" ], "problem_v1": "The director of an alumni association for a university wants to look at the relationship between the number of years since graduation and the amount of monetary contribution an alumnus makes to the university. He collects data on 50 alumni who have made contributions this year. The number of years since graduation has a mean of 7, and the amount of contribution has a mean of 208 dollars. Based on the least squares regression line fitted to the data, it is found that for every year increase since graduation, the contribution is expected to drop by 33 dollars. Predict the amount of contribution made by an alumnus who graduated 6 years ago. Do not round in intermediate steps. The predicted contribution of an alumnus who graduated 6 years ago is: [ANS] (in dollars, rounded to one decimal place.)", "answer_v1": [ "241" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The director of an alumni association for a university wants to look at the relationship between the number of years since graduation and the amount of monetary contribution an alumnus makes to the university. He collects data on 50 alumni who have made contributions this year. The number of years since graduation has a mean of 4, and the amount of contribution has a mean of 244 dollars. Based on the least squares regression line fitted to the data, it is found that for every year increase since graduation, the contribution is expected to drop by 23 dollars. Predict the amount of contribution made by an alumnus who graduated 4 years ago. Do not round in intermediate steps. The predicted contribution of an alumnus who graduated 4 years ago is: [ANS] (in dollars, rounded to one decimal place.)", "answer_v2": [ "244" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The director of an alumni association for a university wants to look at the relationship between the number of years since graduation and the amount of monetary contribution an alumnus makes to the university. He collects data on 50 alumni who have made contributions this year. The number of years since graduation has a mean of 5, and the amount of contribution has a mean of 211 dollars. Based on the least squares regression line fitted to the data, it is found that for every year increase since graduation, the contribution is expected to drop by 25 dollars. Predict the amount of contribution made by an alumnus who graduated 5 years ago. Do not round in intermediate steps. The predicted contribution of an alumnus who graduated 5 years ago is: [ANS] (in dollars, rounded to one decimal place.)", "answer_v3": [ "211" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0385", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Prediction", "level": "3", "keywords": [ "statistics", "regression", "least squares" ], "problem_v1": "A Statistics professor assigned 10 quizzes over the course of the semester. He wanted to see if there was a relationship between the total mark of all 10 quizzes and the final exam mark. There were 276 students who completed all the quizzes and wrote the final exam. The standard deviation of the total quiz marks was 13, and that of the final exam was 19. The correlation between the total quiz mark and the final exam was 0.75. Based on the least squares regression line fitted to the data of the 276 students, if a student scored 18 points above the mean of total quiz marks, then how many points above the mean on the final would you predict her final exam grade to be? The predicted final exam grade is [ANS] above the mean on the final. Round your answer to one decimal place, but do not round in intermediate steps.", "answer_v1": [ "19.7307692307692" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A Statistics professor assigned 10 quizzes over the course of the semester. He wanted to see if there was a relationship between the total mark of all 10 quizzes and the final exam mark. There were 208 students who completed all the quizzes and wrote the final exam. The standard deviation of the total quiz marks was 15, and that of the final exam was 16. The correlation between the total quiz mark and the final exam was 0.67. Based on the least squares regression line fitted to the data of the 208 students, if a student scored 25 points above the mean of total quiz marks, then how many points above the mean on the final would you predict her final exam grade to be? The predicted final exam grade is [ANS] above the mean on the final. Round your answer to one decimal place, but do not round in intermediate steps.", "answer_v2": [ "17.8666666666667" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A Statistics professor assigned 10 quizzes over the course of the semester. He wanted to see if there was a relationship between the total mark of all 10 quizzes and the final exam mark. There were 231 students who completed all the quizzes and wrote the final exam. The standard deviation of the total quiz marks was 14, and that of the final exam was 17. The correlation between the total quiz mark and the final exam was 0.71. Based on the least squares regression line fitted to the data of the 231 students, if a student scored 17 points above the mean of total quiz marks, then how many points above the mean on the final would you predict her final exam grade to be? The predicted final exam grade is [ANS] above the mean on the final. Round your answer to one decimal place, but do not round in intermediate steps.", "answer_v3": [ "14.6564285714286" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0386", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Residuals", "level": "3", "keywords": [ "statistics", "regression" ], "problem_v1": "Consider simple linear regression with $n$ pairs of numbers $x_i,y_i$. Let ${\\hat\\beta}_0+{\\hat\\beta}_1x$ be the least squares line where ${\\hat\\beta}_0={\\overline y}-{\\hat\\beta}_1{\\overline x}$ and ${\\hat\\beta}_1=r_{xy}s_y/s_x$. In terms of the summary statistics, derive a simple expression for the residual standard deviation $[\\sum_{i=1}^n e_i^2/(n-2)]^{1/2}$, where $\\sum_{i=1}^n e_i^2=\\sum_{i=1}^n (y_i-{\\hat\\beta}_0-{\\hat\\beta_1}x_i)^2$ For a question like this one that involves a derivation, after you formulate an algebraic solution, check its validity on some numerical regression examples with small data sets. If you match numerically in some instances, your answer is likely correct. If your theoretical answer doesn't match the numerical cases, go back to review your \"derivation\".\nPart a) To validate whether you have the correct expression, suppose $n=45$, $r_{xy}=0.7$, $s_y=1.3$ and $s_x=1.5$. What is your value of the residual SD: [ANS]", "answer_v1": [ "0.939118834890724" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Consider simple linear regression with $n$ pairs of numbers $x_i,y_i$. Let ${\\hat\\beta}_0+{\\hat\\beta}_1x$ be the least squares line where ${\\hat\\beta}_0={\\overline y}-{\\hat\\beta}_1{\\overline x}$ and ${\\hat\\beta}_1=r_{xy}s_y/s_x$. In terms of the summary statistics, derive a simple expression for the residual standard deviation $[\\sum_{i=1}^n e_i^2/(n-2)]^{1/2}$, where $\\sum_{i=1}^n e_i^2=\\sum_{i=1}^n (y_i-{\\hat\\beta}_0-{\\hat\\beta_1}x_i)^2$ For a question like this one that involves a derivation, after you formulate an algebraic solution, check its validity on some numerical regression examples with small data sets. If you match numerically in some instances, your answer is likely correct. If your theoretical answer doesn't match the numerical cases, go back to review your \"derivation\".\nPart a) To validate whether you have the correct expression, suppose $n=31$, $r_{xy}=0.9$, $s_y=0.4$ and $s_x=0.8$. What is your value of the residual SD: [ANS]", "answer_v2": [ "0.177336617375174" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Consider simple linear regression with $n$ pairs of numbers $x_i,y_i$. Let ${\\hat\\beta}_0+{\\hat\\beta}_1x$ be the least squares line where ${\\hat\\beta}_0={\\overline y}-{\\hat\\beta}_1{\\overline x}$ and ${\\hat\\beta}_1=r_{xy}s_y/s_x$. In terms of the summary statistics, derive a simple expression for the residual standard deviation $[\\sum_{i=1}^n e_i^2/(n-2)]^{1/2}$, where $\\sum_{i=1}^n e_i^2=\\sum_{i=1}^n (y_i-{\\hat\\beta}_0-{\\hat\\beta_1}x_i)^2$ For a question like this one that involves a derivation, after you formulate an algebraic solution, check its validity on some numerical regression examples with small data sets. If you match numerically in some instances, your answer is likely correct. If your theoretical answer doesn't match the numerical cases, go back to review your \"derivation\".\nPart a) To validate whether you have the correct expression, suppose $n=36$, $r_{xy}=0.7$, $s_y=0.7$ and $s_x=1.2$. What is your value of the residual SD: [ANS]", "answer_v3": [ "0.50719818611663" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0387", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Hypothesis tests", "level": "4", "keywords": [ "Hypothesis", "Degree", "Freedom", "Test Statistic", "Rejection", "Region" ], "problem_v1": "Consider the data set below.\n$\\begin{array}{ccccccc}\\hline x & 8 & 6 & 4 & 6 & 4 & 7 \\\\ \\hline y & 6 & 7 & 3 & 6 & 5 & 3 \\\\ \\hline \\end{array}$\nFor a hypothesis test, where $H_0:\\beta_1=0$ and $H_1:\\beta_1 \\ne 0$, and using $\\alpha=0.05$, give the following:\n(a) $\\ $ The test statistic $t=$ [ANS]\n(b) $\\ $ The degree of freedom $df=$ [ANS]\n(c) $\\ $ The rejection region $|t| >$ [ANS]\nThe final conclustion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that $\\beta_1=0$. B. We can reject the null hypothesis that $\\beta_1=0$ and accept that $\\beta_1 \\ne 0$.", "answer_v1": [ "0.625330869650362", "4", "2.77630", "A" ], "answer_type_v1": [ "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B" ] ], "problem_v2": "Consider the data set below.\n$\\begin{array}{ccccccc}\\hline x & 2 & 3 & 9 & 3 & 6 & 7 \\\\ \\hline y & 9 & 4 & 3 & 3 & 1 & 4 \\\\ \\hline \\end{array}$\nFor a hypothesis test, where $H_0:\\beta_1=0$ and $H_1:\\beta_1 \\ne 0$, and using $\\alpha=0.05$, give the following:\n(a) $\\ $ The test statistic $t=$ [ANS]\n(b) $\\ $ The degree of freedom $df=$ [ANS]\n(c) $\\ $ The rejection region $|t| >$ [ANS]\nThe final conclustion is [ANS] A. There is not sufficient evidence to reject the null hypothesis that $\\beta_1=0$. B. We can reject the null hypothesis that $\\beta_1=0$ and accept that $\\beta_1 \\ne 0$.", "answer_v2": [ "-1.28564869306645", "4", "2.77630", "A" ], "answer_type_v2": [ "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B" ] ], "problem_v3": "Consider the data set below.\n$\\begin{array}{ccccccc}\\hline x & 4 & 4 & 3 & 8 & 8 & 4 \\\\ \\hline y & 6 & 5 & 4 & 9 & 2 & 3 \\\\ \\hline \\end{array}$\nFor a hypothesis test, where $H_0:\\beta_1=0$ and $H_1:\\beta_1 \\ne 0$, and using $\\alpha=0.01$, give the following:\n(a) $\\ $ The test statistic $t=$ [ANS]\n(b) $\\ $ The degree of freedom $df=$ [ANS]\n(c) $\\ $ The rejection region $|t| >$ [ANS]\nThe final conclustion is [ANS] A. We can reject the null hypothesis that $\\beta_1=0$ and accept that $\\beta_1 \\ne 0$. B. There is not sufficient evidence to reject the null hypothesis that $\\beta_1=0$.", "answer_v3": [ "0.457208429518308", "4", "4.60224", "B" ], "answer_type_v3": [ "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B" ] ] }, { "id": "Statistics_0388", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Hypothesis tests", "level": "2", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "In testing the hypothesis: $H_0: \\beta_1=0$ $H_0: \\beta_1 \\not=0$ the following statistics are available: $n=167,\\quad b_0=-1.99,\\quad b_1=5.16,\\quad s_{b_1}=2.24,\\quad \\hat{y}=8$ Determine the value of the test statistic. Test Statistic=[ANS]", "answer_v1": [ "2.30357142857143" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "In testing the hypothesis: $H_0: \\beta_1=0$ $H_0: \\beta_1 \\not=0$ the following statistics are available: $n=120,\\quad b_0=-4.67,\\quad b_1=5.87,\\quad s_{b_1}=1.3,\\quad \\hat{y}=12$ Determine the value of the test statistic. Test Statistic=[ANS]", "answer_v2": [ "4.51538461538462" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "In testing the hypothesis: $H_0: \\beta_1=0$ $H_0: \\beta_1 \\not=0$ the following statistics are available: $n=146,\\quad b_0=-3.75,\\quad b_1=5.21,\\quad s_{b_1}=1.56,\\quad \\hat{y}=8$ Determine the value of the test statistic. Test Statistic=[ANS]", "answer_v3": [ "3.33974358974359" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0389", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Hypothesis tests", "level": "2", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "For the data set (-1,-3), (2,2), (5,5), (9,7), (10,10), carry out the hypothesis test $ \\begin{array}{cll} \\qquad & H_0 & \\beta_1=1 \\\\ \\qquad & H_1 & \\beta_1 \\not=1 \\\\ \\end{array}$ Determine the value of the test statistic and the associated p-value. Test Statistic=[ANS]\np-Value=[ANS]", "answer_v1": [ "0.347804171820127", "0.75096" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "For the data set (-3,-3), (3,0), (4,4), (8,7), (12,8), carry out the hypothesis test $ \\begin{array}{cll} \\qquad & H_0 & \\beta_1=1 \\\\ \\qquad & H_1 & \\beta_1 \\not=1 \\\\ \\end{array}$ Determine the value of the test statistic and the associated p-value. Test Statistic=[ANS]\np-Value=[ANS]", "answer_v2": [ "-1.48220661585365", "0.234898" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "For the data set (-3,-2), (2,3), (4,6), (8,8), (9,9), carry out the hypothesis test $ \\begin{array}{cll} \\qquad & H_0 & \\beta_1=1 \\\\ \\qquad & H_1 & \\beta_1 \\not=1 \\\\ \\end{array}$ Determine the value of the test statistic and the associated p-value. Test Statistic=[ANS]\np-Value=[ANS]", "answer_v3": [ "-1.15485896611387", "0.331786" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0390", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Hypothesis tests", "level": "3", "keywords": [ "statistics", "Two Populations", "Inference" ], "problem_v1": "In simple linear regression, most often we perform a two-tail test of the population slope $\\beta_1$ to determine whether there is sufficient evidence to infer that a linear relationship exists. The null hypothesis is stated as: [ANS] A. $H_0: \\beta_1=0$ B. $H_0: \\beta_1=\\rho_s$ C. $H_0: \\beta_1=b_1$ D. $H_0: \\beta_1=r$\nGiven the least squares regression line $\\hat{y}$=5-2x: [ANS] A. as x decreases, so does y B. the relationship between x and y is positive C. the relationship between x and y is negative D. as x increases, so does y", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "In simple linear regression, most often we perform a two-tail test of the population slope $\\beta_1$ to determine whether there is sufficient evidence to infer that a linear relationship exists. The null hypothesis is stated as: [ANS] A. $H_0: \\beta_1=b_1$ B. $H_0: \\beta_1=\\rho_s$ C. $H_0: \\beta_1=r$ D. $H_0: \\beta_1=0$\nTesting whether the slope of the population regression line could be zero is equivalent to testing whether the: [ANS] A. population coefficient of correlation could be zero B. sum of squares for error could be zero C. sample coefficient of correlation could be zero D. standard error of estimate could be zero", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "In simple linear regression, most often we perform a two-tail test of the population slope $\\beta_1$ to determine whether there is sufficient evidence to infer that a linear relationship exists. The null hypothesis is stated as: [ANS] A. $H_0: \\beta_1=b_1$ B. $H_0: \\beta_1=0$ C. $H_0: \\beta_1=\\rho_s$ D. $H_0: \\beta_1=r$\nGiven a specific value of x and confidence level, which of the following statements is correct? [ANS] A. The confidence interval estimate of the expected value of y will be narrower than the prediction interval. B. The confidence interval estimate of the expected value of y will be wider than the prediction interval. C. The confidence interval estimate of the expected value of y can be calculated but the prediction interval of y for the given value of x cannot be calculated. D. The prediction interval of y for the given value of x can be calculated but the confidence interval estimate of the expected value of y cannot be calculated.", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0391", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Confidence intervals", "level": "3", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "For the data set (-1,-3), (2,2), (5,5), (9,7), (10,10), find interval estimates (at a 96.2\\% significance level) for single values and for the mean value of $y$ corresponding to $x=3$. Note: For each part below, your answer should use interval notation. Interval Estimate for Single Value=[ANS]\nInterval Estimate for Mean Value=[ANS]", "answer_v1": [ "(-2.81297576413121,7.0269292525033)", "(-0.0809433069614509,4.29489679533354)" ], "answer_type_v1": [ "INT", "INT" ], "options_v1": [ [], [] ], "problem_v2": "For the data set (-3,-3), (3,0), (4,4), (8,7), (12,8), find interval estimates (at a 96.1\\% significance level) for single values and for the mean value of $y$ corresponding to $x=4$. Note: For each part below, your answer should use interval notation. Interval Estimate for Single Value=[ANS]\nInterval Estimate for Mean Value=[ANS]", "answer_v2": [ "(-3.58525748918201,8.72090417687917)", "(0.029671398889338,5.10597528880782)" ], "answer_type_v2": [ "INT", "INT" ], "options_v2": [ [], [] ], "problem_v3": "For the data set (-3,-2), (2,3), (4,6), (8,8), (9,9), find interval estimates (at a 93.3\\% significance level) for single values and for the mean value of $y$ corresponding to $x=3$. Note: For each part below, your answer should use interval notation. Interval Estimate for Single Value=[ANS]\nInterval Estimate for Mean Value=[ANS]", "answer_v3": [ "(1.40588228478356,6.38560707691856)", "(2.85717220141088,4.93431716029124)" ], "answer_type_v3": [ "INT", "INT" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0392", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Diagnostics", "level": "3", "keywords": [ "Correlation", "Curve" ], "problem_v1": "For each paired data set, construct a scatterplot and identify the mathematical model that best fits the given data. $\\begin{array}{c|ccccccc} x & 1 & 2 & 3 & 4 & 5 & 6 & 7\\cr \\hline y & 1 & 2.83 & 5.2 & 8 & 11.18 & 14.7 & 18.52 \\cr \\end{array}$ [ANS] A. Quadratic B. Power C. Logarithmic D. Logistic E. Linear $\\begin{array}{c|ccccccc} x & 1 & 2 & 3 & 4 & 5 & 6 & 7\\cr \\hline y & 4.5 & 7.5 & 10.5 & 13.5 & 16.5 & 19.5 & 22.5 \\cr \\end{array}$ [ANS] A. Power B. Logistic C. Linear D. Quadratic E. Exponential", "answer_v1": [ "B", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "For each paired data set, construct a scatterplot and identify the mathematical model that best fits the given data. $\\begin{array}{c|ccccccc} x & 1 & 2 & 3 & 4 & 5 & 6 & 7\\cr \\hline y & 16.5 & 9.5 & 4.5 & 1.5 & 0.5 & 1.5 & 4.5 \\cr \\end{array}$ [ANS] A. Exponential B. Power C. Logarithmic D. Quadratic E. Linear $\\begin{array}{c|ccccccc} x & 1 & 2 & 3 & 4 & 5 & 6 & 7\\cr \\hline y & 1.9 & 3.61 & 6.86 & 13.03 & 24.76 & 47.05 & 89.39 \\cr \\end{array}$ [ANS] A. Logarithmic B. Quadratic C. Linear D. Logistic E. Exponential", "answer_v2": [ "D", "E" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "For each paired data set, construct a scatterplot and identify the mathematical model that best fits the given data. $\\begin{array}{c|ccccccc} x & 1 & 2 & 3 & 4 & 5 & 6 & 7\\cr \\hline y & 1.4 & 1.96 & 2.74 & 3.84 & 5.38 & 7.53 & 10.54 \\cr \\end{array}$ [ANS] A. Logistic B. Exponential C. Linear D. Logarithmic E. Quadratic $\\begin{array}{c|ccccccc} x & 1 & 2 & 3 & 4 & 5 & 6 & 7\\cr \\hline y & 1 & 1.97 & 2.54 & 2.94 & 3.25 & 3.51 & 3.72 \\cr \\end{array}$ [ANS] A. Logistic B. Quadratic C. Power D. Exponential E. Logarithmic", "answer_v3": [ "B", "E" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Statistics_0393", "subject": "Statistics", "topic": "Simple linear regression", "subtopic": "Diagnostics", "level": "1", "keywords": [ "statistic", "regression", "correlation" ], "problem_v1": "For each problem, select the best response.\n(a) A lurking variable is [ANS] A. the true cause of a response. B. and variable that produces a large residual. C. the true variable that is explained by the explanatory variable. D. a variable that is not among the variables studied but that affects the response variable. E. None of the above.\n(b) A plot of the residuals will indicate if a line is a good fit to the data if the plot [ANS] A. shows large residuals in a symmetric pattern. B. shows a curved pattern. C. shows increasing or decreasing spread about a line. D. has no systematic pattern. E. None of the above.\n(c) Suppose a straight line is fit to data having response variable $y$ and explanatory variable $x$. Predicting values of $y$ for values of $x$ outside the range of the observed data is called [ANS] A. extrapolation. B. correlation. C. causation. D. contingency. E. None of the above.", "answer_v1": [ "D", "D", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "For each problem, select the best response.\n(a) Suppose a straight line is fit to data having response variable $y$ and explanatory variable $x$. Predicting values of $y$ for values of $x$ outside the range of the observed data is called [ANS] A. causation. B. correlation. C. contingency. D. extrapolation. E. None of the above.\n(b) A lurking variable is [ANS] A. the true cause of a response. B. a variable that is not among the variables studied but that affects the response variable. C. the true variable that is explained by the explanatory variable. D. and variable that produces a large residual. E. None of the above.\n(c) A plot of the residuals will indicate if a line is a good fit to the data if the plot [ANS] A. has no systematic pattern. B. shows increasing or decreasing spread about a line. C. shows large residuals in a symmetric pattern. D. shows a curved pattern. E. None of the above.", "answer_v2": [ "D", "B", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "For each problem, select the best response.\n(a) Suppose a straight line is fit to data having response variable $y$ and explanatory variable $x$. Predicting values of $y$ for values of $x$ outside the range of the observed data is called [ANS] A. extrapolation. B. contingency. C. causation. D. correlation. E. None of the above.