Mind on Statistics. Chapter 4

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1 Mind on Statistics Chapter 4 Sections 4.1 Questions 1 to 4: The table below shows the counts by gender and highest degree attained for 498 respondents in the General Social Survey. Highest Degree Gender No High School Degree High School Degree Junior College Bachelor Degree Graduate Degree Total Male Female Total What percent of the sample were males? A. 43.6% B. 48.5% C. 56.4% D. 77.2% 2. What percent of the sample were males with no high school degree? A. 9.8% B. 20.3% C. 22.6% D. 48.5% 3. What percent of the sample did not graduate from high school? A. 18.5% B. 20.3% C. 22.6% D. 52.4% 4. What percent of females had a graduate degree? A. 2.4% B. 4.3% C. 7.0% D. 56.4% 65

2 Questions 5 to 8: The table below shows the number of Olympic medals won by the three countries with the most medals during the 2000 Olympics in Sydney, Australia. There were a total of 244 medals won by the three countries. Type of Medal Country Gold Silver Bronze Total USA Russia China Total What percent of the medals won among the three countries were won by the USA? A. 39.4% B. 39.8% C. 40.2% D. 40.6% 6. What percent of the medals won among the three countries were gold? A. 39.4% B. 39.8% C. 40.2% D. 40.6% KEY: D 7. What percent of the medals won by the USA were gold? A. 39.4% B. 39.8% C. 40.2% D. 40.6% 8. What percent of the medals won by China were silver? A. 6.6% B. 24.2% C. 27.1% D. 28.3% 66

3 Questions 9 to 13: A researcher conducted a study on college students to see if there was a link between gender and how often they have cheated on an exam. She asked two questions on a survey: (1) What is your gender? Male Female (2) How many times have you cheated on an exam while in college? Never 1 or 2 times 3 or more times A two-way table of observed counts follows: Cheated on an exam? Gender Never 1 or 2 times 3 or more times Total Male Female Total What percent of the students in the study never cheated on an exam? A. 60% B. 120% C. 50% D. 30% 10. What percent of the students in the study cheated 1 or 2 times on an exam? A. 20% B. 30% C. 50% D. 25% KEY: D 11. What percent of the students in the study cheated 3 or more times on an exam? A. 20% B. 10% C. 30% D. 15% KEY: D 12. What percent of the males in the study cheated 3 or more times on an exam? A. 20% B. 10% C. 30% D. 15% 13. What percent of the females in the study cheated 3 or more times on an exam? A. 20% B. 10% C. 30% D. 15% 67

4 Questions 14 to 20: The table below shows the opinions of 908 respondents in the General Social Survey to the question Do you believe there is life after death? 14. What percent of males believe in life after death? KEY: 72.1% Life After Death? Gender Yes No Total Male Female Total What percent of females believe in life after death? KEY: 78.9% 16. What percent of respondents in the sample believe in life after death? KEY: 76.0% Questions 12 to 15: The table shows the opinions of 953 respondents in the General Social Survey to the question If your party nominated a woman for President, would you vote for her if she were qualified for the job? Vote for Qualified Woman? Gender Yes No Total Male Female Total What percent of males would vote for a woman president? KEY: 84.0% 18. What percent of females would vote for a woman president? KEY: 88.1% 19. What percent of respondents in the sample would vote for a woman president? KEY: 86.4% 20. What percent of respondents in the sample were female? KEY: 58.1% 68

6 Section If there is no difference in the risk of a disease for smokers and non-smokers, then the relative risk of the disease for smokers compared to non-smokers would be: A. 0 B. 0.5 C. 1 D. It depends on the actual risks. 26. The relative risk of a disease for group 1 compared to group 2 is close to the odds ratio under which of these circumstances? A. When the numbers with the disease in each group are small compared to the total in each group. B. When the numbers with the disease are about equal to the numbers without the disease in each group. C. When the numbers with the disease in each group are almost equal to the total in each group. D. The relative risk and the odds ratio are never close to each other. 27. If the relative risk of having an accident while eating when driving (compared to not eating) is 5, and the risk of having an accident when not eating is 1 in 10000, then the risk of having an accident while eating is: A. 1 in 5000 B. 1 in C. 1 in 2000 D. None of the above. 70