\n(b) A lurking variable is [ANS] A. and variable that produces a large residual. B. a variable that is not among the variables studied but that affects the response variable. C. the true variable that is explained by the explanatory variable. D. the true cause of a response. E. None of the above.\n(c) A plot of the residuals will indicate if a line is a good fit to the data if the plot [ANS] A. has no systematic pattern. B. shows increasing or decreasing spread about a line. C. shows a curved pattern. D. shows large residuals in a symmetric pattern. E. None of the above.", "answer_v3": [ "A", "B", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Statistics_0394", "subject": "Statistics", "topic": "Time series", "subtopic": "Autocorrelation", "level": "4", "keywords": [], "problem_v1": "Suppose {$Z_t$} is white noise with mean zero. Define the stochastic process {$X_t$}, $t=1,2,3\\ldots$ by X_t=\\alpha X_{t-1}+Z_t. Consider a sample $X_1,X_2,$ $\\dots,$ $X_{200}$ taken from {$X_t$} on 200 consecutive time points. Let the mean of these observations be $\\bar{X}$.\nPart (a) Suppose $\\alpha=$. Compared to the sample mean of a sample of size 200 taken at random from a random variable with the same variance as {$X_t$}, $\\bar{X}$ will have [ANS] A. the same variance. B. lower variance. C. higher variance.\nPart (b) Compared to the sample mean of a sample of size 100 taken at random from a random variable with the same variance as {$X_t$}, $\\bar{X}$ will have [ANS] A. B. C.\nPart (c) Now suppose $\\alpha=$. Compared to the sample mean of a sample of size 200 taken at random from a random variable with the same variance as {$X_t$}, $\\bar{X}$ will have [ANS] A. lower variance. B. the same variance. C. higher variance.", "answer_v1": [ "C", "A", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v2": "Suppose {$Z_t$} is white noise with mean zero. Define the stochastic process {$X_t$}, $t=1,2,3\\ldots$ by X_t=\\alpha X_{t-1}+Z_t. Consider a sample $X_1,X_2,$ $\\dots,$ $X_{200}$ taken from {$X_t$} on 200 consecutive time points. Let the mean of these observations be $\\bar{X}$.\nPart (a) Suppose $\\alpha=$. Compared to the sample mean of a sample of size 200 taken at random from a random variable with the same variance as {$X_t$}, $\\bar{X}$ will have [ANS] A. higher variance. B. lower variance. C. the same variance.\nPart (b) Compared to the sample mean of a sample of size 100 taken at random from a random variable with the same variance as {$X_t$}, $\\bar{X}$ will have [ANS] A. B. C.\nPart (c) Now suppose $\\alpha=$. Compared to the sample mean of a sample of size 200 taken at random from a random variable with the same variance as {$X_t$}, $\\bar{X}$ will have [ANS] A. higher variance. B. lower variance. C. the same variance.", "answer_v2": [ "A", "A", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v3": "Suppose {$Z_t$} is white noise with mean zero. Define the stochastic process {$X_t$}, $t=1,2,3\\ldots$ by X_t=\\alpha X_{t-1}+Z_t. Consider a sample $X_1,X_2,$ $\\dots,$ $X_{200}$ taken from {$X_t$} on 200 consecutive time points. Let the mean of these observations be $\\bar{X}$.\nPart (a) Suppose $\\alpha=$. Compared to the sample mean of a sample of size 200 taken at random from a random variable with the same variance as {$X_t$}, $\\bar{X}$ will have [ANS] A. the same variance. B. higher variance. C. lower variance.\nPart (b) Compared to the sample mean of a sample of size 100 taken at random from a random variable with the same variance as {$X_t$}, $\\bar{X}$ will have [ANS] A. B. C.\nPart (c) Now suppose $\\alpha=$. Compared to the sample mean of a sample of size 200 taken at random from a random variable with the same variance as {$X_t$}, $\\bar{X}$ will have [ANS] A. lower variance. B. higher variance. C. the same variance.", "answer_v3": [ "B", "B", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ] }, { "id": "Statistics_0395", "subject": "Statistics", "topic": "Time series", "subtopic": "ARIMA models", "level": "4", "keywords": [], "problem_v1": "This question concerns the model X(t)=0.5X(t-1)+Z(t)-Z(t-1)+Z(t-2) in which $Z(t)$ is white noise whith mean 0 and variance $\\sigma^2$. It is assumed that $E(X(t))=0$ for all t.\nPart (a) Is this model stationary? [ANS] A. Impossible to determine. B. No. C. Yes.\nPart (b) Show the following: E(X(t)Z(t))=\\sigma^2 E(X(t)Z(t-1))=-0.5\\sigma^2 E(X(t)Z(t-2))=0.75\\sigma^2 [ANS]\nPart (c) With $\\gamma(k)=E(X(t)X(t-k)),$ show that \\gamma(0)=0.5\\gamma(1)+2.25 \\sigma^2, \\gamma(1)=0.5\\gamma(0)-1.5\\sigma^2. [ANS]\nPart (d) Find the variance of $X(t)$ in terms of $\\sigma^2$, and the autocorrelation function of $X(t)$ at lag 1. [ANS]\nPart (e) Writing the model in the form X(t)=\\sum_{j=1}^{\\infty} \\psi_j Z(t-j) find $\\psi_0, \\psi_1, \\psi_2.$ [ANS] [ANS] [ANS]\nPart (f) After de-trending the data, a environmental scientist fits the model to the daily noon temparatures (in Celsius) in Whitehorse over sixty consecutive days. A portion of the data $x(t)$ and the fitted values $\\hat{x}(t)$ are given below (to 2 decimal places):\n\\begin{array}{c|ccc} t & x(t) & \\hat{x}(t) \\\\ \\hline 2 &-0.13 & 0.12 \\\\ \\vdots & \\vdots & \\vdots \\\\ 57 &-2.00 & 1.02 \\\\ 58 &-0.51 & 1.91 \\\\ 59 &-1.04 &-0.85 \\\\ 60 &-5.22 &-2.75 \\end{array} Use the model and the above information to forecast the next two values of the time series, showing your working clearly. [ANS] [ANS]\nPart (g) Given that the estimate of from the fitted model is 1.01, provide approximate 95\\% confidence intervals for each of your forecasts in (f). [ANS]", "answer_v1": [ "C", "0", "0", "-0.25", "1", "-0.5", "0.75", "-0.33", "-2.64", "0" ], "answer_type_v1": [ "MCS", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [ "A", "B", "C" ], [], [], [], [], [], [], [], [], [] ], "problem_v2": "This question concerns the model X(t)=0.5X(t-1)+Z(t)-Z(t-1)+Z(t-2) in which $Z(t)$ is white noise whith mean 0 and variance $\\sigma^2$. It is assumed that $E(X(t))=0$ for all t.\nPart (a) Is this model stationary? [ANS] A. Yes. B. No. C. Impossible to determine.\nPart (b) Show the following: E(X(t)Z(t))=\\sigma^2 E(X(t)Z(t-1))=-0.5\\sigma^2 E(X(t)Z(t-2))=0.75\\sigma^2 [ANS]\nPart (c) With $\\gamma(k)=E(X(t)X(t-k)),$ show that \\gamma(0)=0.5\\gamma(1)+2.25 \\sigma^2, \\gamma(1)=0.5\\gamma(0)-1.5\\sigma^2. [ANS]\nPart (d) Find the variance of $X(t)$ in terms of $\\sigma^2$, and the autocorrelation function of $X(t)$ at lag 1. [ANS]\nPart (e) Writing the model in the form X(t)=\\sum_{j=1}^{\\infty} \\psi_j Z(t-j) find $\\psi_0, \\psi_1, \\psi_2.$ [ANS] [ANS] [ANS]\nPart (f) After de-trending the data, a environmental scientist fits the model to the daily noon temparatures (in Celsius) in Whitehorse over sixty consecutive days. A portion of the data $x(t)$ and the fitted values $\\hat{x}(t)$ are given below (to 2 decimal places):\n\\begin{array}{c|ccc} t & x(t) & \\hat{x}(t) \\\\ \\hline 2 &-0.13 & 0.12 \\\\ \\vdots & \\vdots & \\vdots \\\\ 57 &-2.00 & 1.02 \\\\ 58 &-0.51 & 1.91 \\\\ 59 &-1.04 &-0.85 \\\\ 60 &-5.22 &-2.75 \\end{array} Use the model and the above information to forecast the next two values of the time series, showing your working clearly. [ANS] [ANS]\nPart (g) Given that the estimate of from the fitted model is 1.01, provide approximate 95\\% confidence intervals for each of your forecasts in (f). [ANS]", "answer_v2": [ "A", "0", "0", "-0.25", "1", "-0.5", "0.75", "-0.33", "-2.64", "0" ], "answer_type_v2": [ "MCS", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [ "A", "B", "C" ], [], [], [], [], [], [], [], [], [] ], "problem_v3": "This question concerns the model X(t)=0.5X(t-1)+Z(t)-Z(t-1)+Z(t-2) in which $Z(t)$ is white noise whith mean 0 and variance $\\sigma^2$. It is assumed that $E(X(t))=0$ for all t.\nPart (a) Is this model stationary? [ANS] A. Impossible to determine. B. Yes. C. No.\nPart (b) Show the following: E(X(t)Z(t))=\\sigma^2 E(X(t)Z(t-1))=-0.5\\sigma^2 E(X(t)Z(t-2))=0.75\\sigma^2 [ANS]\nPart (c) With $\\gamma(k)=E(X(t)X(t-k)),$ show that \\gamma(0)=0.5\\gamma(1)+2.25 \\sigma^2, \\gamma(1)=0.5\\gamma(0)-1.5\\sigma^2. [ANS]\nPart (d) Find the variance of $X(t)$ in terms of $\\sigma^2$, and the autocorrelation function of $X(t)$ at lag 1. [ANS]\nPart (e) Writing the model in the form X(t)=\\sum_{j=1}^{\\infty} \\psi_j Z(t-j) find $\\psi_0, \\psi_1, \\psi_2.$ [ANS] [ANS] [ANS]\nPart (f) After de-trending the data, a environmental scientist fits the model to the daily noon temparatures (in Celsius) in Whitehorse over sixty consecutive days. A portion of the data $x(t)$ and the fitted values $\\hat{x}(t)$ are given below (to 2 decimal places):\n\\begin{array}{c|ccc} t & x(t) & \\hat{x}(t) \\\\ \\hline 2 &-0.13 & 0.12 \\\\ \\vdots & \\vdots & \\vdots \\\\ 57 &-2.00 & 1.02 \\\\ 58 &-0.51 & 1.91 \\\\ 59 &-1.04 &-0.85 \\\\ 60 &-5.22 &-2.75 \\end{array} Use the model and the above information to forecast the next two values of the time series, showing your working clearly. [ANS] [ANS]\nPart (g) Given that the estimate of from the fitted model is 1.01, provide approximate 95\\% confidence intervals for each of your forecasts in (f). [ANS]", "answer_v3": [ "B", "0", "0", "-0.25", "1", "-0.5", "0.75", "-0.33", "-2.64", "0" ], "answer_type_v3": [ "MCS", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [ "A", "B", "C" ], [], [], [], [], [], [], [], [], [] ] }, { "id": "Statistics_0396", "subject": "Statistics", "topic": "Point estimation", "subtopic": "Consistency", "level": "3", "keywords": [ "statistics", "Introduction to Estimation" ], "problem_v1": "If there are two unbiased estimators of a population parameter, the one whose variance is smaller is said to be: [ANS] A. relatively efficient B. relatively unbiased C. consistent D. a biased estimator\nWhich of the following statements is (are) true? [ANS] A. The sample mean is relatively more efficient than the sample median B. The sample variance is relatively more efficient than the sample standard deviation C. The sample median is relatively more efficient than the sample mean D. All of the above statements are true", "answer_v1": [ "A", "A" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "If there are two unbiased estimators of a population parameter, the one whose variance is smaller is said to be: [ANS] A. consistent B. relatively unbiased C. a biased estimator D. relatively efficient\nAn estimator is said to be consistent if [ANS] A. it is an unbiased estimator and the difference between the estimator and the population parameter grows smaller as the sample size grows larger B. the expected value of the estimator is known and positive C. it is an unbiased estimator D. the variance of the estimator is close to one", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "If there are two unbiased estimators of a population parameter, the one whose variance is smaller is said to be: [ANS] A. consistent B. relatively efficient C. relatively unbiased D. a biased estimator\nAs its name suggests, the objective of estimation is to determine the approximate value of [ANS] A. a population parameter on the basis of a sample statistic B. the sample mean C. a sample statistic on the basis of a population parameter D. the sample variance", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0397", "subject": "Statistics", "topic": "Multiple regression", "subtopic": "Nonlinear regression", "level": "4", "keywords": [ "Correlation", "Multiple", "Regression" ], "problem_v1": "In some cases, the best-fitting multiple regression equation is of the form $\\hat{y}=b_0+b_1x+b_2x^2+b_3x^3.$ The graph of such an equation is called a cubic. Using the data set given below, and letting $x_1=x,$ $x_2=x^2,$ and $x_3=x^3,$ find the multiple regression equation for the cubic that best fits the given data. \\begin{array}{c|ccccccc} x &-8 &-6 &-3 & 1 & 2 & 5 & 9 \\cr \\hline y & 31.8 & 17.8 & 6.6 & 1.1 & 0.2 &-3.8 &-19.2 \\end{array} The equation is $\\hat{y}=$ [ANS] $+$ [ANS] $x+$ [ANS] $x^2+$ [ANS] $x^3.$", "answer_v1": [ "1.89848336912669", "-0.941315611416559", "0.11098102262374", "-0.0296778570473115" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "In some cases, the best-fitting multiple regression equation is of the form $\\hat{y}=b_0+b_1x+b_2x^2+b_3x^3.$ The graph of such an equation is called a cubic. Using the data set given below, and letting $x_1=x,$ $x_2=x^2,$ and $x_3=x^3,$ find the multiple regression equation for the cubic that best fits the given data. \\begin{array}{c|ccccccc} x &-10 &-5 &-4 & 0 & 4 & 5 & 8 \\cr \\hline y & 48.9 & 13.8 & 9.6 &-3.7 &-15.2 &-16.5 &-25.7 \\end{array} The equation is $\\hat{y}=$ [ANS] $+$ [ANS] $x+$ [ANS] $x^2+$ [ANS] $x^3.$", "answer_v2": [ "-4.32413650998259", "-2.76797816705935", "0.119191385801636", "-0.0135490129683546" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "In some cases, the best-fitting multiple regression equation is of the form $\\hat{y}=b_0+b_1x+b_2x^2+b_3x^3.$ The graph of such an equation is called a cubic. Using the data set given below, and letting $x_1=x,$ $x_2=x^2,$ and $x_3=x^3,$ find the multiple regression equation for the cubic that best fits the given data. \\begin{array}{c|ccccccc} x &-10 &-6 &-4 & 0 & 2 & 6 & 10 \\cr \\hline y &-10.4 &-7.4 &-3.3 & 9.9 & 15.3 & 19.5 & 8.3 \\end{array} The equation is $\\hat{y}=$ [ANS] $+$ [ANS] $x+$ [ANS] $x^2+$ [ANS] $x^3.$", "answer_v3": [ "9.75440444708697", "3.04135426645859", "-0.107807046313019", "-0.0210988379769622" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Statistics_0398", "subject": "Statistics", "topic": "Multiple regression", "subtopic": "Nonlinear regression", "level": "4", "keywords": [ "statistics", "principal component analysis", "covariance" ], "problem_v1": "This is a question on interpretation of principal component analysis when the input could be a correlation matrix (standardized variables) or a covariance matrix (no scaling, in which case, pay attention to the units and range of the variables). A data set consists of women national track records in the year 1984 for over 50 countries: variables are record times for 100m, 200m, 400m in seconds, and for 800m, 1500m, 3000m, marathon (42195m) in minutes. Your data variables are:\nm100=c(11.76, 12.25, 12.23, 12.25, 11.01, 11.79, 12.03, 11.43, 11.85, 11.96, 11.58, 11.84, 12.14, 11.73, 11.09, 11.2, 11.89, 12.9, 11.45, 11.96, 11.25, 11.16, 11, 11.46, 11.55, 11.98, 12.74, 11, 11.29, 11.76, 11.81, 11.73, 11.41, 11.22, 11.75, 11.44, 11.6, 11.42, 12, 11.61, 11.15, 11.95, 11.31, 11.8, 11.79, 11.45, 12.3, 11.45, 11.13, 11.95, 11.43, 11.13, 10.79, 10.81)\nm200=c(23.54, 25.78, 24.21, 25.07, 22.39, 24.08, 24.96, 23.09, 24.24, 24.49, 23.31, 24.54, 24.47, 23.88, 21.97, 22.35, 23.62, 27.1, 23.06, 24.6, 22.81, 22.82, 22.13, 23.05, 23.13, 24.44, 25.85, 22.25, 23, 25.08, 24.22, 24, 23.04, 22.62, 24.46, 23.46, 24, 23.52, 24.52, 22.94, 22.59, 24.41, 23.17, 23.98, 24.05, 23.57, 25, 23.31, 22.21, 24.28, 23.51, 22.39, 21.83, 21.71)\nm400=c(54.6, 51.2, 55.09, 56.96, 49.75, 54.93, 56.1, 50.62, 55.34, 55.7, 53.12, 56.09, 55, 52.7, 47.99, 51.08, 53.76, 60.4, 51.5, 58.25, 52.38, 51.79, 50.46, 53.3, 51.6, 56.45, 58.73, 50.06, 52.01, 58.1, 54.3, 53.73, 52, 52.5, 55.8, 51.2, 53.26, 53.6, 54.9, 54.5, 51.73, 54.97, 52.8, 53.59, 56.05, 54.9, 55.08, 53.11, 49.29, 53.6, 53.24, 50.14, 50.62, 48.16)\nm800=c(2.19, 1.97, 2.19, 2.24, 1.95, 2.07, 2.07, 1.99, 2.22, 2.15, 2.03, 2.28, 2.18, 2, 1.89, 1.98, 2.04, 2.3, 2.01, 2.21, 1.99, 2.02, 1.98, 2.16, 2.02, 2.15, 2.33, 2, 1.96, 2.27, 2.09, 2.09, 2, 2.1, 2.2, 1.92, 2.11, 2.03, 2.05, 2.15, 2, 2.08, 2.1, 2.05, 2.24, 2.1, 2.12, 2.02, 1.95, 2.1, 2.05, 2.03, 1.96, 1.93)\nm1500=c(4.6, 4.25, 4.69, 4.84, 4.03, 4.35, 4.38, 4.22, 4.61, 4.42, 4.01, 4.86, 4.45, 4.15, 4.14, 4.13, 4.25, 4.84, 4.14, 4.68, 4.06, 4.12, 4.03, 4.58, 4.18, 4.37, 5.81, 4.06, 3.98, 4.79, 4.16, 4.35, 4.14, 4.38, 4.72, 3.96, 4.35, 4.18, 4.23, 4.43, 4.14, 4.33, 4.49, 4.14, 4.74, 4.25, 4.52, 4.07, 3.99, 4.32, 4.11, 4.1, 3.95, 3.96)\nm3000=c(10.16, 9.35, 10.46, 10.69, 8.59, 9.87, 9.64, 9.34, 10.02, 9.62, 8.53, 10.54, 9.51, 9.2, 8.92, 9.08, 9.59, 11.1, 8.98, 10.43, 9.01, 8.84, 8.62, 9.81, 8.76, 9.38, 13.04, 8.81, 8.63, 10.9, 8.84, 9.2, 8.88, 9.63, 10.28, 8.53, 9.46, 8.71, 9.37, 9.79, 8.98, 9.31, 9.77, 9.02, 9.89, 9.37, 9.94, 8.77, 8.97, 9.98, 8.89, 8.92, 8.5, 8.75)\nmarathon=c(200.37, 179.17, 182.17, 233, 148.53, 182.2, 174.68, 159.37, 201.28, 164.65, 145.48, 215.08, 191.02, 181.05, 158.85, 152.37, 158.53, 233.22, 156.37, 171.8, 152.48, 154.48, 149.72, 169.98, 145.48, 201.08, 306, 149.45, 151.82, 261.13, 151.2, 150.5, 157.85, 177.87, 168.45, 165.45, 165.42, 151.75, 171.38, 178.52, 155.27, 168.48, 168.75, 162.6, 203.88, 160.48, 182.77, 153.42, 160.82, 188.03, 149.38, 154.23, 142.72, 157.68)\ncountry=c('philippi', 'dprkorea', 'malaysia', 'png', 'frg', 'greece', 'luxembou', 'austria', 'indonesi', 'korea', 'norway', 'guatemal', 'burma', 'kenya', 'czech', 'australi', 'mexico', 'cookis', 'hungary', 'costa', 'netherla', 'sweden', 'gbni', 'bermuda', 'nz', 'turkey', 'wsamoa', 'canada', 'italy', 'mauritiu', 'portugal', 'japan', 'belgium', 'taipei', 'thailand', 'rumania', 'columbia', 'denmark', 'chile', 'argentin', 'france', 'china', 'brazil', 'spain', 'domrep', 'israel', 'singapor', 'switzerl', 'poland', 'india', 'ireland', 'finland', 'usa', 'gdr')\nMake a data.frame with track=data.frame(m100,m200,m400,m800,m1500,m3000,marathon)\nPrincipal component analysis is to be applied using both the sample covariance and sample correlation matrix of the data as given, also principal component analysis is applied to the velocity or speed data (with unit of m/s). To convert a record time in seconds to m/s, take the distance in metres and divide by the record time. To convert a record time in minutes to m/s, take the distance in metres and divide by the record time and then divide by 60. For the interpretation questions, you may want to plot the scores of the first two principal components with an abbreviated country name as a plotting symbol.\nPart a) The number of principal components to achieve 90\\% of the variation is [ANS] for the covariance matrix of record times [ANS] for the correlation matrix of record times [ANS] for the covariance matrix of velocities\nPart b) The absolute value of the coefficient of marathon in the first principal component is: [ANS] for the covariance matrix of record times [ANS] for the correlation matrix of record times [ANS] for the covariance matrix of velocities\nPart c) For the covariance matrix of record times, the interpretation of the first principal component (linear combination with most variation) is: (There can be more than one correct answer) [ANS] A. random weighting B. contrast of long and short distances versus medium distances C. contrast of long distances versus short distances D. contrast of marathon distances versus other distances E. marathon F. contrast of long distances versus medium distances G. weighted sum of the variables H. None of the above\nPart d) For the correlation matrix of record times, the interpretation of the first principal component is: (There can be more than one correct answer) [ANS] A. random weighting B. contrast of marathon distances versus other distances C. contrast of long and short distances versus medium distances D. contrast of long distances versus short distances E. marathon F. contrast of long distances versus medium distances G. weighted sum of the variables H. None of the above\nPart e) For the covariance matrix of velocities, the interpretation of the second principal component is: (There can be more than one correct answer) [ANS] A. marathon B. contrast of long distances versus medium distances C. contrast of marathon distances versus other distances D. weighted sum of the variables E. random weighting F. contrast of long distances versus short distances G. contrast of long and short distances versus medium distances H. None of the above\nPart f) Why is it appropriate to do principal component analysis on the covariance matrix of the velocities but not the covariance matrix of the actual record times. For the velocity data, which of the following are relevant for principal component analysis? (There can be more than one correct answer) [ANS] A. the median values of the 7 variables are about the same B. the variables have the same order of magnitude C. the values are decreasing in most rows D. units of the variables are the same E. the standard deviations of the 7 variables are about the same F. the mean values of the 7 variables are about the same G. None of the above", "answer_v1": [ "1", "2", "2", "0.997447338985308", "0.368669523221834", "0.53968207819344", "E", "G", "F", "BD" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "MCS", "MCS", "MCS", "MCM" ], "options_v1": [ [], [], [], [], [], [], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v2": "This is a question on interpretation of principal component analysis when the input could be a correlation matrix (standardized variables) or a covariance matrix (no scaling, in which case, pay attention to the units and range of the variables). A data set consists of women national track records in the year 1984 for over 50 countries: variables are record times for 100m, 200m, 400m in seconds, and for 800m, 1500m, 3000m, marathon (42195m) in minutes. Your data variables are:\nm100=c(11.46, 11.98, 11.95, 11, 10.79, 11.41, 11.42, 11.84, 12.23, 11.96, 11.55, 11.73, 11.8, 11.79, 11.09, 11.85, 11.6, 11.95, 11.73, 11.58, 11.45, 11.79, 12.14, 12.25, 11.01, 11.45, 11.2, 11.76, 11.13, 11.43, 11.06, 11.76, 11.44, 11.61, 11, 11.81, 11.31, 11.75, 12.9, 12.3, 11.96, 11.15, 11.43, 12, 10.81, 11.16, 11.89, 12.74, 11.29, 11.13, 12.03, 12.25, 11.22, 11.45)\nm200=c(23.05, 24.44, 24.41, 22.13, 21.83, 23.04, 23.52, 24.54, 24.21, 24.6, 23.13, 23.88, 23.98, 24.08, 21.97, 24.24, 24, 24.28, 24, 23.31, 23.06, 24.05, 24.47, 25.78, 22.39, 23.57, 22.35, 25.08, 22.39, 23.09, 22.19, 23.54, 23.46, 22.94, 22.25, 24.22, 23.17, 24.46, 27.1, 25, 24.49, 22.59, 23.51, 24.52, 21.71, 22.82, 23.62, 25.85, 23, 22.21, 24.96, 25.07, 22.62, 23.31)\nm400=c(53.3, 56.45, 54.97, 50.46, 50.62, 52, 53.6, 56.09, 55.09, 58.25, 51.6, 52.7, 53.59, 54.93, 47.99, 55.34, 53.26, 53.6, 53.73, 53.12, 51.5, 56.05, 55, 51.2, 49.75, 54.9, 51.08, 58.1, 50.14, 50.62, 49.19, 54.6, 51.2, 54.5, 50.06, 54.3, 52.8, 55.8, 60.4, 55.08, 55.7, 51.73, 53.24, 54.9, 48.16, 51.79, 53.76, 58.73, 52.01, 49.29, 56.1, 56.96, 52.5, 53.11)\nm800=c(2.16, 2.15, 2.08, 1.98, 1.96, 2, 2.03, 2.28, 2.19, 2.21, 2.02, 2, 2.05, 2.07, 1.89, 2.22, 2.11, 2.1, 2.09, 2.03, 2.01, 2.24, 2.18, 1.97, 1.95, 2.1, 1.98, 2.27, 2.03, 1.99, 1.89, 2.19, 1.92, 2.15, 2, 2.09, 2.1, 2.2, 2.3, 2.12, 2.15, 2, 2.05, 2.05, 1.93, 2.02, 2.04, 2.33, 1.96, 1.95, 2.07, 2.24, 2.1, 2.02)\nm1500=c(4.58, 4.37, 4.33, 4.03, 3.95, 4.14, 4.18, 4.86, 4.69, 4.68, 4.18, 4.15, 4.14, 4.35, 4.14, 4.61, 4.35, 4.32, 4.35, 4.01, 4.14, 4.74, 4.45, 4.25, 4.03, 4.25, 4.13, 4.79, 4.1, 4.22, 3.87, 4.6, 3.96, 4.43, 4.06, 4.16, 4.49, 4.72, 4.84, 4.52, 4.42, 4.14, 4.11, 4.23, 3.96, 4.12, 4.25, 5.81, 3.98, 3.99, 4.38, 4.84, 4.38, 4.07)\nm3000=c(9.81, 9.38, 9.31, 8.62, 8.5, 8.88, 8.71, 10.54, 10.46, 10.43, 8.76, 9.2, 9.02, 9.87, 8.92, 10.02, 9.46, 9.98, 9.2, 8.53, 8.98, 9.89, 9.51, 9.35, 8.59, 9.37, 9.08, 10.9, 8.92, 9.34, 8.45, 10.16, 8.53, 9.79, 8.81, 8.84, 9.77, 10.28, 11.1, 9.94, 9.62, 8.98, 8.89, 9.37, 8.75, 8.84, 9.59, 13.04, 8.63, 8.97, 9.64, 10.69, 9.63, 8.77)\nmarathon=c(169.98, 201.08, 168.48, 149.72, 142.72, 157.85, 151.75, 215.08, 182.17, 171.8, 145.48, 181.05, 162.6, 182.2, 158.85, 201.28, 165.42, 188.03, 150.5, 145.48, 156.37, 203.88, 191.02, 179.17, 148.53, 160.48, 152.37, 261.13, 154.23, 159.37, 151.22, 200.37, 165.45, 178.52, 149.45, 151.2, 168.75, 168.45, 233.22, 182.77, 164.65, 155.27, 149.38, 171.38, 157.68, 154.48, 158.53, 306, 151.82, 160.82, 174.68, 233, 177.87, 153.42)\ncountry=c('bermuda', 'turkey', 'china', 'gbni', 'usa', 'belgium', 'denmark', 'guatemal', 'malaysia', 'costa', 'nz', 'kenya', 'spain', 'greece', 'czech', 'indonesi', 'columbia', 'india', 'japan', 'norway', 'hungary', 'domrep', 'burma', 'dprkorea', 'frg', 'israel', 'australi', 'mauritiu', 'finland', 'austria', 'ussr', 'philippi', 'rumania', 'argentin', 'canada', 'portugal', 'brazil', 'thailand', 'cookis', 'singapor', 'korea', 'france', 'ireland', 'chile', 'gdr', 'sweden', 'mexico', 'wsamoa', 'italy', 'poland', 'luxembou', 'png', 'taipei', 'switzerl')\nMake a data.frame with track=data.frame(m100,m200,m400,m800,m1500,m3000,marathon)\nPrincipal component analysis is to be applied using both the sample covariance and sample correlation matrix of the data as given, also principal component analysis is applied to the velocity or speed data (with unit of m/s). To convert a record time in seconds to m/s, take the distance in metres and divide by the record time. To convert a record time in minutes to m/s, take the distance in metres and divide by the record time and then divide by 60. For the interpretation questions, you may want to plot the scores of the first two principal components with an abbreviated country name as a plotting symbol.