7 Questions 28 to 31: The table below shows the responses from a sample of 680 people in the General Social Survey to the question, Do you sometimes drink more than you think you should? Drink more than should? Gender Yes No Total Male Female Total What is the risk (or percentage) of men thinking they drank more than they should? A. 22.2% B. 35.7% C. 46.0% D. 62.1% 29. What is the risk (or percentage) of women thinking they drank more than they should? A. 13.5% B. 26.1% C. 35.7% D. 37.9% 30. What is the relative risk for women thinking they drank more than they should compared to men? A B C D What is the odds ratio for women thinking they drank more than they should compared to men? A B C D

8 Questions 32 to 37: The table below summarizes, by gender of respondent, the responses from 1,033 people to the question, Do you smoke? Smoke? Gender Yes No Total Male Female Total What are the odds of smoking (to not smoking) for a man? A B C D KEY: D 33. What are the odds of smoking (to not smoking) for a woman? A B C D What is the odds ratio for women smoking (to not smoking) compared to men? A B C D What is the odds ratio for men smoking (to not smoking) compared to women? A B C D KEY: D 36. What is the relative risk for women smoking compared to men smoking? A B C D Based on the National Household Survey on Drug Abuse, the percentage of 17-year olds who ever tried cigarette smoking is 56.2%. The relative risk of ever smoking for a 17-year old versus a 12-year old is 3.6. What is the risk of smoking for a 12-year-old (i.e. what was the percentage of 12-year olds who ever tried smoking)? A. 14.1% B. 15.6% C. 50.0% D. 56.2% 72

9 Questions 38 to 40: A survey asked people how often they exceed speed limits. The data are then categorized into the following contingency table of counts showing the relationship between age group and response. Exceed Limit if Possible? Age Always Not Always Total Under Over Total Among people with age over 30, what's the "risk" of always exceeding the speed limit? A B C D Among people with age under 30 what are the odds that they always exceed the speed limit? A. 1 to 2 B. 2 to 1 C. 1 to 1 D. 50% 40. What is the relative risk of always exceeding the speed limit for people under 30 compared to people over 30? A. 2.5 B. 0.4 C. 0.5 D. 30% Questions 41 and 42: A newspaper article reported that "Children who routinely compete in vigorous after-school sports on smoggy days are three times more likely to get asthma than their non-athletic peers." (Sacramento Bee, Feb 1, 2002, p. A1) 41. Of the following, which is the most important additional information that would be useful before making a decision about participation in school sports? A. Where was the study conducted? B. How many students in the study participated in after-school sports? C. What is the baseline risk for getting asthma? D. Who funded the study? 42. The newspaper also reported that "The number of children in the study who contracted asthma was relatively small of 3,535." From this information and the information given in the original quote, which of the following could not be computed? A. The baseline risk of getting asthma without participating in after-school sports. B. The overall risk of getting asthma for the children in this study. C. The relative risk of getting asthma for children who routinely participate in vigorous after-school sports on smoggy days and their non-athletic peers. D. All of the above could be computed. 73

10 Questions 43 and 44: A study done by the Center for Academic Integrity at Rutgers University surveyed 2116 students at 21 colleges and universities. Some of the schools had an "honor code" and others did not. Of the students at schools with an honor code, 7% reported having plagiarized a paper via the Internet, while at schools with no honor code, 13% did so. (Sacramento Bee, Feb 29, 2000, D1.) 43. For this study, the relative risk of a student having plagiarized a paper via the Internet at a school with no honor code, compared to a school with an honor code is: A. 13/7 = B. 7/13 = C. 13/87 = D. 87/13 = Which of the following statements about percent increase in risk is correct for this study? A. There is a 185.7% increase in the risk of plagiarism at a school with no honor code, compared to a school with an honor code. B. There is an 85.7% increase in the risk of plagiarism at a school with no honor code, compared to a school with an honor code. C. There is a 53.8% increase in the risk of plagiarism at a school with no honor code, compared to a school with an honor code. D. There is a 6% increase in the risk of plagiarism at a school with no honor code, compared to a school with an honor code. 45. A study on the use of seat belts versus belted booster seats for children ages 4 and 5 reported that Using seat belts instead of booster seats was associated with increased risk for serious injury in an accident; the relative risk was 2.4." Based on this, it can be concluded that for this study: A. Children ages 4 and 5 in a booster seat were 2.4 times more likely to have serious injuries in an accident than were children wearing seatbelts. B. Children ages 4 and 5 wearing seatbelts were 2.4 times more likely to have serious injuries in an accident than were children in a booster seat. C. The percent of children ages 4 and 5 in a booster seat was 2.4 times higher than the percent of children wearing seatbelts. D. The percent of children ages 4 and 5 wearing seatbelts was 2.4 times higher than the percent of children in a booster seat. 46. The relative risk of allergies for children of parents who smoke compared to children of parents who don t smoke is 3.0. Suppose that the risk of allergies for the children of non-smokers is 0.15 (15%). What is the risk of allergies for the children of smokers? A. 3% B. 5% C. 30% D. 45% KEY: D 74

11 47. An article in the Sacramento Bee (May 8, 2000, p A12) reported that "Early results from a study of more than 2000 pregnant women in Alabama show that pregnant mothers with severe gum disease are up to eight times more likely to have underweight premature babies than moms with healthy mouths." The phrase "eight times more likely" indicates what in our statistical language? relative risk of eight. 48. A 1995 Washington Post-ABC News Poll on Affirmative Action reported that 76 percent of men and 69 percent of women opposed giving preferences to women in hiring, promotions and college admissions. What are the odds that a woman was opposed to giving preferences versus not opposed? KEY: The odds of being opposed to not being opposed for women are 69 opposed to 31 not opposed. 49. The relative risk of a 25-year-old woman developing invasive breast cancer within 5 years is 5.72 times greater than for a 20-year-old woman. If the 5-year risk of breast cancer for a 20 year old woman is 0.007%, what is the risk for a 25 year old woman? KEY: 0.040% Questions 50 to 52: The table below shows the opinions of 908 respondents in the General Social Survey to the question Do you believe there is life after death? Life After Death? Gender Yes No Total Male Female Total What are the odds of believing in life after death for males? KEY: 282/109 = What are the odds of believing in life after death for females? KEY: 408/109 = What is the odds ratio for males versus females who believe in life after death? KEY: 2.59/3.74 =

12 Section Simpson's Paradox occurs when A. no baseline risk is given, so it is not know whether or not a high relative risk has practical importance. B. a confounding variable rather than the explanatory variable is responsible for a change in the response variable. C. the direction of the relationship between two variables changes when the categories of a confounding variable are taken into account. D. the results of a test are statistically significant but are really due to chance. Questions 54 to 59: The table below shows the ratings given by 200 students in an introductory statistics course. There were two instructors for the course, each with 100 students. Rating Instructor High Quality Not High Quality Total Jones Smith Total What percent of Jones students rated the teaching as high quality? A. 38% B. 62% C. 50% 55. What percent of Smith s students rated the teaching as high quality? A. 38% B. 62% C. 50% 56. Which instructor has the highest percentage of high quality teaching ratings? A. Jones B. Smith C. They had the same percentages. 76

13 Questions 54 to 59 continued: Another question on the evaluation was whether or not the student considered themselves as being good at math or not. The tables below show the ratings for each instructor for these two groups of students. Good at Math Rating Instructor High Quality Not High Quality Total Jones Smith Total Not Good at Math Rating Instructor High Quality Not High Quality Total Jones Smith Total Among those students who consider themselves good at math, which instructor has the highest percentage of high quality teaching ratings? A. Jones B. Smith C. They had the same percentages. 58. Among those students who consider themselves good at math, which instructor has the highest percentage of high quality teaching ratings? A. Jones B. Smith C. They had the same percentages. 59. Which phrases best complete this sentence? Those students who are at math generally gave higher instructor ratings and instructor had a higher percentage of these type of students, giving the inaccurate impression that this instructor was best overall. A. good, Jones B. good, Smith C. not good, Jones D. not good, Smith 77