\nPart a) The number of principal components to achieve 90\\% of the variation is [ANS] for the covariance matrix of record times [ANS] for the correlation matrix of record times [ANS] for the covariance matrix of velocities\nPart b) The absolute value of the coefficient of marathon in the first principal component is: [ANS] for the covariance matrix of record times [ANS] for the correlation matrix of record times [ANS] for the covariance matrix of velocities\nPart c) For the covariance matrix of record times, the interpretation of the first principal component (linear combination with most variation) is: (There can be more than one correct answer) [ANS] A. weighted sum of the variables B. contrast of long distances versus short distances C. contrast of long and short distances versus medium distances D. contrast of marathon distances versus other distances E. contrast of long distances versus medium distances F. marathon G. random weighting H. None of the above\nPart d) For the correlation matrix of record times, the interpretation of the first principal component is: (There can be more than one correct answer) [ANS] A. weighted sum of the variables B. marathon C. contrast of marathon distances versus other distances D. random weighting E. contrast of long and short distances versus medium distances F. contrast of long distances versus medium distances G. contrast of long distances versus short distances H. None of the above\nPart e) For the covariance matrix of velocities, the interpretation of the second principal component is: (There can be more than one correct answer) [ANS] A. contrast of long distances versus short distances B. marathon C. contrast of long distances versus medium distances D. contrast of marathon distances versus other distances E. weighted sum of the variables F. random weighting G. contrast of long and short distances versus medium distances H. None of the above\nPart f) Why is it appropriate to do principal component analysis on the covariance matrix of the velocities but not the covariance matrix of the actual record times. For the velocity data, which of the following are relevant for principal component analysis? (There can be more than one correct answer) [ANS] A. the variables have the same order of magnitude B. the median values of the 7 variables are about the same C. units of the variables are the same D. the mean values of the 7 variables are about the same E. the standard deviations of the 7 variables are about the same F. the values are decreasing in most rows G. None of the above", "answer_v2": [ "1", "2", "2", "0.997345669385532", "0.366795612994789", "0.530209351981312", "F", "A", "A", "AC" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "MCS", "MCS", "MCS", "MCM" ], "options_v2": [ [], [], [], [], [], [], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v3": "This is a question on interpretation of principal component analysis when the input could be a correlation matrix (standardized variables) or a covariance matrix (no scaling, in which case, pay attention to the units and range of the variables). A data set consists of women national track records in the year 1984 for over 50 countries: variables are record times for 100m, 200m, 400m in seconds, and for 800m, 1500m, 3000m, marathon (42195m) in minutes. Your data variables are:\nm100=c(11.15, 12.03, 11.13, 11.96, 11.42, 11.84, 12.3, 11.75, 11.22, 10.81, 11.45, 11.31, 11.09, 11.25, 12.74, 11.8, 11.58, 11.79, 11.29, 11.13, 11.44, 11.55, 11.76, 10.79, 11.45, 11, 11.01, 11.81, 11.06, 11.95, 11.2, 11, 11.95, 12, 11.61, 11.16, 11.41, 11.43, 11.79, 12.23, 11.43, 12.14, 11.73, 11.45, 11.89, 11.6, 12.9, 12.25, 11.73, 11.98, 11.76, 11.46, 12.25, 11.96)\nm200=c(22.59, 24.96, 22.39, 24.49, 23.52, 24.54, 25, 24.46, 22.62, 21.71, 23.06, 23.17, 21.97, 22.81, 25.85, 23.98, 23.31, 24.08, 23, 22.21, 23.46, 23.13, 25.08, 21.83, 23.31, 22.25, 22.39, 24.22, 22.19, 24.28, 22.35, 22.13, 24.41, 24.52, 22.94, 22.82, 23.04, 23.09, 24.05, 24.21, 23.51, 24.47, 23.88, 23.57, 23.62, 24, 27.1, 25.07, 24, 24.44, 23.54, 23.05, 25.78, 24.6)\nm400=c(51.73, 56.1, 50.14, 55.7, 53.6, 56.09, 55.08, 55.8, 52.5, 48.16, 51.5, 52.8, 47.99, 52.38, 58.73, 53.59, 53.12, 54.93, 52.01, 49.29, 51.2, 51.6, 58.1, 50.62, 53.11, 50.06, 49.75, 54.3, 49.19, 53.6, 51.08, 50.46, 54.97, 54.9, 54.5, 51.79, 52, 50.62, 56.05, 55.09, 53.24, 55, 52.7, 54.9, 53.76, 53.26, 60.4, 56.96, 53.73, 56.45, 54.6, 53.3, 51.2, 58.25)\nm800=c(2, 2.07, 2.03, 2.15, 2.03, 2.28, 2.12, 2.2, 2.1, 1.93, 2.01, 2.1, 1.89, 1.99, 2.33, 2.05, 2.03, 2.07, 1.96, 1.95, 1.92, 2.02, 2.27, 1.96, 2.02, 2, 1.95, 2.09, 1.89, 2.1, 1.98, 1.98, 2.08, 2.05, 2.15, 2.02, 2, 1.99, 2.24, 2.19, 2.05, 2.18, 2, 2.1, 2.04, 2.11, 2.3, 2.24, 2.09, 2.15, 2.19, 2.16, 1.97, 2.21)\nm1500=c(4.14, 4.38, 4.1, 4.42, 4.18, 4.86, 4.52, 4.72, 4.38, 3.96, 4.14, 4.49, 4.14, 4.06, 5.81, 4.14, 4.01, 4.35, 3.98, 3.99, 3.96, 4.18, 4.79, 3.95, 4.07, 4.06, 4.03, 4.16, 3.87, 4.32, 4.13, 4.03, 4.33, 4.23, 4.43, 4.12, 4.14, 4.22, 4.74, 4.69, 4.11, 4.45, 4.15, 4.25, 4.25, 4.35, 4.84, 4.84, 4.35, 4.37, 4.6, 4.58, 4.25, 4.68)\nm3000=c(8.98, 9.64, 8.92, 9.62, 8.71, 10.54, 9.94, 10.28, 9.63, 8.75, 8.98, 9.77, 8.92, 9.01, 13.04, 9.02, 8.53, 9.87, 8.63, 8.97, 8.53, 8.76, 10.9, 8.5, 8.77, 8.81, 8.59, 8.84, 8.45, 9.98, 9.08, 8.62, 9.31, 9.37, 9.79, 8.84, 8.88, 9.34, 9.89, 10.46, 8.89, 9.51, 9.2, 9.37, 9.59, 9.46, 11.1, 10.69, 9.2, 9.38, 10.16, 9.81, 9.35, 10.43)\nmarathon=c(155.27, 174.68, 154.23, 164.65, 151.75, 215.08, 182.77, 168.45, 177.87, 157.68, 156.37, 168.75, 158.85, 152.48, 306, 162.6, 145.48, 182.2, 151.82, 160.82, 165.45, 145.48, 261.13, 142.72, 153.42, 149.45, 148.53, 151.2, 151.22, 188.03, 152.37, 149.72, 168.48, 171.38, 178.52, 154.48, 157.85, 159.37, 203.88, 182.17, 149.38, 191.02, 181.05, 160.48, 158.53, 165.42, 233.22, 233, 150.5, 201.08, 200.37, 169.98, 179.17, 171.8)\ncountry=c('france', 'luxembou', 'finland', 'korea', 'denmark', 'guatemal', 'singapor', 'thailand', 'taipei', 'gdr', 'hungary', 'brazil', 'czech', 'netherla', 'wsamoa', 'spain', 'norway', 'greece', 'italy', 'poland', 'rumania', 'nz', 'mauritiu', 'usa', 'switzerl', 'canada', 'frg', 'portugal', 'ussr', 'india', 'australi', 'gbni', 'china', 'chile', 'argentin', 'sweden', 'belgium', 'austria', 'domrep', 'malaysia', 'ireland', 'burma', 'kenya', 'israel', 'mexico', 'columbia', 'cookis', 'png', 'japan', 'turkey', 'philippi', 'bermuda', 'dprkorea', 'costa')\nMake a data.frame with track=data.frame(m100,m200,m400,m800,m1500,m3000,marathon)\nPrincipal component analysis is to be applied using both the sample covariance and sample correlation matrix of the data as given, also principal component analysis is applied to the velocity or speed data (with unit of m/s). To convert a record time in seconds to m/s, take the distance in metres and divide by the record time. To convert a record time in minutes to m/s, take the distance in metres and divide by the record time and then divide by 60. For the interpretation questions, you may want to plot the scores of the first two principal components with an abbreviated country name as a plotting symbol.\nPart a) The number of principal components to achieve 90\\% of the variation is [ANS] for the covariance matrix of record times [ANS] for the correlation matrix of record times [ANS] for the covariance matrix of velocities\nPart b) The absolute value of the coefficient of marathon in the first principal component is: [ANS] for the covariance matrix of record times [ANS] for the correlation matrix of record times [ANS] for the covariance matrix of velocities\nPart c) For the covariance matrix of record times, the interpretation of the first principal component (linear combination with most variation) is: (There can be more than one correct answer) [ANS] A. contrast of long distances versus short distances B. contrast of marathon distances versus other distances C. random weighting D. weighted sum of the variables E. contrast of long distances versus medium distances F. marathon G. contrast of long and short distances versus medium distances H. None of the above\nPart d) For the correlation matrix of record times, the interpretation of the first principal component is: (There can be more than one correct answer) [ANS] A. contrast of marathon distances versus other distances B. marathon C. contrast of long distances versus short distances D. random weighting E. weighted sum of the variables F. contrast of long and short distances versus medium distances G. contrast of long distances versus medium distances H. None of the above\nPart e) For the covariance matrix of velocities, the interpretation of the second principal component is: (There can be more than one correct answer) [ANS] A. weighted sum of the variables B. contrast of long and short distances versus medium distances C. contrast of long distances versus medium distances D. marathon E. random weighting F. contrast of long distances versus short distances G. contrast of marathon distances versus other distances H. None of the above\nPart f) Why is it appropriate to do principal component analysis on the covariance matrix of the velocities but not the covariance matrix of the actual record times. For the velocity data, which of the following are relevant for principal component analysis? (There can be more than one correct answer) [ANS] A. units of the variables are the same B. the standard deviations of the 7 variables are about the same C. the median values of the 7 variables are about the same D. the variables have the same order of magnitude E. the mean values of the 7 variables are about the same F. the values are decreasing in most rows G. None of the above", "answer_v3": [ "1", "2", "2", "0.997352731812442", "0.36649391462826", "0.5281977738486", "F", "E", "F", "AD" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "MCS", "MCS", "MCS", "MCM" ], "options_v3": [ [], [], [], [], [], [], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G" ] ] }, { "id": "Statistics_0399", "subject": "Statistics", "topic": "Multiple regression", "subtopic": "Parameter estimates", "level": "4", "keywords": [ "Regression", "Multiple" ], "problem_v1": "Find the multiple regression equation for the data given below. \\begin{array}{r|rrrrr} x_1 &-2 &-1 & 0 & 2 & 2 \\cr \\hline x_2 &-4 & 2 & 0 &-2 & 2 \\cr \\hline y & 6 &-13 & 0 & 18 & 1 \\end{array} The equation is $\\hat{y}=$ [ANS] $+$ [ANS] $x_1+$ [ANS] $x_2.$", "answer_v1": [ "-0.203125", "4.890625", "-4.0625" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the multiple regression equation for the data given below. \\begin{array}{r|rrrrr} x_1 &-3 &-1 &-1 & 1 & 3 \\cr \\hline x_2 &-4 & 1 &-1 &-1 & 2 \\cr \\hline y &-2 &-7 &-1 & 10 & 10 \\end{array} The equation is $\\hat{y}=$ [ANS] $+$ [ANS] $x_1+$ [ANS] $x_2.$", "answer_v2": [ "1.13953488372093", "5.18604651162791", "-3.16279069767442" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the multiple regression equation for the data given below. \\begin{array}{r|rrrrr} x_1 &-3 &-1 &-1 & 2 & 2 \\cr \\hline x_2 &-4 & 2 & 0 &-1 & 2 \\cr \\hline y & 1 &-14 &-6 & 9 &-4 \\end{array} The equation is $\\hat{y}=$ [ANS] $+$ [ANS] $x_1+$ [ANS] $x_2.$", "answer_v3": [ "-2.82935779816514", "3.7045871559633", "-3.85137614678899" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Statistics_0400", "subject": "Statistics", "topic": "Multiple regression", "subtopic": "Parameter estimates", "level": "4", "keywords": [], "problem_v1": "This is a question on interpretation of regression equations which have categorical explanatory variables where slopes on non-categorical variables do not depend on the category. This model assumes that hyperplanes are parallel for different categories, and the regression coefficients for the binary dummy variables can be used to determine distances between hyperplanes for different categories.\nFor your subset of the cereal data set, the response variable is: calories=c(140, 160, 70, 110, 110, 100, 120, 110, 100, 90, 110, 120, 90, 110, 100, 70, 100, 110, 50, 100, 100, 110, 110, 110, 150, 90, 110, 110, 120, 100, 110, 90, 120, 90, 120, 90, 90, 150, 140, 110, 110, 120, 80)\nThe explanatory variables are: (i) protein=c(3, 3, 4, 2, 2, 3, 3, 3, 2, 3, 2, 3, 3, 1, 3, 4, 3, 6, 4, 2, 3, 2, 1, 1, 4, 2, 3, 2, 3, 2, 2, 3, 3, 2, 3, 3, 2, 4, 3, 2, 1, 2, 2) (ii) fat=c(1, 2, 1, 0, 1, 1, 1, 3, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 3, 1, 0, 0, 0, 0, 1, 0, 1, 0, 3, 0, 0, 3, 2, 0, 0, 1, 0) (iii) fiber=c(2, 3, 10, 1, 1, 3, 6, 4, 0, 3, 1, 5, 5, 1, 1, 9, 3, 1, 14, 1, 3, 0, 0, 0, 3, 4, 3, 1, 5, 1, 1, 3, 5, 2, 3, 4, 3, 3, 3, 0, 0, 0, 3) (iv) carbo=c(20, 17, 5, 21, 11, 15, 11, 10, 11, 18, 9, 12, 13, 14, 20, 7, 14, 16, 8, 21, 17, 22, 13, 14, 16, 15, 17, 11, 14, 18, 17, 20, 14, 15, 13, 19, 15, 16, 21, 22, 23, 15, 16) (v) sugars=c(9, 13, 6, 3, 13, 5, 14, 7, 15, 2, 15, 10, 5, 11, 3, 5, 7, 3, 0, 2, 3, 3, 12, 11, 11, 6, 3, 14, 12, 5, 6, 0, 12, 6, 4, 0, 5, 11, 7, 3, 2, 9, 0) (vi) mfr=c('K', 'K', 'N', 'K', 'K', 'P', 'P', 'K', 'P', 'K', 'K', 'P', 'P', 'K', 'K', 'K', 'K', 'K', 'K', 'K', 'R', 'K', 'P', 'P', 'R', 'R', 'P', 'K', 'K', 'R', 'K', 'N', 'K', 'K', 'P', 'N', 'N', 'R', 'K', 'R', 'R', 'K', 'N')\nYou are to fit a multiple regression model with the response variable 'calories' and 6 explanatory variables protein, fat, fiber, carbo, sugars, mfr. After you have copied the above R vectors into your R session, you can get a dataframe with\ncereal=data.frame(cbind(calories,protein,fat,fiber,carbo,sugars)) cereal\\$ mfr=mfr\nPlease use 3 decimal places for the answers below which are not integer-valued.\nFor the regression being requested, you should find the most or all of the coefficients for protein,fat,fiber,carbo,sugars to be statistically significant. Some of the manufacturers might be significantly different from others but not all pairs of manufacturers are significantly different from each other.\nTo answer the parts\n(a) and (b) below, two separate regressions could be done (with 2 different manufactors as the baseline categories). If you want to challenge yourself to answer them both based on one application of lm(), you need to use the cov.unscaled component of the summary of an lm object.\nPart a) The estimate of the signed distance of the hyperplane for manufacturer K relative to P is [ANS] and its SE is [ANS]\nPart b) The estimate of the signed distance of the hyperplane for manufacturer N relative to R is [ANS] and its SE is [ANS]\nPart c) What is the adjusted $R^2$? [ANS]\nPart d) What is the residual SD (residual SE in R)? [ANS]\nPart e) If interaction of mfr and sugars (mfr:sugars) were added to the lm statement, how many betas would be in the regression equation? [ANS]", "answer_v1": [ "-2.26939", "2.31461", "3.15118", "3.67596", "0.930124", "5.60856", "12" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "This is a question on interpretation of regression equations which have categorical explanatory variables where slopes on non-categorical variables do not depend on the category. This model assumes that hyperplanes are parallel for different categories, and the regression coefficients for the binary dummy variables can be used to determine distances between hyperplanes for different categories.\nFor your subset of the cereal data set, the response variable is: calories=c(110, 100, 110, 110, 110, 110, 110, 110, 100, 110, 100, 70, 110, 110, 110, 110, 120, 100, 120, 110, 110, 110, 110, 110, 120, 110, 110, 110, 50, 150, 120, 140, 110, 100, 120, 100, 100, 130, 110, 100, 110, 100, 110, 120, 140, 100, 90, 100, 90, 110, 110, 120, 90, 110, 160, 150, 100, 110, 130, 90)\nThe explanatory variables are: (i) protein=c(1, 2, 2, 2, 2, 1, 3, 2, 3, 3, 3, 4, 1, 3, 6, 2, 3, 2, 3, 2, 3, 1, 1, 2, 1, 1, 2, 2, 4, 4, 2, 3, 2, 3, 3, 3, 2, 3, 1, 3, 2, 3, 6, 3, 3, 3, 2, 2, 3, 2, 1, 3, 2, 1, 3, 4, 2, 2, 3, 3) (ii) fat=c(0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 0, 1, 0, 0, 2, 1, 2, 0, 1, 2, 2, 1, 0, 1, 3, 1, 0, 1, 0, 3, 1, 1, 1, 2, 0, 0, 0, 2, 1, 1, 0, 1, 0, 1, 2, 1, 1, 1, 0, 1, 0, 3, 0, 1, 2, 3, 0, 1, 2, 0) (iii) fiber=c(0, 2, 0, 0, 0, 0, 1.5, 1, 3, 4, 3, 9, 0, 3, 2, 1, 5, 1, 6, 1.5, 2, 0, 1, 0, 0, 0, 1, 1, 14, 3, 0, 2, 1, 2.5, 5, 1, 1, 2, 0, 3, 0, 3, 1, 5, 3, 3, 4, 2, 3, 0, 1, 3, 2, 0, 3, 3, 0, 1, 1.5, 5) (iv) carbo=c(23, 11, 21, 22, 21, 12, 11.5, 21, 16, 10, 14, 7, 14, 17, 17, 9, 12, 21, 11, 10.5, 13, 12, 14, 12, 13, 13, 11, 17, 8, 16, 15, 20, 16, 10.5, 14, 20, 18, 18, 15, 17, 22, 17, 16, 14, 21, 15, 15, 15, 18, 21, 13, 13, 15, 13, 17, 16, 11, 11, 13.5, 13) (v) sugars=c(2, 10, 3, 3, 3, 13, 10, 3, 3, 7, 7, 5, 11, 3, 1, 15, 10, 2, 14, 10, 7, 13, 11, 12, 9, 12, 14, 6, 0, 11, 9, 9, 8, 8, 12, 3, 5, 8, 9, 3, 3, 3, 3, 12, 7, 5, 6, 6, 2, 3, 12, 4, 6, 12, 13, 11, 15, 13, 10, 5) (vi) mfr=c('R', 'G', 'G', 'R', 'G', 'G', 'G', 'K', 'G', 'K', 'K', 'K', 'P', 'P', 'G', 'K', 'P', 'K', 'P', 'G', 'G', 'G', 'K', 'G', 'G', 'P', 'K', 'K', 'K', 'R', 'K', 'K', 'G', 'G', 'K', 'K', 'R', 'G', 'G', 'G', 'K', 'R', 'K', 'K', 'K', 'P', 'R', 'G', 'K', 'G', 'K', 'P', 'K', 'G', 'K', 'R', 'P', 'K', 'G', 'P')\nYou are to fit a multiple regression model with the response variable 'calories' and 6 explanatory variables protein, fat, fiber, carbo, sugars, mfr. After you have copied the above R vectors into your R session, you can get a dataframe with\ncereal=data.frame(cbind(calories,protein,fat,fiber,carbo,sugars)) cereal\\$ mfr=mfr\nPlease use 3 decimal places for the answers below which are not integer-valued.\nFor the regression being requested, you should find the most or all of the coefficients for protein,fat,fiber,carbo,sugars to be statistically significant. Some of the manufacturers might be significantly different from others but not all pairs of manufacturers are significantly different from each other.\nTo answer the parts\n(a) and (b) below, two separate regressions could be done (with 2 different manufactors as the baseline categories). If you want to challenge yourself to answer them both based on one application of lm(), you need to use the cov.unscaled component of the summary of an lm object.\nPart a) The estimate of the signed distance of the hyperplane for manufacturer G relative to P is [ANS] and its SE is [ANS]\nPart b) The estimate of the signed distance of the hyperplane for manufacturer K relative to R is [ANS] and its SE is [ANS]\nPart c) What is the adjusted $R^2$? [ANS]\nPart d) What is the residual SD (residual SE in R)? [ANS]\nPart e) If interaction of mfr and sugars (mfr:sugars) were added to the lm statement, how many betas would be in the regression equation? [ANS]", "answer_v2": [ "-5.28797", "2.16553", "3.62662", "2.32075", "0.9125", "5.02113", "12" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "This is a question on interpretation of regression equations which have categorical explanatory variables where slopes on non-categorical variables do not depend on the category. This model assumes that hyperplanes are parallel for different categories, and the regression coefficients for the binary dummy variables can be used to determine distances between hyperplanes for different categories.\nFor your subset of the cereal data set, the response variable is: calories=c(110, 100, 110, 100, 110, 90, 110, 110, 110, 110, 100, 100, 130, 90, 100, 80, 100, 110, 90, 120, 90, 140, 150, 120, 150, 110, 110, 100, 120, 70, 100, 120, 110, 100, 130, 110, 110, 110, 110, 110, 110, 110)\nThe explanatory variables are: (i) protein=c(6, 3, 3, 3, 1, 3, 2, 1, 1, 2, 2, 3, 3, 2, 2, 2, 2, 1, 3, 3, 3, 3, 4, 3, 4, 2, 3, 3, 3, 4, 3, 1, 3, 2, 3, 2, 2, 2, 1, 1, 2, 1) (ii) fat=c(2, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 2, 1, 0, 0, 1, 0, 0, 1, 0, 1, 3, 3, 3, 1, 2, 2, 2, 1, 1, 3, 0, 0, 2, 2, 1, 1, 1, 1, 1, 1) (iii) fiber=c(2, 3, 1.5, 3, 0, 4, 0, 0, 0, 0, 2, 3, 2, 4, 1, 3, 2, 0, 3, 6, 5, 4, 3, 3, 3, 0, 2, 2.5, 5, 10, 3, 0, 3, 0, 1.5, 1.5, 0, 1, 0, 0, 0, 0) (iv) carbo=c(17, 17, 11.5, 16, 15, 19, 22, 23, 12, 12, 11, 15, 18, 15, 18, 16, 15, 14, 20, 11, 13, 15, 16, 13, 16, 21, 13, 10.5, 12, 5, 17, 13, 17, 11, 13.5, 10.5, 21, 16, 13, 12, 21, 13) (v) sugars=c(1, 3, 10, 3, 9, 0, 3, 2, 13, 12, 10, 5, 8, 6, 5, 0, 6, 11, 0, 14, 5, 14, 11, 4, 11, 3, 7, 8, 10, 6, 3, 9, 3, 15, 10, 10, 3, 8, 12, 13, 3, 12) (vi) mfr=c('G', 'G', 'G', 'G', 'G', 'N', 'R', 'R', 'G', 'G', 'G', 'P', 'G', 'R', 'R', 'N', 'G', 'P', 'N', 'P', 'P', 'G', 'R', 'P', 'R', 'G', 'G', 'G', 'P', 'N', 'R', 'G', 'P', 'P', 'G', 'G', 'G', 'G', 'G', 'G', 'G', 'P')\nYou are to fit a multiple regression model with the response variable 'calories' and 6 explanatory variables protein, fat, fiber, carbo, sugars, mfr. After you have copied the above R vectors into your R session, you can get a dataframe with\ncereal=data.frame(cbind(calories,protein,fat,fiber,carbo,sugars)) cereal\\$ mfr=mfr\nPlease use 3 decimal places for the answers below which are not integer-valued.