15 Section 4.4 Questions 66 to 68: The table below shows the opinions of 321 respondents from the General Social Survey by whether they owned a gun (or not) and whether they favored (or opposed) a law requiring a permit to own a gun. Chi-Square = 9.137, DF = 1, P-Value = Gun Law Opinion Own Gun? Oppose Favor Total Yes No Total The percentage of gun owners in favor of the gun law was A. 34% B. 41% C. 76% D. 83% 67. The percentage of non-gun owners in favor of the gun law was A. 49% B. 59% C. 83% D. 89% KEY: D 68. Based on the chi-square statistic and p-value, one can conclude that A. the difference between the support for the gun law between gun owners and non-gun owners is not statistically significant. B. the difference between the support for the gun law between gun owners and non-gun owners is statistically significant. C. the difference between the support for the gun law between gun owners and non-gun owners is not practically significant. D. the difference between the support for the gun law between gun owners and non-gun owners is practically significant. 79

16 Questions 69 to 71: A researcher conducted a study on college students to see if there was a link between gender and how often they have cheated on an exam. She asked two questions on a survey: (1) What is your gender? Male Female (2) How many times have you cheated on an exam while in college? Never 1 or 2 times 3 or more times A two-way table of observed counts follows: Cheated on an exam? Gender Never 1 or 2 times 3 or more times Total Male Female Total Chi-Square = 5.33, DF = 2, P-Value = The percentage of males who cheated 1 or 2 times was A. 20% B. 30% C. 50% D. 25% 70. The percentage of females who cheated 1 or 2 times was A. 20% B. 30% C. 50% D. 25% 71. Based on the chi-square statistic and p-value, what conclusion can be made about the relationship between gender and response to the question about cheating, using α = 0.05? A. The result is not statistically significant so there does not appear to be a relationship. B. The result is not statistically significant so there does appear to be a relationship. C. The result is statistically significant so there does not appear to be relationship. D. The result is statistically significant so there does appear to be a relationship. 80

17 Questions 72 to 74: In the General Social Survey, respondents were asked Would you approve of an adult male punching a stranger if the stranger was drunk and bumped into the man and his wife on the street? A two-way table summarizing the results for 1006 respondents, by gender, is shown below: Would you approve of punch? Gender Yes No Total Female Male Total Chi-Square = 0.002, DF = 2, P-Value = The percentage of males who responded Yes was A. 8.1% B. 4.6% C. 8.0% D. 3.5% 73. The percentage of females who responded Yes was A. 8.1% B. 4.6% C. 8.0% D. 3.5% 74. Based on the chi-square statistic and p-value, what conclusion can be made about the relationship between gender and response to the question about approving of a punch, using α = 0.05? A. The result is not statistically significant so there does not appear to be a relationship. B. The result is not statistically significant so there does appear to be a relationship. C. The result is statistically significant so there does not appear to be relationship. D. The result is statistically significant so there does appear to be a relationship. 81

18 Questions 75 and 76: Students in a statistics class were asked, With whom do you find it easier to make friends: person of the same sex, person of opposite sex, or no preference? A table summarizing the responses by gender is given below. Minitab results for a chi-square test for these data were Chi-Sq = 7.15 DF=2 P-value = With whom is it easier to make friends? Gender no preference opposite sex same sex Total Male Female Total Among females, what percent said same sex? A. 15% B. 25% C. 30% D. 60% 76. Assume these students represent a random sample of all students. Based on the chi-square test, what conclusion can be made about the relationship between gender and response to the question about friends, using α = 0.05? A. The result is not statistically significant so there does not appear to be a relationship. B. The result is not statistically significant so there does appear to be a relationship. C. The result is statistically significant so there does not appear to be relationship. D. The result is statistically significant so there does appear to be a relationship. KEY: D 77. A study done by the Center for Academic Integrity at Rutgers University surveyed 2116 students at 21 colleges and universities. Some of the schools had an "honor code" and others did not. Of the students at schools with an honor code, 7% reported having plagiarized a paper via the Internet, while at schools with no honor code, 13% did so. (Sacramento Bee, Feb 29, 2000, D1.) Although the data provided are not sufficient to carry out a chisquare test of the relationship between whether or not a school has an honor code and whether or not a student would plagiarize a paper via the Internet, suppose such a test were to show a statistically significant relationship on the basis of this study. What would be the correct conclusion? A. Because this is an observational study, it can be concluded that implementing an honor code at a college or university will reduce the risk of plagiarism. B. Because this is a randomized experiment, it can be concluded that implementing an honor code at a college or university will reduce the risk of plagiarism. C. Because this is an observational study and confounding variables are likely, it cannot be concluded that implementing an honor code at a college or university will reduce the risk of plagiarism. D. Because this is a randomized experiment and confounding variables are likely, it cannot be concluded that implementing an honor code at a college or university will reduce the risk of plagiarism. 78. The statistical significance of the association or relationship between two categorical variables is examined using a value known as the chi-square statistic, and a corresponding p-value that assesses the chance of getting this value for the Chi-square statistic or one even larger. Suppose the p-value of the test turns out to be In this case, we should decide that A. there is only an 18% chance that the observed relationship occurred by chance, so we can say that the relationship is statistically significant. B. the observed relationship most likely did not occur by chance, so we can say that the relationship is statistically significant. C. the observed relationship most likely did not occur by chance, so we cannot say that the relationship is statistically significant. D. the observed relationship could have occurred by chance, so we cannot say that the relationship is statistically significant KEY: D 82

19 79. The following two-way cross-tabulation (contingency table) of counts summarizes the relationship between gender and smoking status: Of the males, what percentage smokes? A. 600/1000, or 60% B. 150/1000, or 15%. C. 150/600, or 25%. D. 150/250, or 60%. Smoking Status Gender Smoker Non-smoker Total Male Female Total A statistically significant relationship between two categorical variables is illustrated in the sample as one that A. is small enough that it is likely to have occurred in the observed sample even if there is no relationship in the population. B. is small enough that it is unlikely to have occurred in the observed sample if there is no relationship in the population. C. is large enough that it is likely to have occurred in the observed sample even if there is no relationship in the population D. is large enough that it is unlikely to have occurred in the observed sample if there is no relationship in the population KEY: D 81. What is the primary purpose of doing a chi-square test? A. to determine if there is a significant relationship between two quantitative variables B. to determine if there is a significant relationship between two categorical variables C. to determine if there is a significant relationship between two continuous variables D. to estimate a population proportion 82. A chi-square test involves a set of counts called expected counts. What are the expected counts? A. Hypothetical counts that would occur if the alternative hypothesis were true. B. Hypothetical counts that would occur on average if the null hypothesis were true. C. The actual counts that did occur in the observed data. D. The long-run counts that would be expected if the observed counts are representative 83. Pick the choice that best completes the following sentence. If a relationship between two variables is called statistically significant, it means the investigators think the variables are A. related in the population represented by the sample. B. not related in the population represented by the sample. C. related in the sample due to chance alone. D. very important. 83