\nFor the regression being requested, you should find the most or all of the coefficients for protein,fat,fiber,carbo,sugars to be statistically significant. Some of the manufacturers might be significantly different from others but not all pairs of manufacturers are significantly different from each other.\nTo answer the parts\n(a) and (b) below, two separate regressions could be done (with 2 different manufactors as the baseline categories). If you want to challenge yourself to answer them both based on one application of lm(), you need to use the cov.unscaled component of the summary of an lm object.\nPart a) The estimate of the signed distance of the hyperplane for manufacturer G relative to P is [ANS] and its SE is [ANS]\nPart b) The estimate of the signed distance of the hyperplane for manufacturer N relative to R is [ANS] and its SE is [ANS]\nPart c) What is the adjusted $R^2$? [ANS]\nPart d) What is the residual SD (residual SE in R)? [ANS]\nPart e) If interaction of mfr and sugars (mfr:sugars) were added to the lm statement, how many betas would be in the regression equation? [ANS]", "answer_v3": [ "-4.27846", "2.30948", "0.47496", "4.18731", "0.893165", "5.16565", "12" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Statistics_0401", "subject": "Statistics", "topic": "Multiple regression", "subtopic": "Parameter estimates", "level": "4", "keywords": [ "statistics", "regression", "covariance matrix of estimator" ], "problem_v1": "You are being given the covariance matrix of a multiple regression with ${\\hat\\beta}_0,{\\hat\\beta}_1,\\ldots,{\\hat\\beta}_5,{\\hat\\beta}_6, \\ldots,{\\hat\\beta}_9$ where $x_1,\\ldots,x_5$ are numerical explanatory variables and $x_6,\\ldots,x_9$ are 4 binary dummy variables for 4 different categories relative to the baseline category. The categorical explanatory variable has 5 categories.\nYou are given that the estimated covariance matrix is: $\\left[\\begin{array}{cccccccccc} 0.33347 &-0.00123 &0.000379 &-0.00635 &-0.007631 &-0.000744 &-0.036518 &-0.054614 &-0.049256 &-0.043163\\cr-0.00123 &1.1\\times 10^{-5} &3\\times 10^{-6} &0 &3.6\\times 10^{-5} &1\\times 10^{-6} &1.9\\times 10^{-5} &0.000121 &0.0002 &9.9\\times 10^{-5}\\cr 0.000379 &3\\times 10^{-6} &1.9\\times 10^{-5} &-3.5\\times 10^{-5} &2.6\\times 10^{-5} &-1.1\\times 10^{-5} &-7.2\\times 10^{-5} &1.2\\times 10^{-5} &3.2\\times 10^{-5} &-1.6\\times 10^{-5}\\cr-0.00635 &0 &-3.5\\times 10^{-5} &0.000281 &7.7\\times 10^{-5} &1.9\\times 10^{-5} &0.000796 &0.000944 &0.000292 &0.000309\\cr-0.007631 &3.6\\times 10^{-5} &2.6\\times 10^{-5} &7.7\\times 10^{-5} &0.00051 &-4\\times 10^{-5} &0.001221 &0.002007 &0.001827 &0.001396\\cr-0.000744 &1\\times 10^{-6} &-1.1\\times 10^{-5} &1.9\\times 10^{-5} &-4\\times 10^{-5} &1.6\\times 10^{-5} &-3.2\\times 10^{-5} &-6.9\\times 10^{-5} &-2.1\\times 10^{-5} &6.8\\times 10^{-5}\\cr-0.036518 &1.9\\times 10^{-5} &-7.2\\times 10^{-5} &0.000796 &0.001221 &-3.2\\times 10^{-5} &0.022795 &0.016794 &0.014729 &0.014576\\cr-0.054614 &0.000121 &1.2\\times 10^{-5} &0.000944 &0.002007 &-6.9\\times 10^{-5} &0.016794 &0.030437 &0.017899 &0.016878\\cr-0.049256 &0.0002 &3.2\\times 10^{-5} &0.000292 &0.001827 &-2.1\\times 10^{-5} &0.014729 &0.017899 &0.025177 &0.017092\\cr-0.043163 &9.9\\times 10^{-5} &-1.6\\times 10^{-5} &0.000309 &0.001396 &6.8\\times 10^{-5} &0.014576 &0.016878 &0.017092 &0.029926 \\end{array}\\right]$ or in R: acov=matrix(c(0.33347,-0.00123, 0.000379,-0.00635,-0.007631,-0.000744,-0.036518,-0.054614,-0.049256,-0.043163,-0.00123, 1.1e-05, 3e-06, 0, 3.6e-05, 1e-06, 1.9e-05, 0.000121, 0.0002, 9.9e-05, 0.000379, 3e-06, 1.9e-05,-3.5e-05, 2.6e-05,-1.1e-05,-7.2e-05, 1.2e-05, 3.2e-05,-1.6e-05,-0.00635, 0,-3.5e-05, 0.000281, 7.7e-05, 1.9e-05, 0.000796, 0.000944, 0.000292, 0.000309,-0.007631, 3.6e-05, 2.6e-05, 7.7e-05, 0.00051,-4e-05, 0.001221, 0.002007, 0.001827, 0.001396,-0.000744, 1e-06,-1.1e-05, 1.9e-05,-4e-05, 1.6e-05,-3.2e-05,-6.9e-05,-2.1e-05, 6.8e-05,-0.036518, 1.9e-05,-7.2e-05, 0.000796, 0.001221,-3.2e-05, 0.022795, 0.016794, 0.014729, 0.014576,-0.054614, 0.000121, 1.2e-05, 0.000944, 0.002007,-6.9e-05, 0.016794, 0.030437, 0.017899, 0.016878,-0.049256, 0.0002, 3.2e-05, 0.000292, 0.001827,-2.1e-05, 0.014729, 0.017899, 0.025177, 0.017092,-0.043163, 9.9e-05,-1.6e-05, 0.000309, 0.001396, 6.8e-05, 0.014576, 0.016878, 0.017092, 0.029926),10,10)\nPlease use 3 decimal places for the answers below which are not integer-valued.\nPart a) Based on the (estimated) covariance matrix, what is the SE of ${\\hat\\beta}_{9}-{\\hat\\beta}_{8}$? Note that ${\\hat\\beta}_{9}-{\\hat\\beta}_{8}$ corresponds to the estimated distance of the hyperplanes for two different categories where the binary dummy variables are indexed as $x_{9}$ and $x_{8}$. [ANS]\nPart b) Based on the (estimated) covariance matrix, what is the SE of ${\\hat\\beta}_{9}+{\\hat\\beta}_{8}-2{\\hat\\beta}_{6}$? Note that ${\\hat\\beta}_{9}+{\\hat\\beta}_{8}-2{\\hat\\beta}_{6}$ corresponds to a contrast to two categories (where the binary dummy variables are indexed as $x_{9}$) and $x_{8}$ with a third category (where the binary dummy variable is indexed as $x_{6}$). [ANS]", "answer_v1": [ "0.144634", "0.25149" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "You are being given the covariance matrix of a multiple regression with ${\\hat\\beta}_0,{\\hat\\beta}_1,\\ldots,{\\hat\\beta}_5,{\\hat\\beta}_6, \\ldots,{\\hat\\beta}_9$ where $x_1,\\ldots,x_5$ are numerical explanatory variables and $x_6,\\ldots,x_9$ are 4 binary dummy variables for 4 different categories relative to the baseline category. The categorical explanatory variable has 5 categories.\nYou are given that the estimated covariance matrix is: $\\left[\\begin{array}{cccccccccc} 0.33347 &-0.00123 &0.000379 &-0.00635 &-0.007631 &-0.000744 &-0.036518 &-0.054614 &-0.049256 &-0.043163\\cr-0.00123 &1.1\\times 10^{-5} &3\\times 10^{-6} &0 &3.6\\times 10^{-5} &1\\times 10^{-6} &1.9\\times 10^{-5} &0.000121 &0.0002 &9.9\\times 10^{-5}\\cr 0.000379 &3\\times 10^{-6} &1.9\\times 10^{-5} &-3.5\\times 10^{-5} &2.6\\times 10^{-5} &-1.1\\times 10^{-5} &-7.2\\times 10^{-5} &1.2\\times 10^{-5} &3.2\\times 10^{-5} &-1.6\\times 10^{-5}\\cr-0.00635 &0 &-3.5\\times 10^{-5} &0.000281 &7.7\\times 10^{-5} &1.9\\times 10^{-5} &0.000796 &0.000944 &0.000292 &0.000309\\cr-0.007631 &3.6\\times 10^{-5} &2.6\\times 10^{-5} &7.7\\times 10^{-5} &0.00051 &-4\\times 10^{-5} &0.001221 &0.002007 &0.001827 &0.001396\\cr-0.000744 &1\\times 10^{-6} &-1.1\\times 10^{-5} &1.9\\times 10^{-5} &-4\\times 10^{-5} &1.6\\times 10^{-5} &-3.2\\times 10^{-5} &-6.9\\times 10^{-5} &-2.1\\times 10^{-5} &6.8\\times 10^{-5}\\cr-0.036518 &1.9\\times 10^{-5} &-7.2\\times 10^{-5} &0.000796 &0.001221 &-3.2\\times 10^{-5} &0.022795 &0.016794 &0.014729 &0.014576\\cr-0.054614 &0.000121 &1.2\\times 10^{-5} &0.000944 &0.002007 &-6.9\\times 10^{-5} &0.016794 &0.030437 &0.017899 &0.016878\\cr-0.049256 &0.0002 &3.2\\times 10^{-5} &0.000292 &0.001827 &-2.1\\times 10^{-5} &0.014729 &0.017899 &0.025177 &0.017092\\cr-0.043163 &9.9\\times 10^{-5} &-1.6\\times 10^{-5} &0.000309 &0.001396 &6.8\\times 10^{-5} &0.014576 &0.016878 &0.017092 &0.029926 \\end{array}\\right]$ or in R: acov=matrix(c(0.33347,-0.00123, 0.000379,-0.00635,-0.007631,-0.000744,-0.036518,-0.054614,-0.049256,-0.043163,-0.00123, 1.1e-05, 3e-06, 0, 3.6e-05, 1e-06, 1.9e-05, 0.000121, 0.0002, 9.9e-05, 0.000379, 3e-06, 1.9e-05,-3.5e-05, 2.6e-05,-1.1e-05,-7.2e-05, 1.2e-05, 3.2e-05,-1.6e-05,-0.00635, 0,-3.5e-05, 0.000281, 7.7e-05, 1.9e-05, 0.000796, 0.000944, 0.000292, 0.000309,-0.007631, 3.6e-05, 2.6e-05, 7.7e-05, 0.00051,-4e-05, 0.001221, 0.002007, 0.001827, 0.001396,-0.000744, 1e-06,-1.1e-05, 1.9e-05,-4e-05, 1.6e-05,-3.2e-05,-6.9e-05,-2.1e-05, 6.8e-05,-0.036518, 1.9e-05,-7.2e-05, 0.000796, 0.001221,-3.2e-05, 0.022795, 0.016794, 0.014729, 0.014576,-0.054614, 0.000121, 1.2e-05, 0.000944, 0.002007,-6.9e-05, 0.016794, 0.030437, 0.017899, 0.016878,-0.049256, 0.0002, 3.2e-05, 0.000292, 0.001827,-2.1e-05, 0.014729, 0.017899, 0.025177, 0.017092,-0.043163, 9.9e-05,-1.6e-05, 0.000309, 0.001396, 6.8e-05, 0.014576, 0.016878, 0.017092, 0.029926),10,10)\nPlease use 3 decimal places for the answers below which are not integer-valued.\nPart a) Based on the (estimated) covariance matrix, what is the SE of ${\\hat\\beta}_{6}-{\\hat\\beta}_{9}$? Note that ${\\hat\\beta}_{6}-{\\hat\\beta}_{9}$ corresponds to the estimated distance of the hyperplanes for two different categories where the binary dummy variables are indexed as $x_{6}$ and $x_{9}$. [ANS]\nPart b) Based on the (estimated) covariance matrix, what is the SE of ${\\hat\\beta}_{6}+{\\hat\\beta}_{9}-2{\\hat\\beta}_{8}$? Note that ${\\hat\\beta}_{6}+{\\hat\\beta}_{9}-2{\\hat\\beta}_{8}$ corresponds to a contrast to two categories (where the binary dummy variables are indexed as $x_{6}$) and $x_{9}$ with a third category (where the binary dummy variable is indexed as $x_{8}$). [ANS]", "answer_v2": [ "0.153522", "0.235153" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "You are being given the covariance matrix of a multiple regression with ${\\hat\\beta}_0,{\\hat\\beta}_1,\\ldots,{\\hat\\beta}_5,{\\hat\\beta}_6, \\ldots,{\\hat\\beta}_9$ where $x_1,\\ldots,x_5$ are numerical explanatory variables and $x_6,\\ldots,x_9$ are 4 binary dummy variables for 4 different categories relative to the baseline category. The categorical explanatory variable has 5 categories.\nYou are given that the estimated covariance matrix is: $\\left[\\begin{array}{cccccccccc} 0.33347 &-0.00123 &0.000379 &-0.00635 &-0.007631 &-0.000744 &-0.036518 &-0.054614 &-0.049256 &-0.043163\\cr-0.00123 &1.1\\times 10^{-5} &3\\times 10^{-6} &0 &3.6\\times 10^{-5} &1\\times 10^{-6} &1.9\\times 10^{-5} &0.000121 &0.0002 &9.9\\times 10^{-5}\\cr 0.000379 &3\\times 10^{-6} &1.9\\times 10^{-5} &-3.5\\times 10^{-5} &2.6\\times 10^{-5} &-1.1\\times 10^{-5} &-7.2\\times 10^{-5} &1.2\\times 10^{-5} &3.2\\times 10^{-5} &-1.6\\times 10^{-5}\\cr-0.00635 &0 &-3.5\\times 10^{-5} &0.000281 &7.7\\times 10^{-5} &1.9\\times 10^{-5} &0.000796 &0.000944 &0.000292 &0.000309\\cr-0.007631 &3.6\\times 10^{-5} &2.6\\times 10^{-5} &7.7\\times 10^{-5} &0.00051 &-4\\times 10^{-5} &0.001221 &0.002007 &0.001827 &0.001396\\cr-0.000744 &1\\times 10^{-6} &-1.1\\times 10^{-5} &1.9\\times 10^{-5} &-4\\times 10^{-5} &1.6\\times 10^{-5} &-3.2\\times 10^{-5} &-6.9\\times 10^{-5} &-2.1\\times 10^{-5} &6.8\\times 10^{-5}\\cr-0.036518 &1.9\\times 10^{-5} &-7.2\\times 10^{-5} &0.000796 &0.001221 &-3.2\\times 10^{-5} &0.022795 &0.016794 &0.014729 &0.014576\\cr-0.054614 &0.000121 &1.2\\times 10^{-5} &0.000944 &0.002007 &-6.9\\times 10^{-5} &0.016794 &0.030437 &0.017899 &0.016878\\cr-0.049256 &0.0002 &3.2\\times 10^{-5} &0.000292 &0.001827 &-2.1\\times 10^{-5} &0.014729 &0.017899 &0.025177 &0.017092\\cr-0.043163 &9.9\\times 10^{-5} &-1.6\\times 10^{-5} &0.000309 &0.001396 &6.8\\times 10^{-5} &0.014576 &0.016878 &0.017092 &0.029926 \\end{array}\\right]$ or in R: acov=matrix(c(0.33347,-0.00123, 0.000379,-0.00635,-0.007631,-0.000744,-0.036518,-0.054614,-0.049256,-0.043163,-0.00123, 1.1e-05, 3e-06, 0, 3.6e-05, 1e-06, 1.9e-05, 0.000121, 0.0002, 9.9e-05, 0.000379, 3e-06, 1.9e-05,-3.5e-05, 2.6e-05,-1.1e-05,-7.2e-05, 1.2e-05, 3.2e-05,-1.6e-05,-0.00635, 0,-3.5e-05, 0.000281, 7.7e-05, 1.9e-05, 0.000796, 0.000944, 0.000292, 0.000309,-0.007631, 3.6e-05, 2.6e-05, 7.7e-05, 0.00051,-4e-05, 0.001221, 0.002007, 0.001827, 0.001396,-0.000744, 1e-06,-1.1e-05, 1.9e-05,-4e-05, 1.6e-05,-3.2e-05,-6.9e-05,-2.1e-05, 6.8e-05,-0.036518, 1.9e-05,-7.2e-05, 0.000796, 0.001221,-3.2e-05, 0.022795, 0.016794, 0.014729, 0.014576,-0.054614, 0.000121, 1.2e-05, 0.000944, 0.002007,-6.9e-05, 0.016794, 0.030437, 0.017899, 0.016878,-0.049256, 0.0002, 3.2e-05, 0.000292, 0.001827,-2.1e-05, 0.014729, 0.017899, 0.025177, 0.017092,-0.043163, 9.9e-05,-1.6e-05, 0.000309, 0.001396, 6.8e-05, 0.014576, 0.016878, 0.017092, 0.029926),10,10)\nPlease use 3 decimal places for the answers below which are not integer-valued.\nPart a) Based on the (estimated) covariance matrix, what is the SE of ${\\hat\\beta}_{7}-{\\hat\\beta}_{8}$? Note that ${\\hat\\beta}_{7}-{\\hat\\beta}_{8}$ corresponds to the estimated distance of the hyperplanes for two different categories where the binary dummy variables are indexed as $x_{7}$ and $x_{8}$. [ANS]\nPart b) Based on the (estimated) covariance matrix, what is the SE of ${\\hat\\beta}_{7}+{\\hat\\beta}_{8}-2{\\hat\\beta}_{6}$? Note that ${\\hat\\beta}_{7}+{\\hat\\beta}_{8}-2{\\hat\\beta}_{6}$ corresponds to a contrast to two categories (where the binary dummy variables are indexed as $x_{7}$) and $x_{8}$ with a third category (where the binary dummy variable is indexed as $x_{6}$). [ANS]", "answer_v3": [ "0.140769", "0.237697" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0403", "subject": "Statistics", "topic": "Multiple regression", "subtopic": "Parameter estimates", "level": "4", "keywords": [ "statistics", "regression" ], "problem_v1": "For this question and others that might involve derivation of a formula, after you formulate an algebraic solution, check its validity on some numerical regression examples with small data sets. If you match numerically in some instances, your answer is likely correct. If your theoretical answer doesn't match the numerical cases, go back to review your \"derivation\".\nThe raw data are $y_i,x_{i1},x_{i2}$ for $i=1,\\ldots,n$. Let ${\\hat\\beta}_0, {\\hat\\beta}_1, {\\hat\\beta}_2$ be the least squares estimates when fitting a plane.\nPart a) Suppose the $y$ variable has been rescaled to $y^*$ where $y^*=10 y$. Let ${\\hat\\beta}^*_0, {\\hat\\beta}^*_1, {\\hat\\beta}^*_2$ be the least squares estimates for the data $y_i^*,x_{i1},x_{i2}, i=1,\\ldots,n$. Which of the following are correct? Possibly more than one item is correct. [ANS] A. ${\\hat\\beta}^*_0=10{\\hat\\beta}_0$ B. ${\\hat\\beta}^*_0={\\hat\\beta}_0/10$ C. ${\\hat\\beta}^*_2=10{\\hat\\beta}_2$ D. ${\\hat\\beta}^*_1={\\hat\\beta}_1/10$ E. ${\\hat\\beta}^*_2={\\hat\\beta}_2/10$ F. ${\\hat\\beta}^*_1=10{\\hat\\beta}_1$ G. None of the above\nPart b) Suppose the $x_1$ variable has been rescaled to $x_1^*$ where $x_1^*=10x_1$. Let ${\\hat\\beta}^*_0, {\\hat\\beta}^*_1, {\\hat\\beta}^*_2$ be the least squares estimates for the data $y_i,x_{i1}^*,x_{i2}, i=1,\\ldots,n$. Which of the following are correct? Possibly more than one item is correct. [ANS] A. ${\\hat\\beta}^*_2={\\hat\\beta}_2/10$ B. ${\\hat\\beta}^*_0=10{\\hat\\beta}_0$ C. ${\\hat\\beta}^*_2=10{\\hat\\beta}_2$ D. ${\\hat\\beta}^*_0={\\hat\\beta}_0/10$ E. ${\\hat\\beta}^*_1=10{\\hat\\beta}_1$ F. ${\\hat\\beta}^*_1={\\hat\\beta}_1/10$ G. None of the above\nPart c) Suppose the $x_1$ variable has been shifted to $x_1^*$ where $x_1^*=x_1-7$. Let ${\\hat\\beta}^*_0, {\\hat\\beta}^*_1, {\\hat\\beta}^*_2$ be the least squares estimates for the data $y_i,x_{i1}^*,x_{i2}, i=1,\\ldots,n$. Which of the following are correct? There might be more than one correct answer. [ANS] A. ${\\hat\\beta}^*_0={\\hat\\beta}_0$ B. ${\\hat\\beta}^*_1\\ne {\\hat\\beta}_1$ C. ${\\hat\\beta}^*_1={\\hat\\beta}_1$ D. ${\\hat\\beta}^*_2\\ne {\\hat\\beta}_2$ E. ${\\hat\\beta}^*_2={\\hat\\beta}_2$ F. ${\\hat\\beta}^*_0\\ne {\\hat\\beta}_0$ G. None of the above", "answer_v1": [ "ACF", "F", "CEF" ], "answer_type_v1": [ "MCM", "MCS", "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v2": "For this question and others that might involve derivation of a formula, after you formulate an algebraic solution, check its validity on some numerical regression examples with small data sets. If you match numerically in some instances, your answer is likely correct. If your theoretical answer doesn't match the numerical cases, go back to review your \"derivation\".\nThe raw data are $y_i,x_{i1},x_{i2}$ for $i=1,\\ldots,n$. Let ${\\hat\\beta}_0, {\\hat\\beta}_1, {\\hat\\beta}_2$ be the least squares estimates when fitting a plane.\nPart a) Suppose the $y$ variable has been rescaled to $y^*$ where $y^*=0.1 y$. Let ${\\hat\\beta}^*_0, {\\hat\\beta}^*_1, {\\hat\\beta}^*_2$ be the least squares estimates for the data $y_i^*,x_{i1},x_{i2}, i=1,\\ldots,n$. Which of the following are correct? Possibly more than one item is correct. [ANS] A. ${\\hat\\beta}^*_2=10{\\hat\\beta}_2$ B. ${\\hat\\beta}^*_0={\\hat\\beta}_0/10$ C. ${\\hat\\beta}^*_1={\\hat\\beta}_1/10$ D. ${\\hat\\beta}^*_1=10{\\hat\\beta}_1$ E. ${\\hat\\beta}^*_2={\\hat\\beta}_2/10$ F. ${\\hat\\beta}^*_0=10{\\hat\\beta}_0$ G. None of the above\nPart b) Suppose the $x_1$ variable has been rescaled to $x_1^*$ where $x_1^*=10x_1$. Let ${\\hat\\beta}^*_0, {\\hat\\beta}^*_1, {\\hat\\beta}^*_2$ be the least squares estimates for the data $y_i,x_{i1}^*,x_{i2}, i=1,\\ldots,n$. Which of the following are correct? Possibly more than one item is correct. [ANS] A. ${\\hat\\beta}^*_0={\\hat\\beta}_0/10$ B. ${\\hat\\beta}^*_0=10{\\hat\\beta}_0$ C. ${\\hat\\beta}^*_2=10{\\hat\\beta}_2$ D. ${\\hat\\beta}^*_2={\\hat\\beta}_2/10$ E. ${\\hat\\beta}^*_1={\\hat\\beta}_1/10$ F. ${\\hat\\beta}^*_1=10{\\hat\\beta}_1$ G. None of the above\nPart c) Suppose the $x_1$ variable has been shifted to $x_1^*$ where $x_1^*=x_1-2$. Let ${\\hat\\beta}^*_0, {\\hat\\beta}^*_1, {\\hat\\beta}^*_2$ be the least squares estimates for the data $y_i,x_{i1}^*,x_{i2}, i=1,\\ldots,n$. Which of the following are correct? There might be more than one correct answer. [ANS] A. ${\\hat\\beta}^*_0\\ne {\\hat\\beta}_0$ B. ${\\hat\\beta}^*_1={\\hat\\beta}_1$ C. ${\\hat\\beta}^*_2\\ne {\\hat\\beta}_2$ D. ${\\hat\\beta}^*_2={\\hat\\beta}_2$ E. ${\\hat\\beta}^*_0={\\hat\\beta}_0$ F. ${\\hat\\beta}^*_1\\ne {\\hat\\beta}_1$ G. None of the above", "answer_v2": [ "BCE", "E", "ABD" ], "answer_type_v2": [ "MCM", "MCS", "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v3": "For this question and others that might involve derivation of a formula, after you formulate an algebraic solution, check its validity on some numerical regression examples with small data sets. If you match numerically in some instances, your answer is likely correct. If your theoretical answer doesn't match the numerical cases, go back to review your \"derivation\".\nThe raw data are $y_i,x_{i1},x_{i2}$ for $i=1,\\ldots,n$. Let ${\\hat\\beta}_0, {\\hat\\beta}_1, {\\hat\\beta}_2$ be the least squares estimates when fitting a plane.\nPart a) Suppose the $y$ variable has been rescaled to $y^*$ where $y^*=0.1 y$. Let ${\\hat\\beta}^*_0, {\\hat\\beta}^*_1, {\\hat\\beta}^*_2$ be the least squares estimates for the data $y_i^*,x_{i1},x_{i2}, i=1,\\ldots,n$. Which of the following are correct? Possibly more than one item is correct. [ANS] A. ${\\hat\\beta}^*_2=10{\\hat\\beta}_2$ B. ${\\hat\\beta}^*_1={\\hat\\beta}_1/10$ C. ${\\hat\\beta}^*_0={\\hat\\beta}_0/10$ D. ${\\hat\\beta}^*_1=10{\\hat\\beta}_1$ E. ${\\hat\\beta}^*_0=10{\\hat\\beta}_0$ F. ${\\hat\\beta}^*_2={\\hat\\beta}_2/10$ G. None of the above\nPart b) Suppose the $x_1$ variable has been rescaled to $x_1^*$ where $x_1^*=10x_1$. Let ${\\hat\\beta}^*_0, {\\hat\\beta}^*_1, {\\hat\\beta}^*_2$ be the least squares estimates for the data $y_i,x_{i1}^*,x_{i2}, i=1,\\ldots,n$. Which of the following are correct? Possibly more than one item is correct. [ANS] A. ${\\hat\\beta}^*_0={\\hat\\beta}_0/10$ B. ${\\hat\\beta}^*_2={\\hat\\beta}_2/10$ C. ${\\hat\\beta}^*_0=10{\\hat\\beta}_0$ D. ${\\hat\\beta}^*_1=10{\\hat\\beta}_1$ E. ${\\hat\\beta}^*_1={\\hat\\beta}_1/10$ F. ${\\hat\\beta}^*_2=10{\\hat\\beta}_2$ G. None of the above\nPart c) Suppose the $x_1$ variable has been shifted to $x_1^*$ where $x_1^*=x_1-3$. Let ${\\hat\\beta}^*_0, {\\hat\\beta}^*_1, {\\hat\\beta}^*_2$ be the least squares estimates for the data $y_i,x_{i1}^*,x_{i2}, i=1,\\ldots,n$. Which of the following are correct? There might be more than one correct answer. [ANS] A. ${\\hat\\beta}^*_1={\\hat\\beta}_1$ B. ${\\hat\\beta}^*_2\\ne {\\hat\\beta}_2$ C. ${\\hat\\beta}^*_0={\\hat\\beta}_0$ D. ${\\hat\\beta}^*_2={\\hat\\beta}_2$ E. ${\\hat\\beta}^*_1\\ne {\\hat\\beta}_1$ F. ${\\hat\\beta}^*_0\\ne {\\hat\\beta}_0$ G. None of the above", "answer_v3": [ "BCF", "E", "ADF" ], "answer_type_v3": [ "MCM", "MCS", "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ] ] }, { "id": "Statistics_0404", "subject": "Statistics", "topic": "Multiple regression", "subtopic": "Parameter estimates", "level": "4", "keywords": [ "statistics", "regression", "linear dependent vectors" ], "problem_v1": "This is a problem on linear dependent columns of the data matrix ${\\bf X}$ used in multiple regression. Assume the row dimension $n$ of ${\\bf X}$ is greater than the column dimension $k$ of ${\\bf X}$. ${\\bf X}$ has full column rank of the $k$ columns of ${\\bf X}$ are linearly independent. Let ${\\tt X}_1, \\ldots,{\\tt X}_k$ denote the $k$ columns of ${\\bf X}$. The columns of ${\\bf X}$ are linearly independent if $a_1{\\tt X}_1+\\cdots+a_k{\\tt X}_k={\\bf 0}_n$ implies that $a_1=\\cdots=a_k=0$. The columns of ${\\bf X}$ are linearly dependent if $a_1{\\tt X}_1+\\cdots+a_k{\\tt X}_k={\\bf 0}_n$ for at least one vector ${\\bf a}=(a_1,\\cdots,a_k)^T$ that is not the zero vector. Let ${\\tt A}_1, \\ldots,{\\tt A}_k$ denote the $k$ columns of the $k\\times k$ matrix ${\\bf A}={\\bf X}^T{\\bf X}$. The columns of ${\\bf A}$ are linearly independent if $c_1{\\tt A}_1+\\cdots+c_k{\\tt A}_k={\\bf 0}_k$ implies that $c_1=\\cdots=c_k=0$. The columns of ${\\bf A}$ are linearly dependent if $c_1{\\tt A}_1+\\cdots+c_k{\\tt A}_k={\\bf 0}_k$ for at least one vector ${\\bf c}=(c_1,\\cdots,c_k)^T$ that is not the zero vector. Please check your linear algebra text for the definition of column rank. Notation: ${\\bf 0}_s$ denotes the zero vector in $s$-dimensional space.\nPart a) Suppose the columns of ${\\bf X}$ are linearly dependent based on the coefficient vector ${\\bf a}=(a_1,\\cdots,a_k)^T$, that is, $a_1{\\tt X}_1+\\cdots+a_k{\\tt X}_k={\\bf 0}_n$. Which of the following are correct statements and implications. Possibly more than one item is correct. [ANS] A. ${\\bf X}^T{\\bf X}{\\bf a}={\\bf 0}_k$ B. the dimension of the null space of ${\\bf X}^T{\\bf X}$ is greater than or equal to that of ${\\bf X}$ C. ${\\bf X}{\\bf a}={\\bf 0}_n$ D. the column rank of ${\\bf X}^T{\\bf X}$ is less than or equal to the column rank of ${\\bf X}$ E. ${\\bf a}{\\bf X}={\\bf 0}_n$ F. the dimension of the null space of ${\\bf X}^T{\\bf X}$ is less than or equal to that of ${\\bf X}$ G. ${\\bf a}^T{\\bf X}={\\bf 0}_n$\nPart b) Suppose the columns of ${\\bf A}={\\bf X}^T{\\bf X}$ are linearly dependent based on the coefficient vector ${\\bf c}=(c_1,\\cdots,c_k)^T$, that is, $c_1{\\tt A}_1+\\cdots+c_k{\\tt A}_k={\\bf 0}_k$. Which of the following are correct statements and implications. Possibly more than one item is correct. [ANS] A. ${\\bf A}{\\bf c}={\\bf 0}_n$ B. ${\\bf X}{\\bf c}={\\bf 0}_k$ C. $({\\bf X}^T{\\bf X}){\\bf c}={\\bf 0}_k$ D. the column rank of ${\\bf X}$ is less than or equal to the column rank of ${\\bf X}^T{\\bf X}$ E. ${\\bf c}^T{\\bf X}^T{\\bf X}{\\bf c}=0$ F. ${\\bf X}{\\bf c}={\\bf 0}_n$ G. ${\\bf c}^T{\\bf X}^T{\\bf X}{\\bf c}={\\bf 0}_k$ H. $({\\bf X}{\\bf c})^T({\\bf X}{\\bf c})=0$ I. ${\\bf A}{\\bf c}={\\bf 0}_k$ J. ${\\bf c}{\\bf A}={\\bf 0}_k$\nPart c) What is the column rank of $\\left[\\begin{array}{ccc} 1 &7 &49\\cr 1 &6 &36\\cr 1 &1 &1\\cr 1 &8 &64\\cr 1 &5 &25 \\end{array}\\right]$ [ANS]\nPart d) What is the column rank of $\\left[\\begin{array}{cccc} 1 &7 &49 &70\\cr 1 &6 &36 &54\\cr 1 &1 &1 &4\\cr 1 &8 &64 &88\\cr 1 &5 &25 &40 \\end{array}\\right]$ [ANS]\nPart e) What is the row rank of $\\left[\\begin{array}{cccc} 1 &7 &49 &70\\cr 1 &6 &36 &54\\cr 1 &1 &1 &4\\cr 1 &8 &64 &88\\cr 1 &5 &25 &40 \\end{array}\\right]$ [ANS]\nPart f) Which of the following are ways to determine the column rank of ${\\bf X}$? There might be more than one correct answer. [ANS] A. The number of positive eigenvalues of ${\\bf X}^T{\\bf X}$: ${\\tt eigen}({\\bf X}^T{\\bf X})$ in R B. max {s: column s is linearly independent of columns 1,...,s-1} C. Algorithm of (i) subtracting multiples of a column to get zeroes in first row and k-1 columns, (ii) then subtracting multiples of another column to get zeroes in the second row and k-2 columns, (iii) iterate to row k: column rank is the number of columns that are not the zero vector D. The number of non-zero singular values of ${\\bf X}$: ${\\tt svd}({\\bf X})$ in R E. The number of non-zero diagonal entries of ${\\bf R}$ in the QR decomposition of ${\\bf X}={\\bf Q}{\\bf R}$ where ${\\bf Q}$ is an $n\\times k$ matrix with orthogonal columns, and ${\\bf R}$ is a $k\\times k$ upper triangular matrix: qrobj=${\\tt qr}({\\bf X})$, Q=${\\tt qr.Q}$ (qrobj), R=${\\tt qr.R}$ (qrobj) in R F. None of the above\nFrom the above, write at least two proofs for the statement: \"the rank of ${\\bf X}$ and ${\\bf X}^T{\\bf X}$ are the same\".", "answer_v1": [ "ABCD", "CDEFHI", "3", "3", "3", "ACDE" ], "answer_type_v1": [ "MCM", "MCM", "NV", "NV", "NV", "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J" ], [], [], [], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "This is a problem on linear dependent columns of the data matrix ${\\bf X}$ used in multiple regression. Assume the row dimension $n$ of ${\\bf X}$ is greater than the column dimension $k$ of ${\\bf X}$. ${\\bf X}$ has full column rank of the $k$ columns of ${\\bf X}$ are linearly independent. Let ${\\tt X}_1, \\ldots,{\\tt X}_k$ denote the $k$ columns of ${\\bf X}$. The columns of ${\\bf X}$ are linearly independent if $a_1{\\tt X}_1+\\cdots+a_k{\\tt X}_k={\\bf 0}_n$ implies that $a_1=\\cdots=a_k=0$. The columns of ${\\bf X}$ are linearly dependent if $a_1{\\tt X}_1+\\cdots+a_k{\\tt X}_k={\\bf 0}_n$ for at least one vector ${\\bf a}=(a_1,\\cdots,a_k)^T$ that is not the zero vector. Let ${\\tt A}_1, \\ldots,{\\tt A}_k$ denote the $k$ columns of the $k\\times k$ matrix ${\\bf A}={\\bf X}^T{\\bf X}$. The columns of ${\\bf A}$ are linearly independent if $c_1{\\tt A}_1+\\cdots+c_k{\\tt A}_k={\\bf 0}_k$ implies that $c_1=\\cdots=c_k=0$. The columns of ${\\bf A}$ are linearly dependent if $c_1{\\tt A}_1+\\cdots+c_k{\\tt A}_k={\\bf 0}_k$ for at least one vector ${\\bf c}=(c_1,\\cdots,c_k)^T$ that is not the zero vector. Please check your linear algebra text for the definition of column rank. Notation: ${\\bf 0}_s$ denotes the zero vector in $s$-dimensional space.\nPart a) Suppose the columns of ${\\bf X}$ are linearly dependent based on the coefficient vector ${\\bf a}=(a_1,\\cdots,a_k)^T$, that is, $a_1{\\tt X}_1+\\cdots+a_k{\\tt X}_k={\\bf 0}_n$. Which of the following are correct statements and implications. Possibly more than one item is correct. [ANS] A. ${\\bf X}^T{\\bf X}{\\bf a}={\\bf 0}_k$ B. the dimension of the null space of ${\\bf X}^T{\\bf X}$ is greater than or equal to that of ${\\bf X}$ C. ${\\bf X}{\\bf a}={\\bf 0}_n$ D. ${\\bf a}{\\bf X}={\\bf 0}_n$ E. the column rank of ${\\bf X}^T{\\bf X}$ is less than or equal to the column rank of ${\\bf X}$ F. ${\\bf a}^T{\\bf X}={\\bf 0}_n$ G. the dimension of the null space of ${\\bf X}^T{\\bf X}$ is less than or equal to that of ${\\bf X}$\nPart b) Suppose the columns of ${\\bf A}={\\bf X}^T{\\bf X}$ are linearly dependent based on the coefficient vector ${\\bf c}=(c_1,\\cdots,c_k)^T$, that is, $c_1{\\tt A}_1+\\cdots+c_k{\\tt A}_k={\\bf 0}_k$. Which of the following are correct statements and implications. Possibly more than one item is correct. [ANS] A. ${\\bf X}{\\bf c}={\\bf 0}_k$ B. $({\\bf X}{\\bf c})^T({\\bf X}{\\bf c})=0$ C. ${\\bf c}^T{\\bf X}^T{\\bf X}{\\bf c}={\\bf 0}_k$ D. the column rank of ${\\bf X}$ is less than or equal to the column rank of ${\\bf X}^T{\\bf X}$ E. ${\\bf A}{\\bf c}={\\bf 0}_n$ F. ${\\bf c}{\\bf A}={\\bf 0}_k$ G. $({\\bf X}^T{\\bf X}){\\bf c}={\\bf 0}_k$ H. ${\\bf A}{\\bf c}={\\bf 0}_k$ I. ${\\bf X}{\\bf c}={\\bf 0}_n$ J. ${\\bf c}^T{\\bf X}^T{\\bf X}{\\bf c}=0$\nPart c) What is the column rank of $\\left[\\begin{array}{ccc} 1 &0 &0\\cr 1 &9 &81\\cr 1 &3 &9\\cr 1 &5 &25\\cr 1 &1 &1 \\end{array}\\right]$ [ANS]\nPart d) What is the column rank of $\\left[\\begin{array}{cccc} 1 &0 &0 &0\\cr 1 &9 &81 &108\\cr 1 &3 &9 &18\\cr 1 &5 &25 &40\\cr 1 &1 &1 &4 \\end{array}\\right]$ [ANS]\nPart e) What is the row rank of $\\left[\\begin{array}{cccc} 1 &0 &0 &0\\cr 1 &9 &81 &108\\cr 1 &3 &9 &18\\cr 1 &5 &25 &40\\cr 1 &1 &1 &4 \\end{array}\\right]$ [ANS]\nPart f) Which of the following are ways to determine the column rank of ${\\bf X}$? There might be more than one correct answer. [ANS] A. max {s: column s is linearly independent of columns 1,...,s-1} B. The number of non-zero diagonal entries of ${\\bf R}$ in the QR decomposition of ${\\bf X}={\\bf Q}{\\bf R}$ where ${\\bf Q}$ is an $n\\times k$ matrix with orthogonal columns, and ${\\bf R}$ is a $k\\times k$ upper triangular matrix: qrobj=${\\tt qr}({\\bf X})$, Q=${\\tt qr.Q}$ (qrobj), R=${\\tt qr.R}$ (qrobj) in R C. The number of positive eigenvalues of ${\\bf X}^T{\\bf X}$: ${\\tt eigen}({\\bf X}^T{\\bf X})$ in R D. The number of non-zero singular values of ${\\bf X}$: ${\\tt svd}({\\bf X})$ in R E. Algorithm of (i) subtracting multiples of a column to get zeroes in first row and k-1 columns, (ii) then subtracting multiples of another column to get zeroes in the second row and k-2 columns, (iii) iterate to row k: column rank is the number of columns that are not the zero vector F. None of the above\nFrom the above, write at least two proofs for the statement: \"the rank of ${\\bf X}$ and ${\\bf X}^T{\\bf X}$ are the same\".", "answer_v2": [ "ABCE", "BDGHIJ", "3", "3", "3", "BCDE" ], "answer_type_v2": [ "MCM", "MCM", "NV", "NV", "NV", "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J" ], [], [], [], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "This is a problem on linear dependent columns of the data matrix ${\\bf X}$ used in multiple regression. Assume the row dimension $n$ of ${\\bf X}$ is greater than the column dimension $k$ of ${\\bf X}$. ${\\bf X}$ has full column rank of the $k$ columns of ${\\bf X}$ are linearly independent. Let ${\\tt X}_1, \\ldots,{\\tt X}_k$ denote the $k$ columns of ${\\bf X}$. The columns of ${\\bf X}$ are linearly independent if $a_1{\\tt X}_1+\\cdots+a_k{\\tt X}_k={\\bf 0}_n$ implies that $a_1=\\cdots=a_k=0$. The columns of ${\\bf X}$ are linearly dependent if $a_1{\\tt X}_1+\\cdots+a_k{\\tt X}_k={\\bf 0}_n$ for at least one vector ${\\bf a}=(a_1,\\cdots,a_k)^T$ that is not the zero vector. Let ${\\tt A}_1, \\ldots,{\\tt A}_k$ denote the $k$ columns of the $k\\times k$ matrix ${\\bf A}={\\bf X}^T{\\bf X}$. The columns of ${\\bf A}$ are linearly independent if $c_1{\\tt A}_1+\\cdots+c_k{\\tt A}_k={\\bf 0}_k$ implies that $c_1=\\cdots=c_k=0$. The columns of ${\\bf A}$ are linearly dependent if $c_1{\\tt A}_1+\\cdots+c_k{\\tt A}_k={\\bf 0}_k$ for at least one vector ${\\bf c}=(c_1,\\cdots,c_k)^T$ that is not the zero vector. Please check your linear algebra text for the definition of column rank. Notation: ${\\bf 0}_s$ denotes the zero vector in $s$-dimensional space.\nPart a) Suppose the columns of ${\\bf X}$ are linearly dependent based on the coefficient vector ${\\bf a}=(a_1,\\cdots,a_k)^T$, that is, $a_1{\\tt X}_1+\\cdots+a_k{\\tt X}_k={\\bf 0}_n$. Which of the following are correct statements and implications. Possibly more than one item is correct. [ANS] A. ${\\bf X}{\\bf a}={\\bf 0}_n$ B. the dimension of the null space of ${\\bf X}^T{\\bf X}$ is less than or equal to that of ${\\bf X}$ C. ${\\bf a}^T{\\bf X}={\\bf 0}_n$ D. ${\\bf a}{\\bf X}={\\bf 0}_n$ E. ${\\bf X}^T{\\bf X}{\\bf a}={\\bf 0}_k$ F. the column rank of ${\\bf X}^T{\\bf X}$ is less than or equal to the column rank of ${\\bf X}$ G. the dimension of the null space of ${\\bf X}^T{\\bf X}$ is greater than or equal to that of ${\\bf X}$\nPart b) Suppose the columns of ${\\bf A}={\\bf X}^T{\\bf X}$ are linearly dependent based on the coefficient vector ${\\bf c}=(c_1,\\cdots,c_k)^T$, that is, $c_1{\\tt A}_1+\\cdots+c_k{\\tt A}_k={\\bf 0}_k$. Which of the following are correct statements and implications. Possibly more than one item is correct. [ANS] A. ${\\bf X}{\\bf c}={\\bf 0}_n$ B. ${\\bf c}^T{\\bf X}^T{\\bf X}{\\bf c}=0$ C. $({\\bf X}{\\bf c})^T({\\bf X}{\\bf c})=0$ D. ${\\bf c}{\\bf A}={\\bf 0}_k$ E. the column rank of ${\\bf X}$ is less than or equal to the column rank of ${\\bf X}^T{\\bf X}$ F. ${\\bf c}^T{\\bf X}^T{\\bf X}{\\bf c}={\\bf 0}_k$ G. $({\\bf X}^T{\\bf X}){\\bf c}={\\bf 0}_k$ H. ${\\bf A}{\\bf c}={\\bf 0}_k$ I. ${\\bf A}{\\bf c}={\\bf 0}_n$ J. ${\\bf X}{\\bf c}={\\bf 0}_k$\nPart c) What is the column rank of $\\left[\\begin{array}{ccc} 1 &3 &9\\cr 1 &6 &36\\cr 1 &4 &16\\cr 1 &1 &1\\cr 1 &5 &25 \\end{array}\\right]$ [ANS]\nPart d) What is the column rank of $\\left[\\begin{array}{cccc} 1 &3 &9 &18\\cr 1 &6 &36 &54\\cr 1 &4 &16 &28\\cr 1 &1 &1 &4\\cr 1 &5 &25 &40 \\end{array}\\right]$ [ANS]\nPart e) What is the row rank of $\\left[\\begin{array}{cccc} 1 &3 &9 &18\\cr 1 &6 &36 &54\\cr 1 &4 &16 &28\\cr 1 &1 &1 &4\\cr 1 &5 &25 &40 \\end{array}\\right]$ [ANS]\nPart f) Which of the following are ways to determine the column rank of ${\\bf X}$? There might be more than one correct answer. [ANS] A. max {s: column s is linearly independent of columns 1,...,s-1} B. Algorithm of (i) subtracting multiples of a column to get zeroes in first row and k-1 columns, (ii) then subtracting multiples of another column to get zeroes in the second row and k-2 columns, (iii) iterate to row k: column rank is the number of columns that are not the zero vector C. The number of non-zero singular values of ${\\bf X}$: ${\\tt svd}({\\bf X})$ in R D. The number of positive eigenvalues of ${\\bf X}^T{\\bf X}$: ${\\tt eigen}({\\bf X}^T{\\bf X})$ in R E. The number of non-zero diagonal entries of ${\\bf R}$ in the QR decomposition of ${\\bf X}={\\bf Q}{\\bf R}$ where ${\\bf Q}$ is an $n\\times k$ matrix with orthogonal columns, and ${\\bf R}$ is a $k\\times k$ upper triangular matrix: qrobj=${\\tt qr}({\\bf X})$, Q=${\\tt qr.Q}$ (qrobj), R=${\\tt qr.R}$ (qrobj) in R F. None of the above\nFrom the above, write at least two proofs for the statement: \"the rank of ${\\bf X}$ and ${\\bf X}^T{\\bf X}$ are the same\".", "answer_v3": [ "AEFG", "ABCEGH", "3", "3", "3", "BCDE" ], "answer_type_v3": [ "MCM", "MCM", "NV", "NV", "NV", "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J" ], [], [], [], [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Statistics_0405", "subject": "Statistics", "topic": "Multiple regression", "subtopic": "Parameter estimates", "level": "4", "keywords": [ "statistics", "regression", "quadratic terms" ], "problem_v1": "This is a problem on interpretation of regression equations which have quadratic terms in explanatory variables.\nContext of data set: A paper helicopter experiment (http://www.paperhelicopterexperiment.com/) was run to find some optimal dimensions. Explanatory variables are body length and body width (both in cm) of a piece of paper before the folding/cutting is done to produce the helicopter. The response variable is the flight time (in seconds) to land on floor after release from a height of 2.5 m. You can compare multiple regressions from fitting (i) a plane, (ii) a quadratic with the original variables and (iii) a quadratic with centred variables (len.centered=len-6, wid.centered=wid-1.5). For (ii) and (iii), compare the numerical results with what you would expect based on transformed equations.\nFor your data set, the response variable is: flight time, in seconds, of the paper helicopter.\nflighttime=c(1.94, 1.73, 0.93, 1.98, 1.1, 1.69, 2.08, 1.83, 1.43, 0.93, 2, 1.87, 1.98, 1.85, 1.3, 1.41, 1.87, 1.94, 1.82, 1.25, 1, 1.21, 1.85, 1.86, 1.89)\nThe explanatory variables are: (i) length of paper before folding len=c(6.1, 6.1, 6.6, 5.6, 5.6, 5.6, 6.6, 5.6, 6.1, 6.6, 6.6, 6.1, 6.6, 6.6, 6.1, 6.1, 5.6, 6.6, 6.1, 5.6, 6.6, 5.6, 6.1, 6.6, 6.1)\n(ii) width of paper before folding wid=c(2.1, 1.4, 0.7, 1.4, 0.7, 2.1, 2.1, 1.4, 0.7, 0.7, 2.1, 1.4, 1.4, 1.4, 0.7, 0.7, 1.4, 1.4, 2.1, 0.7, 0.7, 0.7, 2.1, 2.1, 1.4)\nFor the $i$ th case, let $x_{i1}$ be the value of length and $x^*_{i1}=x_{i1}-6$ be value of centered length; let $x_{i2}$ be the value of width and $x^*_{i2}=x_{i2}-1.5$ be value of centered width; and let $y_{i}$ be the value of flight time. Consider two regression models that are quadratic in the original variables and quadratic in the centered variables.\n$y_i=\\beta_0+\\beta_1x_{i1}+\\beta_2x_{i2}+\\beta_3x_{i1}^2+\\beta_4x_{i2}^2+\\beta_5x_{i1}x_{i2}+\\epsilon_i, \\quad i=1,\\ldots,n$ and $y_i=\\beta^*_0+\\beta^*_1x^*_{i1}+\\beta^*_2x^*_{i2}+\\beta^*_3 (x^*_{i1})^2+\\beta^*_4 (x^*_{i2})^2+\\beta^*_5x^*_{i1}x^*_{i2}+\\epsilon_i, \\quad i=1,\\ldots,n.$ To answer the parts below, several separate regressions could be done or you can answer based on theory.\nPart a) Compare the least squares coefficients for the two quadratic models. Maybe some coefficients are invariant to the centering. Which of the following are correct? Possibly more than one item is correct. [ANS] A. ${\\hat\\beta}^*_2={\\hat\\beta}_2$ B. ${\\hat\\beta}^*_4={\\hat\\beta}_4$ C. ${\\hat\\beta}^*_2\\ne{\\hat\\beta}_2$ D. ${\\hat\\beta}^*_3={\\hat\\beta}_3$ E. ${\\hat\\beta}^*_5={\\hat\\beta}_5$ F. ${\\hat\\beta}^*_1\\ne{\\hat\\beta}_1$ G. ${\\hat\\beta}^*_0={\\hat\\beta}_0$ H. ${\\hat\\beta}^*_1={\\hat\\beta}_1$ I. ${\\hat\\beta}^*_3\\ne{\\hat\\beta}_3$ J. ${\\hat\\beta}^*_0\\ne{\\hat\\beta}_0$ K. ${\\hat\\beta}^*_5\\ne{\\hat\\beta}_5$ L. ${\\hat\\beta}^*_4\\ne{\\hat\\beta}_4$\nPart b) What is the relationship of the least squares coefficient ${\\hat\\beta}_1^*$ with centered length and width variables, and the least squares coefficient ${\\hat\\beta}_0,{\\hat\\beta}_1,\\ldots, {\\hat\\beta}_5$ for the original length and width variables. Which of the following are correct? There might be more than one correct answer. [ANS] A. ${\\hat\\beta}^*_1={\\hat\\beta}_1-12{\\hat\\beta}_3$ B. ${\\hat\\beta}^*_1={\\hat\\beta}_1-12{\\hat\\beta}_3-1.5{\\hat\\beta}_5$ C. ${\\hat\\beta}^*_1={\\hat\\beta}_1+12{\\hat\\beta}_3+1.5{\\hat\\beta}_5$ D. ${\\hat\\beta}^*_1={\\hat\\beta}_1+6{\\hat\\beta}_3$ E. ${\\hat\\beta}^*_1={\\hat\\beta}_1$ F. ${\\hat\\beta}^*_1={\\hat\\beta}_1-6{\\hat\\beta}_3$ G. ${\\hat\\beta}^*_1={\\hat\\beta}_1+12{\\hat\\beta}_3$ H. None of the above\nPart c) Which of following would suggest that quadratic model is a better fit to the data? There might be more than one correct answer. [ANS] A. Adjusted R2 is more than marginally larger for the quadratic B. At least one of $\\beta_3,\\beta_4,\\beta_5$ is significant C. From the physics of the experiment, flight time cannot be linear in length and width of the paper over a wide range of inputs D. At least one of $\\beta_1,\\beta_2$ is significant E. At least one of $\\beta_1^*,\\beta_2^*$ is significant F. None of the above\nPart d) If $\\beta_2$ is not significant, then choose appropriate answers below. There might be more than one correct answer. [ANS] A. width should not be dropped from the model for an improved fit B. one cannot make a conclusion because the coefficient of width in the quadratic model is not invariant to shifting/centering C. width can be dropped from the model for an improved fit with regression on $x_1,x_1^2,x_2^2,x_1x_2$ D. None of the above", "answer_v1": [ "BCDEFJ", "C", "ABC", "B" ], "answer_type_v1": [ "MCM", "MCS", "MCM", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D" ] ], "problem_v2": "This is a problem on interpretation of regression equations which have quadratic terms in explanatory variables.\nContext of data set: A paper helicopter experiment (http://www.paperhelicopterexperiment.com/) was run to find some optimal dimensions. Explanatory variables are body length and body width (both in cm) of a piece of paper before the folding/cutting is done to produce the helicopter. The response variable is the flight time (in seconds) to land on floor after release from a height of 2.5 m. You can compare multiple regressions from fitting (i) a plane, (ii) a quadratic with the original variables and (iii) a quadratic with centred variables (len.centered=len-6, wid.centered=wid-1.5). For (ii) and (iii), compare the numerical results with what you would expect based on transformed equations.\nFor your data set, the response variable is: flight time, in seconds, of the paper helicopter.\nflighttime=c(2.08, 1.21, 1.3, 1.69, 1.89, 1.41, 1.86, 1.94, 1.83, 1.1, 1.94, 1.87, 1.85, 1.43, 1.73, 1.98, 1.25, 1.85, 0.93, 1, 0.93, 2, 1.82, 1.69, 1.87)\nThe explanatory variables are: (i) length of paper before folding len=c(6.6, 5.6, 6.1, 5.6, 6.1, 6.1, 6.6, 6.6, 5.6, 5.6, 6.1, 6.1, 6.1, 6.1, 6.1, 5.6, 5.6, 6.6, 6.6, 6.6, 6.6, 6.6, 6.1, 5.6, 5.6)\n(ii) width of paper before folding wid=c(2.1, 0.7, 0.7, 2.1, 1.4, 0.7, 2.1, 1.4, 1.4, 0.7, 2.1, 1.4, 2.1, 0.7, 1.4, 1.4, 0.7, 1.4, 0.7, 0.7, 0.7, 2.1, 2.1, 2.1, 1.4)\nFor the $i$ th case, let $x_{i1}$ be the value of length and $x^*_{i1}=x_{i1}-6$ be value of centered length; let $x_{i2}$ be the value of width and $x^*_{i2}=x_{i2}-1.5$ be value of centered width; and let $y_{i}$ be the value of flight time. Consider two regression models that are quadratic in the original variables and quadratic in the centered variables.\n$y_i=\\beta_0+\\beta_1x_{i1}+\\beta_2x_{i2}+\\beta_3x_{i1}^2+\\beta_4x_{i2}^2+\\beta_5x_{i1}x_{i2}+\\epsilon_i, \\quad i=1,\\ldots,n$ and $y_i=\\beta^*_0+\\beta^*_1x^*_{i1}+\\beta^*_2x^*_{i2}+\\beta^*_3 (x^*_{i1})^2+\\beta^*_4 (x^*_{i2})^2+\\beta^*_5x^*_{i1}x^*_{i2}+\\epsilon_i, \\quad i=1,\\ldots,n.$ To answer the parts below, several separate regressions could be done or you can answer based on theory.\nPart a) Compare the least squares coefficients for the two quadratic models. Maybe some coefficients are invariant to the centering. Which of the following are correct? Possibly more than one item is correct. [ANS] A. ${\\hat\\beta}^*_5\\ne{\\hat\\beta}_5$ B. ${\\hat\\beta}^*_3\\ne{\\hat\\beta}_3$ C. ${\\hat\\beta}^*_4={\\hat\\beta}_4$ D. ${\\hat\\beta}^*_1={\\hat\\beta}_1$ E. ${\\hat\\beta}^*_5={\\hat\\beta}_5$ F. ${\\hat\\beta}^*_0={\\hat\\beta}_0$ G. ${\\hat\\beta}^*_2={\\hat\\beta}_2$ H. ${\\hat\\beta}^*_1\\ne{\\hat\\beta}_1$ I. ${\\hat\\beta}^*_0\\ne{\\hat\\beta}_0$ J. ${\\hat\\beta}^*_4\\ne{\\hat\\beta}_4$ K. ${\\hat\\beta}^*_3={\\hat\\beta}_3$ L. ${\\hat\\beta}^*_2\\ne{\\hat\\beta}_2$\nPart b) What is the relationship of the least squares coefficient ${\\hat\\beta}_1^*$ with centered length and width variables, and the least squares coefficient ${\\hat\\beta}_0,{\\hat\\beta}_1,\\ldots, {\\hat\\beta}_5$ for the original length and width variables. Which of the following are correct? There might be more than one correct answer. [ANS] A. ${\\hat\\beta}^*_1={\\hat\\beta}_1-6{\\hat\\beta}_3$ B. ${\\hat\\beta}^*_1={\\hat\\beta}_1+6{\\hat\\beta}_3$ C. ${\\hat\\beta}^*_1={\\hat\\beta}_1-12{\\hat\\beta}_3-1.5{\\hat\\beta}_5$ D. ${\\hat\\beta}^*_1={\\hat\\beta}_1-12{\\hat\\beta}_3$ E. ${\\hat\\beta}^*_1={\\hat\\beta}_1+12{\\hat\\beta}_3+1.5{\\hat\\beta}_5$ F. ${\\hat\\beta}^*_1={\\hat\\beta}_1+12{\\hat\\beta}_3$ G. ${\\hat\\beta}^*_1={\\hat\\beta}_1$ H. None of the above\nPart c) Which of following would suggest that quadratic model is a better fit to the data? There might be more than one correct answer. [ANS] A. At least one of $\\beta_3,\\beta_4,\\beta_5$ is significant B. At least one of $\\beta_1^*,\\beta_2^*$ is significant C. At least one of $\\beta_1,\\beta_2$ is significant D. From the physics of the experiment, flight time cannot be linear in length and width of the paper over a wide range of inputs E. Adjusted R2 is more than marginally larger for the quadratic F. None of the above\nPart d) If $\\beta_2$ is not significant, then choose appropriate answers below. There might be more than one correct answer. [ANS] A. one cannot make a conclusion because the coefficient of width in the quadratic model is not invariant to shifting/centering B. width can be dropped from the model for an improved fit with regression on $x_1,x_1^2,x_2^2,x_1x_2$ C. width should not be dropped from the model for an improved fit D. None of the above", "answer_v2": [ "CEHIKL", "E", "ADE", "A" ], "answer_type_v2": [ "MCM", "MCS", "MCM", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D" ] ], "problem_v3": "This is a problem on interpretation of regression equations which have quadratic terms in explanatory variables.\nContext of data set: A paper helicopter experiment (http://www.paperhelicopterexperiment.com/) was run to find some optimal dimensions. Explanatory variables are body length and body width (both in cm) of a piece of paper before the folding/cutting is done to produce the helicopter. The response variable is the flight time (in seconds) to land on floor after release from a height of 2.5 m. You can compare multiple regressions from fitting (i) a plane, (ii) a quadratic with the original variables and (iii) a quadratic with centred variables (len.centered=len-6, wid.centered=wid-1.5). For (ii) and (iii), compare the numerical results with what you would expect based on transformed equations.