20 84. A statement that claims that changes in the explanatory variable will have no effect on the response variable is an example of a(n) hypothesis. A. negative B. dependent C. null D. alternative 85. In a newspaper article about whether the regular use of Vitamin C reduces the risk of getting a cold, a researcher is quoted as saying that Vitamin C performed better than placebo in an experiment, but the difference was not larger than what could be explained by chance. In statistical terms, the researcher is saying the results are A. due to non-sampling errors. B. definitely due to chance. C. statistically significant. D. not statistically significant. KEY: D Questions 86 to 89: The table shows the opinions of 1027 respondents in the General Social Survey to the question Is there any area right around here that is, within a mile where you would be afraid to walk alone at night? The purpose of examining the data is to see if there is a gender difference in how people would respond to this question. Area afraid to walk at night? Gender Yes No Total Male Female Total State the null and alternative hypotheses for this study. KEY: Null hypothesis: There is no relationship between gender and response to the question about being afraid to walk alone in one s neighborhood at night. Alternative hypothesis: There is a relationship between gender and response to the question about being afraid to walk alone in one s neighborhood at night. 87. Assuming the null hypothesis is true, calculate the expected number of women in the sample who would be afraid to walk alone in their neighborhood at night. KEY: If the null hypothesis is true, calculate the expected number of men in the sample who would be afraid to walk alone in their neighborhood at night. KEY: To test whether there is a relationship in the population between gender and fear of walking alone at night, the chi-square statistic = and the p-value < Are the results statistically significant or not statistically significant? State your conclusion. KEY: The results are statistically significant. The conclusion is that there is a relationship in the population between gender and response to the question about being afraid to walk alone in one s neighborhood at night. 84

21 Questions 90 to 93: The table below shows the opinions of 1027 respondents in the General Social Survey to the question Everything considered, would you say that in general, you approve or disapprove of wiretapping? The purpose of examining the data is to see if there is a gender difference in how people would respond to this question. Wiretapping Opinion Gender Approve Disapprove Total Male Female Total State the null and alternative hypotheses for this study. KEY: Null hypothesis: There is no relationship between gender and approval of wiretapping. Alternative hypothesis: There is a relationship between gender and approval of wiretapping. 91. Assuming the null hypothesis is true, calculate the expected number of women in the sample who would approve of wiretapping. KEY: Assuming the null hypothesis is true, calculate the expected number of men in the sample who would approve of wiretapping. KEY: To test whether there is a relationship in the population between gender and support for wiretapping, the chisquare statistic = 11.7 and the p-value = Are the results statistically significant or not statistically significant? State your conclusion. KEY: The results are statistically significant. The conclusion is that there is a relationship between gender and approving of the practice of wiretapping. 85

22 Questions 94 to 96: In the General Social Survey, respondents were asked, Do you agree with the following statement? In spite of what some people say, the lot (situation/condition) of the average man is getting worse, not better. The results, for 989 respondents by gender, are shown below. Lot is getting worse Gender Agree Disagree Total Female Male Total How many females in the sample agreed with the statement? KEY: Assuming there is no relationship between gender and opinion about the lot of the average man, how many females would we be expected to agree with the statement? KEY: (556)(590)/989 = To test whether there is a relationship in the population between gender and opinion about the lot of the average man, the chi-square statistic = and the p-value = Are the results statistically significant at the 5% level? State your conclusion. KEY: Yes, the results are statistically significant. The conclusion is that there is a relationship between gender and opinion about the lot of the average man for the population represented by this sample. 86

23 Questions 97 to 100: The table below shows the opinions of 1447 respondents in the General Social Survey to the question Do you favor or oppose the death penalty for persons convicted of murder? The purpose of examining the data is to see if there is a gender difference in how people would respond to this question. Death Penalty Opinion Gender Favor Oppose Total Male Female Total State the null and alternative hypotheses for this study. KEY: Null hypothesis: There is no relationship between gender and opinion on the death penalty. Alternative hypothesis: There is a relationship between gender and opinion on the death penalty. 98. Assuming the null hypothesis is true, calculate the expected number of women in the sample who would favor the death penalty KEY: Assuming the null hypothesis is true, calculate the expected number of men in the sample who would favor the death penalty. KEY: To test whether there is a relationship in the population between gender and support for the death penalty, the chi-square statistic = 20.9 and the p-value < Are the results statistically significant or not statistically significant? State your conclusion. KEY: The results are statistically significant. The conclusion is that there is a relationship between gender and opinion about the death penalty. 87

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