\nFor your data set, the response variable is: flight time, in seconds, of the paper helicopter.\nflighttime=c(1.87, 1.73, 1.98, 1.89, 2.08, 1.69, 1.25, 1.21, 1.85, 1.3, 1.82, 1.43, 1.86, 2, 1.61, 1.87, 0.93, 1.83, 1.94, 1.69, 1.98, 0.93, 1.94, 1.1, 1)\nThe explanatory variables are: (i) length of paper before folding len=c(6.1, 6.1, 5.6, 6.1, 6.6, 5.6, 5.6, 5.6, 6.1, 6.1, 6.1, 6.1, 6.6, 6.6, 5.6, 5.6, 6.6, 5.6, 6.1, 5.6, 6.6, 6.6, 6.6, 5.6, 6.6)\n(ii) width of paper before folding wid=c(1.4, 1.4, 1.4, 1.4, 2.1, 2.1, 0.7, 0.7, 2.1, 0.7, 2.1, 0.7, 2.1, 2.1, 2.1, 1.4, 0.7, 1.4, 2.1, 2.1, 1.4, 0.7, 1.4, 0.7, 0.7)\nFor the $i$ th case, let $x_{i1}$ be the value of length and $x^*_{i1}=x_{i1}-6$ be value of centered length; let $x_{i2}$ be the value of width and $x^*_{i2}=x_{i2}-1.5$ be value of centered width; and let $y_{i}$ be the value of flight time. Consider two regression models that are quadratic in the original variables and quadratic in the centered variables.\n$y_i=\\beta_0+\\beta_1x_{i1}+\\beta_2x_{i2}+\\beta_3x_{i1}^2+\\beta_4x_{i2}^2+\\beta_5x_{i1}x_{i2}+\\epsilon_i, \\quad i=1,\\ldots,n$ and $y_i=\\beta^*_0+\\beta^*_1x^*_{i1}+\\beta^*_2x^*_{i2}+\\beta^*_3 (x^*_{i1})^2+\\beta^*_4 (x^*_{i2})^2+\\beta^*_5x^*_{i1}x^*_{i2}+\\epsilon_i, \\quad i=1,\\ldots,n.$ To answer the parts below, several separate regressions could be done or you can answer based on theory.\nPart a) Compare the least squares coefficients for the two quadratic models. Maybe some coefficients are invariant to the centering. Which of the following are correct? Possibly more than one item is correct. [ANS] A. ${\\hat\\beta}^*_3={\\hat\\beta}_3$ B. ${\\hat\\beta}^*_0={\\hat\\beta}_0$ C. ${\\hat\\beta}^*_2={\\hat\\beta}_2$ D. ${\\hat\\beta}^*_4\\ne{\\hat\\beta}_4$ E. ${\\hat\\beta}^*_1\\ne{\\hat\\beta}_1$ F. ${\\hat\\beta}^*_0\\ne{\\hat\\beta}_0$ G. ${\\hat\\beta}^*_5={\\hat\\beta}_5$ H. ${\\hat\\beta}^*_4={\\hat\\beta}_4$ I. ${\\hat\\beta}^*_2\\ne{\\hat\\beta}_2$ J. ${\\hat\\beta}^*_1={\\hat\\beta}_1$ K. ${\\hat\\beta}^*_3\\ne{\\hat\\beta}_3$ L. ${\\hat\\beta}^*_5\\ne{\\hat\\beta}_5$\nPart b) What is the relationship of the least squares coefficient ${\\hat\\beta}_1^*$ with centered length and width variables, and the least squares coefficient ${\\hat\\beta}_0,{\\hat\\beta}_1,\\ldots, {\\hat\\beta}_5$ for the original length and width variables. Which of the following are correct? There might be more than one correct answer. [ANS] A. ${\\hat\\beta}^*_1={\\hat\\beta}_1$ B. ${\\hat\\beta}^*_1={\\hat\\beta}_1+12{\\hat\\beta}_3$ C. ${\\hat\\beta}^*_1={\\hat\\beta}_1+6{\\hat\\beta}_3$ D. ${\\hat\\beta}^*_1={\\hat\\beta}_1+12{\\hat\\beta}_3+1.5{\\hat\\beta}_5$ E. ${\\hat\\beta}^*_1={\\hat\\beta}_1-6{\\hat\\beta}_3$ F. ${\\hat\\beta}^*_1={\\hat\\beta}_1-12{\\hat\\beta}_3$ G. ${\\hat\\beta}^*_1={\\hat\\beta}_1-12{\\hat\\beta}_3-1.5{\\hat\\beta}_5$ H. None of the above\nPart c) Which of following would suggest that quadratic model is a better fit to the data? There might be more than one correct answer. [ANS] A. From the physics of the experiment, flight time cannot be linear in length and width of the paper over a wide range of inputs B. Adjusted R2 is more than marginally larger for the quadratic C. At least one of $\\beta_1^*,\\beta_2^*$ is significant D. At least one of $\\beta_3,\\beta_4,\\beta_5$ is significant E. At least one of $\\beta_1,\\beta_2$ is significant F. None of the above\nPart d) If $\\beta_2$ is not significant, then choose appropriate answers below. There might be more than one correct answer. [ANS] A. one cannot make a conclusion because the coefficient of width in the quadratic model is not invariant to shifting/centering B. width can be dropped from the model for an improved fit with regression on $x_1,x_1^2,x_2^2,x_1x_2$ C. width should not be dropped from the model for an improved fit D. None of the above", "answer_v3": [ "AEFGHI", "D", "ABD", "A" ], "answer_type_v3": [ "MCM", "MCS", "MCM", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0406", "subject": "Statistics", "topic": "Multiple regression", "subtopic": "Parameter estimates", "level": "3", "keywords": [ "statistics", "multpile regression", "regression" ], "problem_v1": "The graphical depiction of the equation of a multiple regression model with $k$ independent variables $k>1$ is referred to as: [ANS] A. a response surface B. a plane only when $k=3$ C. a response variable D. a straight line\nIf all the points for a multiple regression model with two independent variables were on the regression plane, then the multiple coefficient of determination would equal: [ANS] A. any number between 0 and 2 B. 0 C. 1 D. 2, since there are two independent variables", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "The graphical depiction of the equation of a multiple regression model with $k$ independent variables $k>1$ is referred to as: [ANS] A. a response variable B. a plane only when $k=3$ C. a straight line D. a response surface\nFor the following multiple regression model $ \\qquad \\begin{array}{rrcl} \\hat{y}=2-3x_1+4x_2+5x_3 \\end{array}$ a unit increase in $x_1$, holding $x_2$ and $x_3$ constant, results in: [ANS] A. a decrease of 3 units on average in the value of $y$ B. an increase of 8 units in the value of $y$ C. an increase of 3 units in the value of $y$ D. a decrease of 3 units in the value of $y$", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "The graphical depiction of the equation of a multiple regression model with $k$ independent variables $k>1$ is referred to as: [ANS] A. a response variable B. a response surface C. a plane only when $k=3$ D. a straight line\nThe multiple coefficient of determination is defined as: [ANS] A. $1-(SSE/SST)$ B. $MSE/MSR$ C. $SSE/SST$ D. $1-(MSE/MSR)$", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0407", "subject": "Statistics", "topic": "Multiple regression", "subtopic": "Parameter estimates", "level": "1", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "Below is Excel output from a multiple regression: \\begin{array}{|l|r|r|r|r|r|} \\hline \\multicolumn{2}{|c|}{\\mbox{Regression Statistics}} & & & & \\\\ \\hline \\mbox{Multiple R} & 0.6207 & & & & \\\\ \\hline \\mbox{R Square} & 0.3853 & & & & \\\\ \\hline \\mbox{Adjusted R Square} & 0.3434 & & & & \\\\ \\hline \\mbox{Standard Error} & 5.59 & & & & \\\\ \\hline \\mbox{Observations} & 48 & & & & \\\\ \\hline & & & & & \\\\ \\hline \\mbox{ANOVA} & & & & & \\\\ \\hline & \\mbox{df} &\\mbox{SS} &\\mbox{MS} &\\mbox{F} &\\mbox{Significance F} \\\\ \\hline \\mbox{Regression} & 3 & 862 & 287.33 & 9.19 & 0.0000776348\\\\ \\hline \\mbox{Residual} & 44 & 1375 & 31.25 & & \\\\ \\hline \\mbox{Total} & 47 & 2237 & & & \\\\ \\hline & & & & & \\\\ \\hline & \\mbox{Coefficients} & \\mbox{Standard Error} & \\mbox{t Stat} & \\mbox{P-value} & \\\\ \\hline \\mbox{Intercept} & 4.5 & 3.2 & 1.41 & 0.1656 & \\\\ \\hline \\mbox{x1} &-4 & 2.7 &-1.48 & 0.146 & \\\\ \\hline \\mbox{x2} &-3.4 & 3 &-1.13 & 0.2646 & \\\\ \\hline \\mbox{x3} & 1.4 & 3.3 & 0.42 & 0.6765 & \\\\ \\hline \\end{array} Find the point estimate $\\hat{y}$ when $x_1=-4$, $x_2=-1$, and $x_3=-2$. Point Estimate=[ANS]", "answer_v1": [ "21.1" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Below is Excel output from a multiple regression: \\begin{array}{|l|r|r|r|r|r|} \\hline \\multicolumn{2}{|c|}{\\mbox{Regression Statistics}} & & & & \\\\ \\hline \\mbox{Multiple R} & 0.5959 & & & & \\\\ \\hline \\mbox{R Square} & 0.3551 & & & & \\\\ \\hline \\mbox{Adjusted R Square} & 0.3014 & & & & \\\\ \\hline \\mbox{Standard Error} & 6.41 & & & & \\\\ \\hline \\mbox{Observations} & 40 & & & & \\\\ \\hline & & & & & \\\\ \\hline \\mbox{ANOVA} & & & & & \\\\ \\hline & \\mbox{df} &\\mbox{SS} &\\mbox{MS} &\\mbox{F} &\\mbox{Significance F} \\\\ \\hline \\mbox{Regression} & 3 & 815 & 271.67 & 6.61 & 0.00113109\\\\ \\hline \\mbox{Residual} & 36 & 1480 & 41.11 & & \\\\ \\hline \\mbox{Total} & 39 & 2295 & & & \\\\ \\hline & & & & & \\\\ \\hline & \\mbox{Coefficients} & \\mbox{Standard Error} & \\mbox{t Stat} & \\mbox{P-value} & \\\\ \\hline \\mbox{Intercept} &-3.3 & 2.6 &-1.27 & 0.2122 & \\\\ \\hline \\mbox{x1} & 9 & 3.1 & 2.9 & 0.0063 & \\\\ \\hline \\mbox{x2} &-3.7 & 2.1 &-1.76 & 0.0869 & \\\\ \\hline \\mbox{x3} &-6.4 & 3.3 &-1.94 & 0.0602 & \\\\ \\hline \\end{array} Find the point estimate $\\hat{y}$ when $x_1=-1$, $x_2=5$, and $x_3=-5$. Point Estimate=[ANS]", "answer_v2": [ "1.2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Below is Excel output from a multiple regression: \\begin{array}{|l|r|r|r|r|r|} \\hline \\multicolumn{2}{|c|}{\\mbox{Regression Statistics}} & & & & \\\\ \\hline \\mbox{Multiple R} & 0.6121 & & & & \\\\ \\hline \\mbox{R Square} & 0.3747 & & & & \\\\ \\hline \\mbox{Adjusted R Square} & 0.3266 & & & & \\\\ \\hline \\mbox{Standard Error} & 5.95 & & & & \\\\ \\hline \\mbox{Observations} & 43 & & & & \\\\ \\hline & & & & & \\\\ \\hline \\mbox{ANOVA} & & & & & \\\\ \\hline & \\mbox{df} &\\mbox{SS} &\\mbox{MS} &\\mbox{F} &\\mbox{Significance F} \\\\ \\hline \\mbox{Regression} & 3 & 828 & 276 & 7.79 & 0.000341184\\\\ \\hline \\mbox{Residual} & 39 & 1382 & 35.44 & & \\\\ \\hline \\mbox{Total} & 42 & 2210 & & & \\\\ \\hline & & & & & \\\\ \\hline & \\mbox{Coefficients} & \\mbox{Standard Error} & \\mbox{t Stat} & \\mbox{P-value} & \\\\ \\hline \\mbox{Intercept} & 1 & 3.9 & 0.26 & 0.7962 & \\\\ \\hline \\mbox{x1} &-5.9 & 3.8 &-1.55 & 0.1292 & \\\\ \\hline \\mbox{x2} &-3.1 & 2.4 &-1.29 & 0.2046 & \\\\ \\hline \\mbox{x3} & 6.2 & 2.6 & 2.38 & 0.0223 & \\\\ \\hline \\end{array} Find the point estimate $\\hat{y}$ when $x_1=-4$, $x_2=-8$, and $x_3=1$. Point Estimate=[ANS]", "answer_v3": [ "55.6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Statistics_0408", "subject": "Statistics", "topic": "Multiple regression", "subtopic": "Parameter estimates", "level": "2", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "Below is partial Excel output from a multiple regression, with some values missing: \\begin{array}{|l|r|r|r|r|r|} \\hline \\multicolumn{2}{|c|}{\\mbox{Regression Statistics}} & & & & \\\\ \\hline \\mbox{Multiple R} & & & & & \\\\ \\hline \\mbox{R Square} & & & & & \\\\ \\hline \\mbox{Adjusted R Square} & & & & & \\\\ \\hline \\mbox{Standard Error} & 5.89 & & & & \\\\ \\hline \\mbox{Observations} & 48 & & & & \\\\ \\hline & & & & & \\\\ \\hline \\mbox{ANOVA} & & & & & \\\\ \\hline & \\mbox{df} &\\mbox{SS} &\\mbox{MS} &\\mbox{F} &\\mbox{Significance F} \\\\ \\hline \\mbox{Regression} & 7 & 872 & 124.57 & 3.6 & 0.00425129\\\\ \\hline \\mbox{Residual} & 40 & 1386 & 34.65 & & \\\\ \\hline \\mbox{Total} & 47 & 2258 & & & \\\\ \\hline \\end{array} Determine each of the following from the output: $R^2$=[ANS]\nAdjusted $R^2$=[ANS]", "answer_v1": [ "0.3862", "0.2788" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Below is partial Excel output from a multiple regression, with some values missing: \\begin{array}{|l|r|r|r|r|r|} \\hline \\multicolumn{2}{|c|}{\\mbox{Regression Statistics}} & & & & \\\\ \\hline \\mbox{Multiple R} & & & & & \\\\ \\hline \\mbox{R Square} & & & & & \\\\ \\hline \\mbox{Adjusted R Square} & & & & & \\\\ \\hline \\mbox{Standard Error} & 6.44 & & & & \\\\ \\hline \\mbox{Observations} & 40 & & & & \\\\ \\hline & & & & & \\\\ \\hline \\mbox{ANOVA} & & & & & \\\\ \\hline & \\mbox{df} &\\mbox{SS} &\\mbox{MS} &\\mbox{F} &\\mbox{Significance F} \\\\ \\hline \\mbox{Regression} & 9 & 833 & 92.56 & 2.23 & 0.0481854\\\\ \\hline \\mbox{Residual} & 30 & 1245 & 41.5 & & \\\\ \\hline \\mbox{Total} & 39 & 2078 & & & \\\\ \\hline \\end{array} Determine each of the following from the output: $R^2$=[ANS]\nAdjusted $R^2$=[ANS]", "answer_v2": [ "0.4009", "0.2211" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Below is partial Excel output from a multiple regression, with some values missing: \\begin{array}{|l|r|r|r|r|r|} \\hline \\multicolumn{2}{|c|}{\\mbox{Regression Statistics}} & & & & \\\\ \\hline \\mbox{Multiple R} & & & & & \\\\ \\hline \\mbox{R Square} & & & & & \\\\ \\hline \\mbox{Adjusted R Square} & & & & & \\\\ \\hline \\mbox{Standard Error} & 6.14 & & & & \\\\ \\hline \\mbox{Observations} & 43 & & & & \\\\ \\hline & & & & & \\\\ \\hline \\mbox{ANOVA} & & & & & \\\\ \\hline & \\mbox{df} &\\mbox{SS} &\\mbox{MS} &\\mbox{F} &\\mbox{Significance F} \\\\ \\hline \\mbox{Regression} & 8 & 855 & 106.88 & 2.83 & 0.0160603\\\\ \\hline \\mbox{Residual} & 34 & 1283 & 37.74 & & \\\\ \\hline \\mbox{Total} & 42 & 2138 & & & \\\\ \\hline \\end{array} Determine each of the following from the output: $R^2$=[ANS]\nAdjusted $R^2$=[ANS]", "answer_v3": [ "0.3999", "0.2587" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0409", "subject": "Statistics", "topic": "Multiple regression", "subtopic": "Parameter estimates", "level": "3", "keywords": [ "statistics", "multpile regression", "regression" ], "problem_v1": "A multiple regression model involves 10 independent variables and 30 observations. If we want to test at the 5\\% significance level the parameter $\\beta_4$, the critical value will be: [ANS] A. 2.093 B. 1.729 C. 2.228 D. 1.697\nIn a multiple regression analysis involving $k$ independent variables and $n$ data points, the degrees of freedom associated with the SSE is: [ANS] A. $n-k$ B. $k-1$ C. $n-k-1$ D. $n-1$", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "A multiple regression model involves 10 independent variables and 30 observations. If we want to test at the 5\\% significance level the parameter $\\beta_4$, the critical value will be: [ANS] A. 2.228 B. 1.729 C. 1.697 D. 2.093\nA multiple regression model has the form $ \\qquad \\begin{array}{rrcl} \\hat{y}=8+3x_1+5x_2-4x_3 \\end{array}$ As $x_3$ increases by 1 unit, with $x_1$ and $x_2$ held constant, the $y$, on average, is expected to: [ANS] A. decrease by 4 units B. decrease by 16 units C. increase by 1 unit D. increase by 12 units", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "A multiple regression model involves 10 independent variables and 30 observations. If we want to test at the 5\\% significance level the parameter $\\beta_4$, the critical value will be: [ANS] A. 2.228 B. 2.093 C. 1.729 D. 1.697\nTo test the validity of a multiple regression model, we test the null hypothesis that the regression coefficients are all zero. We apply: [ANS] A. the $z-$ test B. the $F-$ test C. the $t-$ test D. either the $t-$ test or the $z-$ test", "answer_v3": [ "B", "B" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0410", "subject": "Statistics", "topic": "Multiple regression", "subtopic": "Parameter estimates", "level": "", "keywords": [ "statistics", "multiple regression", "regression" ], "problem_v1": "Select True or False, depending on whether the corresponding statement is true or false.\n[ANS] 1. For each $x$ term in the multiple regression equation, the corresponding $\\beta$ is referred to as a partial regression coefficient. [ANS] 2. In reference to the equation $\\hat{y}=-0.80+0.12x_1+0.08x_2$, the value-0.80 is the $y-$ intercept. [ANS] 3. In a multiple regression problem involving 24 observations and three independent variables, the estimated regression equation is $\\hat{y}=72+3.2x_1+1.5x_2-x_3$. For this model, $SST=800$ and $SSE=245$. The value of the $F$ statistic for testing the significance of this model is 15.102. [ANS] 4. In order to test the significance of a multiple regression model involving 4 independent variables and 25 observations, the numerator and denominator degrees of freedom (respectively) for the critical value of $F$ are 3 and 21.", "answer_v1": [ "T", "T", "T", "F" ], "answer_type_v1": [ "TF", "TF", "TF", "TF" ], "options_v1": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v2": "Select True or False, depending on whether the corresponding statement is true or false.\n[ANS] 1. In reference to the equation $\\hat{y}=-0.80+0.12x_1+0.08x_2$, the value-0.80 is the $y-$ intercept. [ANS] 2. In order to test the significance of a multiple regression model involving 4 independent variables and 25 observations, the numerator and denominator degrees of freedom (respectively) for the critical value of $F$ are 3 and 21. [ANS] 3. For each $x$ term in the multiple regression equation, the corresponding $\\beta$ is referred to as a partial regression coefficient. [ANS] 4. In a multiple regression problem, the regression equation is $\\hat{y}=60.6-5.2x_1+0.75x_2$. The estimated value for $y$ when $x_1=3$ and $x_2=4$ is 48.", "answer_v2": [ "T", "F", "T", "T" ], "answer_type_v2": [ "TF", "TF", "TF", "TF" ], "options_v2": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v3": "Select True or False, depending on whether the corresponding statement is true or false.\n[ANS] 1. Multicollinearity is a situation in which two or more independent variables are highly correlated with each other. [ANS] 2. In a multiple regression problem, the regression equation is $\\hat{y}=60.6-5.2x_1+0.75x_2$. The estimated value for $y$ when $x_1=3$ and $x_2=4$ is 48. [ANS] 3. For each $x$ term in the multiple regression equation, the corresponding $\\beta$ is referred to as a partial regression coefficient. [ANS] 4. In reference to the equation $\\hat{y}=-0.80+0.12x_1+0.08x_2$, the value 0.12 is the average change in $y$ per unit change in $x_1$, when $x_2$ is held constant.", "answer_v3": [ "T", "T", "T", "T" ], "answer_type_v3": [ "TF", "TF", "TF", "TF" ], "options_v3": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ] }, { "id": "Statistics_0411", "subject": "Statistics", "topic": "Multiple regression", "subtopic": "Parameter estimates", "level": "3", "keywords": [ "statistics", "multpile regression", "regression" ], "problem_v1": "In a multiple regression analysis involving 40 observations and 5 independent variables, the total variation $SST=350$ and $SSE=50$. The multiple coefficient of determination is: [ANS] A. 0.8571 B. 0.8529 C. 0.8469 D. 0.8408\nIn a multiple regression analysis involving 20 observations and 5 independent variables, the total variation $SST=250$ and $SSE=35$. The multiple coefficient of determination adjusted for degrees of freedom is: [ANS] A. 0.831 B. 0.860 C. 0.810 D. 0.835", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "In a multiple regression analysis involving 40 observations and 5 independent variables, the total variation $SST=350$ and $SSE=50$. The multiple coefficient of determination is: [ANS] A. 0.8469 B. 0.8529 C. 0.8408 D. 0.8571\nIn a regression model involving 50 observations, the following estimated regression model was obtained: $ \\qquad \\begin{array}{rrcl} \\hat{y}=10.5+3.2x_1+5.8x_2+6.5x_3 \\end{array}$ for this model, $SSR=450$ and $SSE=175$. The value of the $MSR$ is: [ANS] A. 150 B. 3.804 C. 12.50 D. 275", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "In a multiple regression analysis involving 40 observations and 5 independent variables, the total variation $SST=350$ and $SSE=50$. The multiple coefficient of determination is: [ANS] A. 0.8469 B. 0.8571 C. 0.8529 D. 0.8408\nIn multiple regression analysis involving 10 independent variables and 100 observations, the critical value $t$ for testing individual coefficients in the model will have: [ANS] A. 89 degrees of freedom B. 10 degrees of freedom C. 100 degrees of freedom D. 9 degrees of freedom", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0412", "subject": "Statistics", "topic": "Multiple regression", "subtopic": "Parameter estimates", "level": "3", "keywords": [ "statistics", "hypothesis testing" ], "problem_v1": "Below is partial Excel output from a multiple regression, with some information missing: \\begin{array}{|l|r|r|r|r|r|} \\hline \\mbox{ANOVA} & & & & & \\\\ \\hline & \\mbox{df} &\\mbox{SS} &\\mbox{MS} &\\mbox{F} &\\mbox{Significance F} \\\\ \\hline \\mbox{Regression} & & 872 & & 18.43 & \\\\ \\hline \\mbox{Residual} & & 262 & & & \\\\ \\hline \\mbox{Total} & 72 & 1134 & & & \\\\ \\hline \\end{array} Determine the degrees of freedom for the regression and degrees of freedom for the error (you may need to round to the nearest integer): Degrees of Freedom for the Regression=[ANS]\nDegrees of Freedom for the Error=[ANS]", "answer_v1": [ "11", "61" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Below is partial Excel output from a multiple regression, with some information missing: \\begin{array}{|l|r|r|r|r|r|} \\hline \\mbox{ANOVA} & & & & & \\\\ \\hline & \\mbox{df} &\\mbox{SS} &\\mbox{MS} &\\mbox{F} &\\mbox{Significance F} \\\\ \\hline \\mbox{Regression} & & 833 & & 9.3 & \\\\ \\hline \\mbox{Residual} & & 215 & & & \\\\ \\hline \\mbox{Total} & 51 & 1048 & & & \\\\ \\hline \\end{array} Determine the degrees of freedom for the regression and degrees of freedom for the error (you may need to round to the nearest integer): Degrees of Freedom for the Regression=[ANS]\nDegrees of Freedom for the Error=[ANS]", "answer_v2": [ "15", "36" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Below is partial Excel output from a multiple regression, with some information missing: \\begin{array}{|l|r|r|r|r|r|} \\hline \\mbox{ANOVA} & & & & & \\\\ \\hline & \\mbox{df} &\\mbox{SS} &\\mbox{MS} &\\mbox{F} &\\mbox{Significance F} \\\\ \\hline \\mbox{Regression} & & 855 & & 16.03 & \\\\ \\hline \\mbox{Residual} & & 228 & & & \\\\ \\hline \\mbox{Total} & 58 & 1083 & & & \\\\ \\hline \\end{array} Determine the degrees of freedom for the regression and degrees of freedom for the error (you may need to round to the nearest integer): Degrees of Freedom for the Regression=[ANS]\nDegrees of Freedom for the Error=[ANS]", "answer_v3": [ "11", "47" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0413", "subject": "Statistics", "topic": "Multiple regression", "subtopic": "Hypothesis tests", "level": "3", "keywords": [ "statistics", "multpile regression", "regression" ], "problem_v1": "A multiple regression analysis involving 3 independent variables and 25 data points results in a value of 0.769 for the unadjusted multiple coefficient of determination. The adjusted multiple coefficient of determination is: [ANS] A. 0.736 B. 0.591 C. 0.877 D. 0.385\nIf multicollinearity exists among the independent variables included in a multiple regression model, the: [ANS] A. multiple coefficient of determination will assume a value close to zero B. regression coefficients will be difficult to interpret C. standard errors of the regression coefficients for the correlated independent variables will increase D. regression coefficients will be difficult to interpret and the standard errors of the regression coefficients for the correlated independent variables will increase", "answer_v1": [ "A", "D" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "A multiple regression analysis involving 3 independent variables and 25 data points results in a value of 0.769 for the unadjusted multiple coefficient of determination. The adjusted multiple coefficient of determination is: [ANS] A. 0.877 B. 0.591 C. 0.385 D. 0.736\nTo test the validity of a multiple regression model involving 2 independent variables, the null hypothesis is that: [ANS] A. $\\beta_1=\\beta_2=0$ B. $\\beta_1\\not=\\beta_2$ C. $\\beta_0=\\beta_1=\\beta_2$ D. $\\beta_1=\\beta_2$", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "A multiple regression analysis involving 3 independent variables and 25 data points results in a value of 0.769 for the unadjusted multiple coefficient of determination. The adjusted multiple coefficient of determination is: [ANS] A. 0.877 B. 0.736 C. 0.591 D. 0.385\nA multiple regression model has the form $ \\qquad \\begin{array}{rrcl} \\hat{y}=b_0+b_1x_1+b_2x_2 \\end{array}$ The coefficient $b_1$ is interpreted as the: [ANS] A. change in the average value of $y$ per unit change in $x_1$, holding $x_2$ constant B. change in $y$ per unit change in $x_1$, when $x_1$ and $x_2$ values are correlated C. change in $y$ per unit change in $x_1$ D. change in $y$ per unit change in $x_1$, holding $x_2$ constant", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0414", "subject": "Statistics", "topic": "Multiple regression", "subtopic": "Multiple selection", "level": "3", "keywords": [ "statistics", "regression" ], "problem_v1": "For multiple regression, the definitions of $R^2$ and adjusted $R^2$ are: $R^2=1-{SS(Res)\\over SS(Total)}$ adjusted $R^2=1-{SS(Res)/(n-k)\\over SS(Total)/(n-1)}$ where $k$ is the number of betas and $n$ is the sample size. Assume that $n>k$.\nPart a) Which of the following is a correct version of adjusted $R^2$ and shows clearly that adjusted $R^2$ is $\\le R^2$. Choose the single best item [ANS] A. $1-{SS(Res)\\over SS(Total)}+{(k-1)SS(Res)\\over (n-1)SS(Total)}$ B. $1-{(n-1){\\hat\\sigma}^2\\over \\sum_i y_i^2}$ C. $1-{(n-1)SS(Res)\\over (n-k)SS(Total)}$ D. $1-{{\\hat\\sigma}^2\\over s_y^2}$ E. $1-{SS(Res)\\over SS(Total)}-{(k-1)SS(Res)\\over (n-k)SS(Total)}$ F. $1-{(n-k)SS(Res)\\over (n-1)SS(Total)}$ G. $1-{SS(Res)\\over SS(Total)}-{(k-1)SS(Res)\\over (n-1)SS(Total)}$ H. None of the above\nPart b) Which of the following is a correct version of adjusted $R^2$ and shows clearly that adjusted $R^2$ increases as ${\\hat\\sigma}$ decreases. Choose the single best item [ANS] A. $1-{(n-1)SS(Res)\\over (n-k)SS(Total)}$ B. $1-{{\\hat\\sigma}^2\\over s_y^2}$ C. $1-{SS(Res)\\over SS(Total)}-{(k-1)SS(Res)\\over (n-1)SS(Total)}$ D. $1-{SS(Res)\\over SS(Total)}+{(k-1)SS(Res)\\over (n-1)SS(Total)}$ E. $1-{(n-1){\\hat\\sigma}^2\\over \\sum_i y_i^2}$ F. $1-{(n-k)SS(Res)\\over (n-1)SS(Total)}$ G. $1-{SS(Res)\\over SS(Total)}-{(k-1)SS(Res)\\over (n-k)SS(Total)}$ H. None of the above", "answer_v1": [ "E", "B" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ] ], "problem_v2": "For multiple regression, the definitions of $R^2$ and adjusted $R^2$ are: $R^2=1-{SS(Res)\\over SS(Total)}$ adjusted $R^2=1-{SS(Res)/(n-k)\\over SS(Total)/(n-1)}$ where $k$ is the number of betas and $n$ is the sample size. Assume that $n>k$.\nPart a) Which of the following is a correct version of adjusted $R^2$ and shows clearly that adjusted $R^2$ is $\\le R^2$. Choose the single best item [ANS] A. $1-{(n-1){\\hat\\sigma}^2\\over \\sum_i y_i^2}$ B. $1-{(n-1)SS(Res)\\over (n-k)SS(Total)}$ C. $1-{SS(Res)\\over SS(Total)}-{(k-1)SS(Res)\\over (n-k)SS(Total)}$ D. $1-{{\\hat\\sigma}^2\\over s_y^2}$ E. $1-{(n-k)SS(Res)\\over (n-1)SS(Total)}$ F. $1-{SS(Res)\\over SS(Total)}+{(k-1)SS(Res)\\over (n-1)SS(Total)}$ G. $1-{SS(Res)\\over SS(Total)}-{(k-1)SS(Res)\\over (n-1)SS(Total)}$ H. None of the above\nPart b) Which of the following is a correct version of adjusted $R^2$ and shows clearly that adjusted $R^2$ increases as ${\\hat\\sigma}$ decreases. Choose the single best item [ANS] A. $1-{SS(Res)\\over SS(Total)}+{(k-1)SS(Res)\\over (n-1)SS(Total)}$ B. $1-{SS(Res)\\over SS(Total)}-{(k-1)SS(Res)\\over (n-1)SS(Total)}$ C. $1-{{\\hat\\sigma}^2\\over s_y^2}$ D. $1-{(n-1)SS(Res)\\over (n-k)SS(Total)}$ E. $1-{(n-k)SS(Res)\\over (n-1)SS(Total)}$ F. $1-{SS(Res)\\over SS(Total)}-{(k-1)SS(Res)\\over (n-k)SS(Total)}$ G. $1-{(n-1){\\hat\\sigma}^2\\over \\sum_i y_i^2}$ H. None of the above", "answer_v2": [ "C", "C" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ] ], "problem_v3": "For multiple regression, the definitions of $R^2$ and adjusted $R^2$ are: $R^2=1-{SS(Res)\\over SS(Total)}$ adjusted $R^2=1-{SS(Res)/(n-k)\\over SS(Total)/(n-1)}$ where $k$ is the number of betas and $n$ is the sample size. Assume that $n>k$.\nPart a) Which of the following is a correct version of adjusted $R^2$ and shows clearly that adjusted $R^2$ is $\\le R^2$. Choose the single best item [ANS] A. $1-{SS(Res)\\over SS(Total)}+{(k-1)SS(Res)\\over (n-1)SS(Total)}$ B. $1-{SS(Res)\\over SS(Total)}-{(k-1)SS(Res)\\over (n-k)SS(Total)}$ C. $1-{SS(Res)\\over SS(Total)}-{(k-1)SS(Res)\\over (n-1)SS(Total)}$ D. $1-{(n-k)SS(Res)\\over (n-1)SS(Total)}$ E. $1-{(n-1)SS(Res)\\over (n-k)SS(Total)}$ F. $1-{(n-1){\\hat\\sigma}^2\\over \\sum_i y_i^2}$ G. $1-{{\\hat\\sigma}^2\\over s_y^2}$ H. None of the above\nPart b) Which of the following is a correct version of adjusted $R^2$ and shows clearly that adjusted $R^2$ increases as ${\\hat\\sigma}$ decreases. Choose the single best item [ANS] A. $1-{(n-1){\\hat\\sigma}^2\\over \\sum_i y_i^2}$ B. $1-{(n-1)SS(Res)\\over (n-k)SS(Total)}$ C. $1-{SS(Res)\\over SS(Total)}+{(k-1)SS(Res)\\over (n-1)SS(Total)}$ D. $1-{{\\hat\\sigma}^2\\over s_y^2}$ E. $1-{(n-k)SS(Res)\\over (n-1)SS(Total)}$ F. $1-{SS(Res)\\over SS(Total)}-{(k-1)SS(Res)\\over (n-1)SS(Total)}$ G. $1-{SS(Res)\\over SS(Total)}-{(k-1)SS(Res)\\over (n-k)SS(Total)}$ H. None of the above", "answer_v3": [ "B", "D" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ] ] }, { "id": "Statistics_0415", "subject": "Statistics", "topic": "Multiple regression", "subtopic": "Model selection", "level": "4", "keywords": [ "statistics", "multiple regression", "cross-validation" ], "problem_v1": "The questions involves the data set for Richmond townhouses obtained on 2014.11.03. For your subset, the response variable is: asking price divided by 10000: askpr=c(41.99, 77.8, 46.8, 54.8, 49.9, 51.68, 50.5, 65.8, 40.9, 60.8, 40.8, 50.8, 79.8, 40.8, 33.7, 57.8, 47.8, 86.8, 56.8, 50.8, 65.99, 47.9, 54.98, 68.5, 26.99, 55.2, 48.8, 53.9, 52.4, 57.8, 74.8, 45.99, 78.8, 58.39, 71.99, 62.9, 53.8, 58.8, 73.8, 108.8, 73.9, 55.8, 61.5, 62.8888, 54.8, 25.9, 59.8, 51.99, 68.5, 44.8) The explanatory variables are: (i) finished floor area divided by 100 ffarea=c(12.9, 16.5, 16.2, 15.46, 15.6, 15.1, 12.26, 13.45, 16.06, 13.2, 14, 16.6, 15.25, 12.26, 12, 12.01, 13.34, 15.08, 15.5, 12.27, 22.78, 12.1, 13.06, 13.59, 10.5, 15.3, 14.8, 11.84, 16.22, 13.84, 17.48, 16.01, 19.48, 15.09, 15.05, 14, 10.95, 17.37, 17.54, 23.98, 15.15, 13.06, 14.5, 15.77, 11.26, 6.1, 17.63, 12.09, 15.76, 9.4) (ii) age=c(44, 3, 30, 41, 20, 20, 3, 1, 25, 3, 38, 23, 3, 29, 28, 0, 32, 1, 23, 17, 35, 7, 1, 2, 37, 9, 50, 15, 25, 10, 5, 25, 11, 8, 8, 5, 18, 26, 9, 16, 0, 0, 7, 6, 0, 11, 26, 7, 4, 14) (iii) monthly maintenance fee divided by 10 mfee=c(23.2, 25.4, 16, 31, 27, 24.5, 18, 18.2, 24.4, 18.9, 23, 19.9, 35, 19.8, 25.9, 14.2, 24.5, 48.8, 17.4, 25.2, 57.4, 18, 19.6, 17, 28, 16.9, 25, 21, 36.4, 16, 29.7, 33.7, 20.4, 20.3, 22.3, 19.6, 24.7, 31, 18.2, 36.9, 22.2, 18.6, 18.7, 35.7, 24.8, 17.1, 32, 18.1, 22.1, 23.3) (iv) number of bedrooms beds=c(3, 4, 4, 3, 3, 3, 3, 3, 2, 3, 3, 4, 2, 3, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 4, 3, 3, 4, 3, 3, 2, 3, 4, 3, 4, 3, 3, 3, 2, 1, 5, 3, 4, 2) After you have copied the above R vectors into your R session, you can get a dataframe with richmondtownh=data.frame(askpr,ffarea,age,mfee,beds)\nThe corresponding vectors for the holdout set are: askpr.ho=c(57.5, 53.8, 68.8, 58.68, 79.99, 56.88, 48.5, 68.8, 81.9) ffarea.ho=c(13.46, 12.22, 15.95, 13.96, 22, 15.78, 14.8, 16.9, 20.95) age.ho=c(10, 9, 18, 9, 20, 17, 24, 8, 19) mfee.ho=c(22.1, 18.5, 23.6, 22, 26.7, 17.3, 16.1, 19.4, 34.8) beds.ho=c(3, 3, 3, 3, 3, 4, 3, 4, 1)\nCreate a second data frame: holdout=data.frame(askpr.ho,ffarea.ho,age.ho,mfee.ho,beds.ho) names(holdout)=names(richmondtownh) [to make variables names the same as before]\nUse the ls.cvrmse() function from the course web site with 3 and 4 explanatory variables, when askpr is the response variable. For 4 explanatory variables, all of the above are included. For 3 explanatory variables, all of the above are included except beds.\nPlease use 3 decimal places for the answers below which are not integer-valued\nPart a) The values of residual SD or residual SE for the regressions with 3 and 4 explanatory variables are respectively: 3 explanatory: [ANS]\n4 explanatory: [ANS]\nPart b) The values of the leave-one-out cross-validation RMS prediction error for the regressions with 3 and 4 explanatory variables are respectively: 3 explanatory: [ANS]\n4 explanatory: [ANS]\nPart c) The values of the holdout RMS prediction error for the regressions with 3 and 4 explanatory variables are respectively: 3 explanatory: [ANS]\n4 explanatory: [ANS]\nPart d) Do you get the same conclusion on which is better between the 3-explanatory versus 4-explanatory models from the summaries in (a), (b), (c). [ANS] (enter either Yes or No).\nPart e) Compare the three values for the models with 3 explanatory variables (respectively 4 explanatory); that is, values of residual SE, leave-one out root mean square error and holdout subset root mean square error for 3 explanatory (respectively, 4 explanatory). What is the best explanation for the root mean square prediction errors to be larger for parts (b) and (c)? [ANS] A. out-of-sample variability is always larger B. sampling variability C. sample size is small D. the RMS prediction error values were not always larger for my data set E. the linear approximation to a non-linear function of the explanatory variables", "answer_v1": [ "7.31285", "7.38789", "8.2397", "8.64226", "3.19839", "3.39998", "Yes", "D" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "TF", "MCS" ], "options_v1": [ [], [], [], [], [], [], [], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "The questions involves the data set for Richmond townhouses obtained on 2014.11.03. For your subset, the response variable is: asking price divided by 10000: askpr=c(54.98, 55.2, 78.8, 79.99, 58.68, 51.68, 50.8, 56.8, 50.5, 53.9, 40.8, 40.8, 47.9, 54.8, 86.8, 40.9, 26.99, 57.8, 33.7, 55.8, 53.8, 48.5, 108.8, 77.8, 68.5, 52.4, 57.8, 57.5, 68.8, 61.5, 53.8, 54.8, 62.9, 73.9, 79.8, 74.8, 50.8, 71.99, 58.8, 62.8888, 49.9, 65.99, 73.8, 41.99, 44.8, 51.99, 81.9, 48.8, 60.8, 46.8) The explanatory variables are: (i) finished floor area divided by 100 ffarea=c(13.06, 15.3, 19.48, 22, 13.96, 15.1, 12.27, 15.5, 12.26, 11.84, 14, 12.26, 12.1, 11.26, 15.08, 16.06, 10.5, 13.84, 12, 13.06, 10.95, 14.8, 23.98, 16.5, 13.59, 16.22, 12.01, 13.46, 15.95, 14.5, 12.22, 15.46, 14, 15.15, 15.25, 17.48, 16.6, 15.05, 17.37, 15.77, 15.6, 22.78, 17.54, 12.9, 9.4, 12.09, 20.95, 14.8, 13.2, 16.2) (ii) age=c(1, 9, 11, 20, 9, 20, 17, 23, 3, 15, 38, 29, 7, 0, 1, 25, 37, 10, 28, 0, 18, 24, 16, 3, 2, 25, 0, 10, 18, 7, 9, 41, 5, 0, 3, 5, 23, 8, 26, 6, 20, 35, 9, 44, 14, 7, 19, 50, 3, 30) (iii) monthly maintenance fee divided by 10 mfee=c(19.6, 16.9, 20.4, 26.7, 22, 24.5, 25.2, 17.4, 18, 21, 23, 19.8, 18, 24.8, 48.8, 24.4, 28, 16, 25.9, 18.6, 24.7, 16.1, 36.9, 25.4, 17, 36.4, 14.2, 22.1, 23.6, 18.7, 18.5, 31, 19.6, 22.2, 35, 29.7, 19.9, 22.3, 31, 35.7, 27, 57.4, 18.2, 23.2, 23.3, 18.1, 34.8, 25, 18.9, 16) (iv) number of bedrooms beds=c(3, 3, 3, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 2, 4, 4, 3, 3, 3, 3, 2, 4, 3, 2, 3, 1, 3, 3, 4) After you have copied the above R vectors into your R session, you can get a dataframe with richmondtownh=data.frame(askpr,ffarea,age,mfee,beds)\nThe corresponding vectors for the holdout set are: askpr.ho=c(65.8, 56.88, 47.8, 59.8, 25.9, 68.5, 58.39, 45.99, 68.8) ffarea.ho=c(13.45, 15.78, 13.34, 17.63, 6.1, 15.76, 15.09, 16.01, 16.9) age.ho=c(1, 17, 32, 26, 11, 4, 8, 25, 8) mfee.ho=c(18.2, 17.3, 24.5, 32, 17.1, 22.1, 20.3, 33.7, 19.4) beds.ho=c(3, 4, 3, 5, 1, 4, 4, 3, 4)\nCreate a second data frame: holdout=data.frame(askpr.ho,ffarea.ho,age.ho,mfee.ho,beds.ho) names(holdout)=names(richmondtownh) [to make variables names the same as before]\nUse the ls.cvrmse() function from the course web site with 3 and 4 explanatory variables, when askpr is the response variable. For 4 explanatory variables, all of the above are included. For 3 explanatory variables, all of the above are included except beds.\nPlease use 3 decimal places for the answers below which are not integer-valued\nPart a) The values of residual SD or residual SE for the regressions with 3 and 4 explanatory variables are respectively: 3 explanatory: [ANS]\n4 explanatory: [ANS]\nPart b) The values of the leave-one-out cross-validation RMS prediction error for the regressions with 3 and 4 explanatory variables are respectively: 3 explanatory: [ANS]\n4 explanatory: [ANS]\nPart c) The values of the holdout RMS prediction error for the regressions with 3 and 4 explanatory variables are respectively: 3 explanatory: [ANS]\n4 explanatory: [ANS]\nPart d) Do you get the same conclusion on which is better between the 3-explanatory versus 4-explanatory models from the summaries in (a), (b), (c). [ANS] (enter either Yes or No).\nPart e) Compare the three values for the models with 3 explanatory variables (respectively 4 explanatory); that is, values of residual SE, leave-one out root mean square error and holdout subset root mean square error for 3 explanatory (respectively, 4 explanatory). What is the best explanation for the root mean square prediction errors to be larger for parts (b) and (c)? [ANS] A. the linear approximation to a non-linear function of the explanatory variables B. out-of-sample variability is always larger C. sample size is small D. sampling variability E. the RMS prediction error values were not always larger for my data set", "answer_v2": [ "7.05163", "7.11756", "7.99005", "8.14992", "5.51629", "5.75384", "Yes", "E" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "TF", "MCS" ], "options_v2": [ [], [], [], [], [], [], [], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "The questions involves the data set for Richmond townhouses obtained on 2014.11.03. For your subset, the response variable is: asking price divided by 10000: askpr=c(81.9, 50.5, 53.8, 60.8, 50.8, 50.8, 51.99, 73.8, 58.39, 68.5, 56.8, 86.8, 56.88, 65.99, 48.8, 61.5, 68.5, 79.99, 58.8, 41.99, 62.8888, 26.99, 46.8, 58.68, 68.8, 71.99, 25.9, 74.8, 53.8, 78.8, 77.8, 62.9, 52.4, 57.5, 108.8, 47.9, 33.7, 55.8, 65.8, 49.9, 54.8, 73.9, 59.8, 51.68, 40.9, 44.8, 47.8, 79.8, 57.8, 54.8) The explanatory variables are: (i) finished floor area divided by 100 ffarea=c(20.95, 12.26, 10.95, 13.2, 12.27, 16.6, 12.09, 17.54, 15.09, 13.59, 15.5, 15.08, 15.78, 22.78, 14.8, 14.5, 15.76, 22, 17.37, 12.9, 15.77, 10.5, 16.2, 13.96, 16.9, 15.05, 6.1, 17.48, 12.22, 19.48, 16.5, 14, 16.22, 13.46, 23.98, 12.1, 12, 13.06, 13.45, 15.6, 15.46, 15.15, 17.63, 15.1, 16.06, 9.4, 13.34, 15.25, 12.01, 11.26) (ii) age=c(19, 3, 18, 3, 17, 23, 7, 9, 8, 2, 23, 1, 17, 35, 50, 7, 4, 20, 26, 44, 6, 37, 30, 9, 8, 8, 11, 5, 9, 11, 3, 5, 25, 10, 16, 7, 28, 0, 1, 20, 41, 0, 26, 20, 25, 14, 32, 3, 0, 0) (iii) monthly maintenance fee divided by 10 mfee=c(34.8, 18, 24.7, 18.9, 25.2, 19.9, 18.1, 18.2, 20.3, 17, 17.4, 48.8, 17.3, 57.4, 25, 18.7, 22.1, 26.7, 31, 23.2, 35.7, 28, 16, 22, 19.4, 22.3, 17.1, 29.7, 18.5, 20.4, 25.4, 19.6, 36.4, 22.1, 36.9, 18, 25.9, 18.6, 18.2, 27, 31, 22.2, 32, 24.5, 24.4, 23.3, 24.5, 35, 14.2, 24.8) (iv) number of bedrooms beds=c(1, 3, 2, 3, 2, 4, 3, 4, 4, 3, 3, 3, 4, 2, 3, 3, 4, 3, 3, 3, 3, 2, 4, 3, 4, 3, 1, 4, 3, 3, 4, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 4, 5, 3, 2, 2, 3, 2, 3, 2) After you have copied the above R vectors into your R session, you can get a dataframe with richmondtownh=data.frame(askpr,ffarea,age,mfee,beds)\nThe corresponding vectors for the holdout set are: askpr.ho=c(48.5, 53.9, 55.2, 40.8, 57.8, 54.98, 40.8, 68.8, 45.99) ffarea.ho=c(14.8, 11.84, 15.3, 12.26, 13.84, 13.06, 14, 15.95, 16.01) age.ho=c(24, 15, 9, 29, 10, 1, 38, 18, 25) mfee.ho=c(16.1, 21, 16.9, 19.8, 16, 19.6, 23, 23.6, 33.7) beds.ho=c(3, 2, 3, 3, 3, 3, 3, 3, 3)\nCreate a second data frame: holdout=data.frame(askpr.ho,ffarea.ho,age.ho,mfee.ho,beds.ho) names(holdout)=names(richmondtownh) [to make variables names the same as before]\nUse the ls.cvrmse() function from the course web site with 3 and 4 explanatory variables, when askpr is the response variable. For 4 explanatory variables, all of the above are included. For 3 explanatory variables, all of the above are included except beds.\nPlease use 3 decimal places for the answers below which are not integer-valued\nPart a) The values of residual SD or residual SE for the regressions with 3 and 4 explanatory variables are respectively: 3 explanatory: [ANS]\n4 explanatory: [ANS]\nPart b) The values of the leave-one-out cross-validation RMS prediction error for the regressions with 3 and 4 explanatory variables are respectively: 3 explanatory: [ANS]\n4 explanatory: [ANS]\nPart c) The values of the holdout RMS prediction error for the regressions with 3 and 4 explanatory variables are respectively: 3 explanatory: [ANS]\n4 explanatory: [ANS]\nPart d) Do you get the same conclusion on which is better between the 3-explanatory versus 4-explanatory models from the summaries in (a), (b), (c). [ANS] (enter either Yes or No).\nPart e) Compare the three values for the models with 3 explanatory variables (respectively 4 explanatory); that is, values of residual SE, leave-one out root mean square error and holdout subset root mean square error for 3 explanatory (respectively, 4 explanatory). What is the best explanation for the root mean square prediction errors to be larger for parts (b) and (c)? [ANS] A. sample size is small B. out-of-sample variability is always larger C. the RMS prediction error values were not always larger for my data set D. the linear approximation to a non-linear function of the explanatory variables E. sampling variability", "answer_v3": [ "6.89382", "6.96836", "7.82356", "7.99881", "6.39568", "6.4164", "Yes", "C" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "TF", "MCS" ], "options_v3": [ [], [], [], [], [], [], [], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Statistics_0416", "subject": "Statistics", "topic": "Multiple regression", "subtopic": "Model selection", "level": "4", "keywords": [ "statistics", "regression", "influential observations" ], "problem_v1": "You have 2 explanatory variables x1 and x2 below, and a response variable y. Please fit the least square hyperplane.\nx1=c(3.8, 3.6, 9.8, 2.5, 2.5, 2.2, 0.8, 4, 2.3, 1.4, 2.7, 3.8, 0.8, 0.7, 2.1, 2.7, 0.5, 4.2, 4.5, 2.3) x2=c(3.9, 2.5, 10.9, 4.3, 3.1, 3.6, 3, 3.3, 5.5, 1.1, 1.8, 5, 2.8, 2.2, 2.3, 3.7, 3.3, 2.6, 5.4, 5.6) and y=c(22.5, 18.4, 53.9, 18.6, 16.5, 16, 11.4, 21.2, 21.9, 8.9, 14.4, 24.3, 11.6, 3.9, 13.3, 17.7, 12, 21, 27.9, 22) Part a) The slope of $x_1$ for the least square hyperplane is ${\\hat \\beta}_1$=[ANS]\nPart b) The value of largest absolute residual is: [ANS]\nPart c) Based on Cook's distances or the dfit statistics, from ls.diag() applied to an lm object, does the observation with the largest absolute residual correspond to an influential observation? (Enter either Yes or No) [ANS]\nPart d) Check the output of Cook's distances or the dfit statistics, and check on how your largest absolute residual compares with the other residuals. Then determine the best answers below on interpretations of influential observations. Possibly more than one item is correct. [ANS] A. An influential observation need not have an extreme residual B. Influential observations sometimes are in the extremes of the space of explanatory variables C. Influential observations should be deleted D. In general, influential observations are extreme residuals (outliers) E. Influential observations have heavy influence on some ${\\hat\\beta}$ F. The number of influential observations can be 0,1,2,..., or n-p-1, where p is the number of explanatory variables G. None of the above", "answer_v1": [ "2.95025", "5.08958", "YES", "ABEF" ], "answer_type_v1": [ "NV", "NV", "TF", "MCM" ], "options_v1": [ [], [], [], [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v2": "You have 2 explanatory variables x1 and x2 below, and a response variable y. Please fit the least square hyperplane.\nx1=c(0.4, 1.7, 0.9, 0.3, 11.1, 1.3, 1.6, 0.6, 2.4, 1.5, 4.8, 4.5, 2.1, 2, 3.7, 1.3, 3.7, 4.7, 1.2, 0) x2=c(5.7, 5.8, 2.6, 4.2, 8.9, 3.9, 3.8, 1.4, 1.4, 2.9, 3.3, 1.3, 4.8, 1.3, 1.3, 1.3, 3.1, 5.2, 2.2, 1.2) and y=c(14.9, 19.3, 11, 12.2, 53.4, 14, 14.6, 7.2, 13.8, 13.7, 24.1, 18.9, 19.8, 11.3, 16.1, 10.5, 20.4, 26.5, 11.2, 1) Part a) The slope of $x_1$ for the least square hyperplane is ${\\hat \\beta}_1$=[ANS]\nPart b) The value of largest absolute residual is: [ANS]\nPart c) Based on Cook's distances or the dfit statistics, from ls.diag() applied to an lm object, does the observation with the largest absolute residual correspond to an influential observation? (Enter either Yes or No) [ANS]\nPart d) Check the output of Cook's distances or the dfit statistics, and check on how your largest absolute residual compares with the other residuals. Then determine the best answers below on interpretations of influential observations. Possibly more than one item is correct. [ANS] A. Influential observations sometimes are in the extremes of the space of explanatory variables B. The number of influential observations can be 0,1,2,..., or n-p-1, where p is the number of explanatory variables C. In general, influential observations are extreme residuals (outliers) D. Influential observations have heavy influence on some ${\\hat\\beta}$ E. An influential observation need not have an extreme residual F. Influential observations should be deleted G. None of the above", "answer_v2": [ "3.04466", "4.01167", "YES", "ABDE" ], "answer_type_v2": [ "NV", "NV", "TF", "MCM" ], "options_v2": [ [], [], [], [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v3": "You have 2 explanatory variables x1 and x2 below, and a response variable y. Please fit the least square hyperplane.\nx1=c(1.5, 2.7, 4.1, 1, 0.2, 4, 8.4, 2.6, 4, 2.9, 0.8, 3.8, 2.3, 2.2, 2.7, 4, 0.1, 4.6, 0.8, 3) x2=c(4, 2, 5.6, 2.4, 3.9, 4.1, 11.2, 3.1, 5.3, 5.8, 3.7, 2, 1.7, 1.9, 1.2, 1.7, 3, 3.4, 1.1, 2.9) and y=c(15, 14.8, 27.3, 10.3, 12.4, 22.4, 51, 17.9, 25.7, 23.9, 13.7, 13.7, 13.3, 13.1, 12.6, 17.9, 9.1, 24, 7.8, 17.7) Part a) The slope of $x_1$ for the least square hyperplane is ${\\hat \\beta}_1$=[ANS]\nPart b) The value of largest absolute residual is: [ANS]\nPart c) Based on Cook's distances or the dfit statistics, from ls.diag() applied to an lm object, does the observation with the largest absolute residual correspond to an influential observation? (Enter either Yes or No) [ANS]\nPart d) Check the output of Cook's distances or the dfit statistics, and check on how your largest absolute residual compares with the other residuals. Then determine the best answers below on interpretations of influential observations. Possibly more than one item is correct. [ANS] A. An influential observation need not have an extreme residual B. In general, influential observations are extreme residuals (outliers) C. Influential observations have heavy influence on some ${\\hat\\beta}$ D. Influential observations sometimes are in the extremes of the space of explanatory variables E. Influential observations should be deleted F. The number of influential observations can be 0,1,2,..., or n-p-1, where p is the number of explanatory variables G. None of the above", "answer_v3": [ "2.6606", "3.65261", "YES", "ACDF" ], "answer_type_v3": [ "NV", "NV", "TF", "MCM" ], "options_v3": [ [], [], [], [ "A", "B", "C", "D", "E", "F", "G" ] ] }, { "id": "Statistics_0417", "subject": "Statistics", "topic": "Multiple regression", "subtopic": "Model selection", "level": "3", "keywords": [ "statistics", "multpile regression", "regression" ], "problem_v1": "When the independent variables are correlated with one another in a multiple regression analysis, this condition is called: [ANS] A. multicollinearity B. elasticity C. homoscedasticity D. heteroscedasticity\nIn multiple regression, the mean of the probability distribution of the error variable $\\epsilon$ is assumed to be: [ANS] A. $k$,where $k$ is the number of independent variables included in the model B. 1.0 C. 0.0 D. Any value greater than 1", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "When the independent variables are correlated with one another in a multiple regression analysis, this condition is called: [ANS] A. homoscedasticity B. elasticity C. heteroscedasticity D. multicollinearity\nIn a multiple regression model, the standard deviation of the error variable $\\epsilon$ is assumed to be: [ANS] A. constant for all values of the independent variables B. not enough information is given to answer this question C. constant for all values of the dependent variables D. 1.0", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "When the independent variables are correlated with one another in a multiple regression analysis, this condition is called: [ANS] A. homoscedasticity B. multicollinearity C. elasticity D. heteroscedasticity\nIn multiple regression analysis, the ratio $MSR/MSE$ yields the: [ANS] A. $F-$ test statistic for testing the validity of the regression equation B. multiple coefficient of determination C. $t-$ test statistic for testing each individual regression coefficient D. adjusted multiple coefficient of determination", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Statistics_0418", "subject": "Statistics", "topic": "Generalized linear methods", "subtopic": "Logistic regression", "level": "3", "keywords": [ "statistics", "logistic regression" ], "problem_v1": "Suppose the probability $\\pi(x)$ of reaching a target (such as getting a ball between goal posts) as a function of distance x (in metres) from the target is well-fitted by a logistic regression equation with $\\log(\\pi(x)/[1-\\pi(x)])=6.7-0.11x$ Please answer below to 3 significant digits.\nPart a) For this prediction model, what is the probability of reaching the target from a distance of 43 metres. [ANS]\nPart b) At what distance is the probability of reaching the target equal to 0.6? [ANS]", "answer_v1": [ "0.877611113176951", "57.223044471744" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose the probability $\\pi(x)$ of reaching a target (such as getting a ball between goal posts) as a function of distance x (in metres) from the target is well-fitted by a logistic regression equation with $\\log(\\pi(x)/[1-\\pi(x)])=6.1-0.13x$ Please answer below to 3 significant digits.\nPart a) For this prediction model, what is the probability of reaching the target from a distance of 33 metres. [ANS]\nPart b) At what distance is the probability of reaching the target equal to 0.6? [ANS]", "answer_v2": [ "0.859361874265866", "43.8041145530141" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose the probability $\\pi(x)$ of reaching a target (such as getting a ball between goal posts) as a function of distance x (in metres) from the target is well-fitted by a logistic regression equation with $\\log(\\pi(x)/[1-\\pi(x)])=6.3-0.11x$ Please answer below to 3 significant digits.\nPart a) For this prediction model, what is the probability of reaching the target from a distance of 35 metres. [ANS]\nPart b) At what distance is the probability of reaching the target equal to 0.6? [ANS]", "answer_v3": [ "0.920561450816022", "53.5866808353803" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Statistics_0419", "subject": "Statistics", "topic": "Generalized linear methods", "subtopic": "Logistic regression", "level": "4", "keywords": [ "statistics", "logistic regression" ], "problem_v1": "This question involves logistic regression analysis of the Pima data set in R on risk factors for diabetes among Pima women. Your training and holding data sets will be subsets of the Pima.tr and Pima.te data sets in the library MASS. The binary response variable is \"type\" (type=Yes for Diabetes, type=No for no diabetes). Get your training set and holdout set with the following in R.\nitrain=c(151, 117, 125, 146, 64, 70, 2, 118, 80, 104, 135, 62, 108, 93, 88, 96, 111, 42, 47, 91, 113, 162, 10, 81, 106, 181, 200, 76, 34, 74, 123, 61, 14, 159, 166, 21, 5, 97, 163, 37, 24, 78, 110, 86, 18, 129, 128, 67, 66, 8, 191, 175, 101, 139, 185, 182, 167, 31, 188, 189, 95, 100, 122, 190, 35, 153, 99, 116, 52, 12, 46, 44, 20, 11, 63, 23, 142, 184, 169, 180, 50, 7, 83, 157, 94, 156, 165, 195, 49, 131, 28, 119, 161, 126, 45, 141, 186, 17, 39, 72, 19, 171, 196, 65, 127, 158, 132, 69, 71, 54, 148, 16, 98, 183, 121, 53, 58, 145, 68, 41, 136, 199, 138, 55, 198, 13, 40, 59, 105, 137, 147, 112, 177, 25, 30, 85, 164, 84, 75, 32, 172, 150, 187, 178, 197, 89, 107, 114, 26, 3, 192, 90, 140, 160, 51, 9, 133, 43, 174, 82, 57, 56, 134, 73, 48, 143, 60, 38, 149, 173)\nihold=c(147, 227, 59, 274, 309, 267, 232, 179, 56, 145, 330, 173, 275, 141, 250, 99, 236, 4, 283, 3, 78, 17, 142, 134, 103, 119, 102, 207, 135, 230, 177, 323, 265, 153, 144, 89, 282, 98, 30, 110, 2, 188, 178, 277, 172, 52, 312, 10, 272, 174, 138, 34, 308, 128, 171, 243, 127, 240, 199, 136, 296, 284, 293, 150, 107, 105, 19, 46, 48, 151, 88, 228, 100, 327, 152, 201, 183, 271, 197, 303, 38, 58, 221, 126, 81, 326, 31, 226, 15, 310, 49, 202, 87, 231, 50, 148, 53, 238, 258, 252, 329, 280, 198, 32, 176, 292, 270, 104, 63, 276, 182, 321, 254, 26, 44, 193, 86, 289, 241, 68, 290, 18, 322, 266, 190, 156, 234, 181, 209, 304, 260, 97, 223, 215, 214, 281, 263, 317, 117, 133, 158, 291, 54, 45, 37, 122, 247, 75, 5, 316, 261, 6, 131, 60, 166, 211, 124, 233, 287, 16, 191, 85, 305, 331, 8, 204, 74, 94, 143, 184, 208, 320, 93, 39, 70, 40, 278, 116, 161, 157, 307, 115, 225, 219, 91, 245, 194, 137, 175, 11, 196, 149, 64, 237, 35, 235, 24, 251, 162, 255)\nlibrary(MASS) data(Pima.tr) mytrain=Pima.tr[itrain,] data(Pima.te) myhold=Pima.te[ihold,]\nNext do the following: (1) Fit the logistic regression model with all 7 explanatory variables npreg, glu, bp, skin, bmi, ped, age. Call this model 1. (2) Fit the logistic regression model with 4 explanatory variables glu, bmi, ped, age (this is best model from backward elimination if all cases of Pima.tr is used). For this model with 4 explanatory variables, call it model 2. (3) Apply both models 1 and 2 to the holdout data set and get the predicted probabilities. Classify a case as diabetes if the predicted probability exceeds (>=) 0.5 and otherwise classify it as non-diabetes. (4) For models 1 and 2, get the total number of misclassifications. Which model is better based on this criterion? (5) For models 1 and 2, compare the misclassification tables if one classifies a case as diabetes if the predicted probability exceeds (>=) 0.3 and otherwise classify it as non-diabetes. Which is the better boundary to use?\nYou will be asked to supply some numbers below from doing the above.\nPart a) For model 1, the regression coefficient for ped is [ANS]\nPart b) For model 2, the regression coefficient for age is [ANS]\nPart c) For the first subject in the holdout set, the predicted probability is: [ANS] for model 1, [ANS] for model 2. Part d) Use a boundary of 0.5 in the predicted probabilities to decide on diabetes (predicted probability greater than or equal to 0.5) or non-diabetes. The total number of misclassifications of the 200 cases in the holdout set is: [ANS] for model 1, [ANS] for model 2. Part e) With a boundary of 0.5 in predicted probabilities, the better model with a lower misclassification rate is model: [ANS] (enter 1 or 2, and enter model 2 in case of a tie).\nPart f) Use a boundary of 0.3 in the predicted probabilities to decide on diabetes (predicted probability greater than or equal to 0.3) or non-diabetes. The total number of misclassifications of the 200 cases in the holdout set is: [ANS] for model 1, [ANS] for model 2. There is no question on the better boundary to use, because that depends on the relative seriousness of the two types of misclassification errors.", "answer_v1": [ "1.58578", "0.0615938", "0.658621", "0.56829", "42", "45", "1", "48", "43" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "This question involves logistic regression analysis of the Pima data set in R on risk factors for diabetes among Pima women. Your training and holding data sets will be subsets of the Pima.tr and Pima.te data sets in the library MASS. The binary response variable is \"type\" (type=Yes for Diabetes, type=No for no diabetes). Get your training set and holdout set with the following in R.\nitrain=c(17, 187, 32, 69, 190, 67, 42, 71, 117, 23, 132, 95, 165, 50, 14, 66, 122, 52, 77, 120, 41, 44, 40, 78, 110, 7, 182, 80, 96, 145, 191, 109, 123, 27, 46, 104, 106, 159, 5, 105, 53, 97, 158, 56, 76, 85, 58, 198, 161, 114, 134, 39, 175, 60, 90, 55, 143, 47, 65, 171, 113, 188, 162, 101, 91, 172, 68, 45, 152, 2, 87, 63, 141, 126, 139, 174, 184, 166, 176, 34, 8, 168, 36, 135, 51, 131, 72, 157, 142, 1, 29, 84, 11, 64, 89, 26, 49, 98, 21, 103, 199, 100, 73, 147, 9, 196, 127, 24, 133, 185, 148, 164, 130, 88, 28, 186, 102, 116, 99, 20, 92, 156, 177, 192, 82, 125, 170, 13, 137, 81, 12, 169, 15, 33, 151, 10, 155, 19, 115, 35, 31, 181, 61, 178, 144, 183, 149, 59, 112, 83, 119, 160, 200, 94, 173, 154, 150, 146, 153, 93, 18, 70, 16, 107, 4, 62, 74, 6, 111, 57)\nihold=c(135, 205, 13, 24, 107, 322, 143, 264, 249, 104, 271, 332, 50, 33, 162, 159, 66, 103, 119, 75, 287, 125, 102, 207, 49, 30, 67, 27, 29, 189, 89, 113, 151, 181, 23, 236, 176, 164, 198, 291, 273, 314, 36, 194, 179, 260, 144, 195, 214, 141, 18, 79, 254, 43, 10, 40, 301, 191, 253, 210, 118, 70, 6, 269, 157, 329, 21, 321, 148, 173, 228, 200, 129, 232, 127, 250, 154, 4, 237, 290, 310, 229, 308, 68, 209, 257, 270, 222, 327, 78, 256, 245, 238, 288, 178, 328, 224, 281, 149, 134, 248, 63, 35, 293, 132, 142, 158, 244, 163, 285, 233, 1, 31, 25, 242, 147, 219, 252, 319, 44, 28, 186, 39, 211, 331, 65, 324, 315, 305, 97, 283, 190, 87, 307, 41, 108, 282, 303, 145, 280, 94, 180, 59, 38, 122, 58, 124, 306, 123, 202, 54, 235, 304, 276, 101, 120, 160, 136, 267, 262, 138, 86, 297, 201, 166, 261, 77, 316, 93, 82, 14, 295, 90, 259, 115, 277, 121, 204, 220, 146, 128, 213, 57, 140, 302, 299, 7, 284, 45, 91, 9, 206, 16, 8, 110, 275, 268, 117, 53, 286)\nlibrary(MASS) data(Pima.tr) mytrain=Pima.tr[itrain,] data(Pima.te) myhold=Pima.te[ihold,]\nNext do the following: (1) Fit the logistic regression model with all 7 explanatory variables npreg, glu, bp, skin, bmi, ped, age. Call this model 1. (2) Fit the logistic regression model with 4 explanatory variables glu, bmi, ped, age (this is best model from backward elimination if all cases of Pima.tr is used). For this model with 4 explanatory variables, call it model 2. (3) Apply both models 1 and 2 to the holdout data set and get the predicted probabilities. Classify a case as diabetes if the predicted probability exceeds (>=) 0.5 and otherwise classify it as non-diabetes. (4) For models 1 and 2, get the total number of misclassifications. Which model is better based on this criterion? (5) For models 1 and 2, compare the misclassification tables if one classifies a case as diabetes if the predicted probability exceeds (>=) 0.3 and otherwise classify it as non-diabetes. Which is the better boundary to use?\nYou will be asked to supply some numbers below from doing the above.\nPart a) For model 1, the regression coefficient for ped is [ANS]\nPart b) For model 2, the regression coefficient for age is [ANS]\nPart c) For the first subject in the holdout set, the predicted probability is: [ANS] for model 1, [ANS] for model 2. Part d) Use a boundary of 0.5 in the predicted probabilities to decide on diabetes (predicted probability greater than or equal to 0.5) or non-diabetes. The total number of misclassifications of the 200 cases in the holdout set is: [ANS] for model 1, [ANS] for model 2. Part e) With a boundary of 0.5 in predicted probabilities, the better model with a lower misclassification rate is model: [ANS] (enter 1 or 2, and enter model 2 in case of a tie).\nPart f) Use a boundary of 0.3 in the predicted probabilities to decide on diabetes (predicted probability greater than or equal to 0.3) or non-diabetes. The total number of misclassifications of the 200 cases in the holdout set is: [ANS] for model 1, [ANS] for model 2. There is no question on the better boundary to use, because that depends on the relative seriousness of the two types of misclassification errors.", "answer_v2": [ "2.48085", "0.0701607", "0.64638", "0.6913", "38", "38", "2", "47", "42" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "This question involves logistic regression analysis of the Pima data set in R on risk factors for diabetes among Pima women. Your training and holding data sets will be subsets of the Pima.tr and Pima.te data sets in the library MASS. The binary response variable is \"type\" (type=Yes for Diabetes, type=No for no diabetes). Get your training set and holdout set with the following in R.\nitrain=c(63, 122, 58, 112, 45, 73, 163, 183, 176, 48, 66, 59, 21, 121, 197, 162, 130, 37, 72, 136, 147, 115, 97, 187, 164, 173, 124, 128, 193, 160, 3, 27, 189, 159, 69, 141, 4, 1, 14, 18, 35, 98, 28, 52, 53, 168, 70, 89, 54, 109, 111, 186, 125, 152, 80, 60, 139, 143, 114, 32, 145, 38, 65, 171, 75, 78, 62, 161, 191, 106, 123, 36, 88, 140, 81, 86, 166, 24, 100, 117, 6, 182, 96, 131, 19, 34, 92, 43, 91, 82, 107, 192, 90, 64, 133, 167, 2, 71, 51, 77, 188, 76, 26, 68, 15, 94, 196, 44, 113, 20, 104, 16, 83, 67, 85, 79, 57, 181, 180, 138, 50, 150, 148, 93, 9, 39, 105, 169, 23, 165, 158, 199, 119, 102, 144, 157, 41, 200, 178, 29, 55, 99, 137, 103, 120, 11, 179, 172, 177, 31, 174, 7, 95, 142, 108, 129, 87, 134, 56, 84, 22, 74, 185, 33, 127, 154, 10, 110, 40, 46)\nihold=c(230, 253, 223, 93, 166, 145, 280, 207, 124, 186, 43, 314, 302, 102, 71, 143, 75, 288, 177, 217, 306, 199, 106, 277, 86, 128, 52, 241, 202, 68, 211, 164, 323, 78, 332, 123, 37, 146, 85, 10, 232, 12, 292, 266, 14, 210, 5, 159, 59, 301, 225, 264, 258, 141, 250, 287, 174, 67, 320, 315, 171, 90, 30, 27, 209, 257, 80, 201, 290, 122, 154, 279, 156, 16, 69, 157, 58, 234, 129, 239, 254, 291, 244, 261, 265, 161, 99, 82, 138, 197, 57, 105, 95, 228, 243, 212, 44, 144, 299, 316, 204, 235, 112, 41, 312, 267, 142, 113, 215, 153, 169, 286, 88, 23, 263, 24, 83, 87, 33, 20, 165, 283, 162, 222, 231, 270, 118, 268, 89, 205, 6, 45, 185, 325, 120, 240, 116, 139, 151, 103, 179, 219, 40, 72, 130, 247, 51, 150, 331, 76, 125, 158, 21, 18, 249, 148, 248, 149, 324, 246, 213, 61, 134, 74, 282, 97, 42, 187, 293, 237, 318, 13, 107, 276, 64, 136, 233, 38, 208, 126, 272, 50, 60, 114, 224, 15, 115, 49, 206, 133, 54, 79, 175, 31, 160, 275, 173, 271, 19, 2)\nlibrary(MASS) data(Pima.tr) mytrain=Pima.tr[itrain,] data(Pima.te) myhold=Pima.te[ihold,]\nNext do the following: (1) Fit the logistic regression model with all 7 explanatory variables npreg, glu, bp, skin, bmi, ped, age. Call this model 1. (2) Fit the logistic regression model with 4 explanatory variables glu, bmi, ped, age (this is best model from backward elimination if all cases of Pima.tr is used). For this model with 4 explanatory variables, call it model 2. (3) Apply both models 1 and 2 to the holdout data set and get the predicted probabilities. Classify a case as diabetes if the predicted probability exceeds (>=) 0.5 and otherwise classify it as non-diabetes. (4) For models 1 and 2, get the total number of misclassifications. Which model is better based on this criterion? (5) For models 1 and 2, compare the misclassification tables if one classifies a case as diabetes if the predicted probability exceeds (>=) 0.3 and otherwise classify it as non-diabetes. Which is the better boundary to use?\nYou will be asked to supply some numbers below from doing the above.\nPart a) For model 1, the regression coefficient for ped is [ANS]\nPart b) For model 2, the regression coefficient for age is [ANS]\nPart c) For the first subject in the holdout set, the predicted probability is: [ANS] for model 1, [ANS] for model 2. Part d) Use a boundary of 0.5 in the predicted probabilities to decide on diabetes (predicted probability greater than or equal to 0.5) or non-diabetes. The total number of misclassifications of the 200 cases in the holdout set is: [ANS] for model 1, [ANS] for model 2. Part e) With a boundary of 0.5 in predicted probabilities, the better model with a lower misclassification rate is model: [ANS] (enter 1 or 2, and enter model 2 in case of a tie).\nPart f) Use a boundary of 0.3 in the predicted probabilities to decide on diabetes (predicted probability greater than or equal to 0.3) or non-diabetes. The total number of misclassifications of the 200 cases in the holdout set is: [ANS] for model 1, [ANS] for model 2. There is no question on the better boundary to use, because that depends on the relative seriousness of the two types of misclassification errors.", "answer_v3": [ "2.183", "0.0650713", "0.499002", "0.485175", "43", "42", "2", "48", "47" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Statistics_0420", "subject": "Statistics", "topic": "Generalized linear methods", "subtopic": "Logistic regression", "level": "3", "keywords": [ "statistics", "regression", "likelihood" ], "problem_v1": "This question concerns the likelihood for logistic regression. Suppose your data consist of $(x_i,y_i),i=1,\\ldots,165$, with values (8,0), (6,1), (7,1), (8,1), (4,0), (4,0),...\nPart a) Consider the likelihood for logistic regression. Which are the following are true? There might be more than one correct answer. [ANS] A. The residual deviance is greater than or equal to 0 B. The residual deviance is less than or equal to 0 C. The likelihood is less than or equal to 0 D. The log-likelihood is less than or equal to 0 E. The likelihood is less than or equal to 1 F. The likelihood is greater than or equal to 1 G. The likelihood is greater than or equal to 0 H. The log-likelihood is greater than or equal to 0 I. Under I have all data values, I cannot determine any of the above\nPart b) The reason for the answer in (a) is because for the likelihood of discrete random variables, the likelihood is a [ANS]. [Fill in one suitable word.]", "answer_v1": [ "ADEG", "PROBABILITY" ], "answer_type_v1": [ "MCM", "OE" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [] ], "problem_v2": "This question concerns the likelihood for logistic regression. Suppose your data consist of $(x_i,y_i),i=1,\\ldots,164$, with values (1,0), (10,1), (2,0), (4,0), (10,1), (4,0),...\nPart a) Consider the likelihood for logistic regression. Which are the following are true? There might be more than one correct answer. [ANS] A. The residual deviance is greater than or equal to 0 B. The log-likelihood is greater than or equal to 0 C. The residual deviance is less than or equal to 0 D. The likelihood is less than or equal to 1 E. The likelihood is less than or equal to 0 F. The likelihood is greater than or equal to 0 G. The likelihood is greater than or equal to 1 H. The log-likelihood is less than or equal to 0 I. Under I have all data values, I cannot determine any of the above\nPart b) The reason for the answer in (a) is because for the likelihood of discrete random variables, the likelihood is a [ANS]. [Fill in one suitable word.]", "answer_v2": [ "ADFH", "PROBABILITY" ], "answer_type_v2": [ "MCM", "OE" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [] ], "problem_v3": "This question concerns the likelihood for logistic regression. Suppose your data consist of $(x_i,y_i),i=1,\\ldots,129$, with values (4,0), (7,1), (3,1), (6,1), (3,1), (4,0),...\nPart a) Consider the likelihood for logistic regression. Which are the following are true? There might be more than one correct answer. [ANS] A. The residual deviance is less than or equal to 0 B. The log-likelihood is less than or equal to 0 C. The likelihood is less than or equal to 1 D. The residual deviance is greater than or equal to 0 E. The likelihood is greater than or equal to 0 F. The likelihood is greater than or equal to 1 G. The log-likelihood is greater than or equal to 0 H. The likelihood is less than or equal to 0 I. Under I have all data values, I cannot determine any of the above\nPart b) The reason for the answer in (a) is because for the likelihood of discrete random variables, the likelihood is a [ANS]. [Fill in one suitable word.]", "answer_v3": [ "BCDE", "PROBABILITY" ], "answer_type_v3": [ "MCM", "OE" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [] ] } ]