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IV.1. We’ve been hacked. An October 2014 Gallup Poll asked a random sample of 1017 adults if they or another household member had information from a credit card used at a store stolen by computer hackers in the previous year. Of these adults, 275 said Yes. We can act as if the sample were an SRS. Give a 95% confidence interval for the proportion of all adults who had information from a credit card used at a store by themselves or another household member stolen by computer hackers in the year prior to October 2014. (Hint: See pages 495–499.)

IV.2. Do you drink? In a July 2014 Gallup Poll a random sample of 1013 adults were asked whether they drank alcoholic beverages (liquor, wine, or beer) or completely abstained from drinking any alcohol. Among respondents, 365 said that they completely abstain from drinking alcohol. Assume that the sample was an SRS. Give a 95% confidence interval for the proportion of all adults who completely abstain from drinking alcoholic beverages.

IV.3. We’ve been hacked. Exercise IV.1 concerns a random sample of 1017 adults. Suppose that (unknown to the pollsters) exactly 25% of all adults had information from a credit card used at a store by themselves or another household member stolen by computer hackers in the year prior to October 2014. Imagine that we take very many SRSs of size 1017 from this population and record the percentage in each sample who claim to have had information from a credit card used at a store by themselves or another household member stolen by computer hackers in the year prior to October 2014. Where would the middle 95% of all values of this percentage lie? (Hint: See pages 495–497.)

IV.4. Do you drink? Exercise IV.2 concerns a random sample of 1013 adults. Suppose that, in the population of all adults, exactly 35% would say that they completely abstain from drinking alcoholic beverages. Imagine that we take a very large number of SRSs of size 1013. For each sample, we record the proportion $\widehat{p}$ of the sample who would say that they totally abstain from alcoholic beverages.

IV.5. Honesty in the media. A Gallup Poll conducted from December 5 to 8, 2013, asked a random sample of 1031 adults to rate the honesty and ethical standards of people in a variety of professions. Among the respondents, 206 rated the honesty and ethical standards of TV reporters as very high or high. Assume that the sample was an SRS. Give a 95% confidence interval for the proportion of all adults who would rate the honesty and ethical standards of TV reporters as very high or high. (Hint: See pages 498–499.)

IV.6. Roulette. A roulette wheel has 18 red slots among its 38 slots. You observe many spins and record the number of times the ball falls in a red slot. Now you want to use these data to test whether the probability p of the ball falling in a red slot has the value that is correct for a fair roulette wheel. State the hypotheses H₀ and H_a that you will test.

IV.7. Why not? Table 11.1 (page 244) records the percentage of residents aged 65 or older in each of the 50 states. You can check that this percentage is 14% or higher in 12 of the states. So the sample proportion of states with at least 14% of elderly residents is $\widehat{p}=12/50=0.24$. Explain why it does not make sense to go on to calculate a 95% confidence interval for the population proportion p. (Hint: See pages 494–495.)

IV.8. Helping welfare mothers. A study compares two groups of mothers with young children who were on welfare two years ago. One group attended a voluntary training program that was offered free of charge at a local vocational school and was advertised in the local news media. The other group did not choose to attend the training program. The study finds a significant difference ($P<0.01$) between the proportions of the mothers in the two groups who are still on welfare. The difference is not only significant but quite large. The report says that with 95% confidence the percentage of the nonattending group still on welfare is 21% $\pm$ 4% higher than that of the group who attended the program. You are on the staff of a member of Congress who is interested in the plight of welfare mothers and who asks you about the report.

IV.9. Beating the system. Some doctors think that health plan rules restrict their ability to treat their patients effectively, so they bend the rules to help patients get reimbursed by their health plans. Here’s a sentence from a study on this topic: “Physicians who agree with the statement ‘Today it is necessary to game the system to provide high-quality care’ reported manipulating reimbursement systems more often than those who did not agree with the statement (64.3% vs 35.7%; P < .001).’’

IV.10. Smoking in the United States. A July 2015 nationwide random survey of 1009 adults asked whether they had smoked cigarettes in the last week. Among the respondents, 192 said they had. Assume that the sample was an SRS.

IV.11. When shall we call you? As you might guess, telephone sample surveys get better response rates during the evening than during the weekday daytime. One study called 2304 randomly chosen telephone numbers on weekday mornings. Of these, 1313 calls were answered and only 207 resulted in interviews. Of 2454 calls on weekday evenings, 1840 were answered and 712 interviews resulted. Give two 95% confidence intervals, for the proportions of all calls that are answered on weekday mornings and on weekday evenings. Are you confident that the proportion is higher in the evening? (Hint: See pages 498–499, 554.)

IV.12. Smoking in the United States. Does the survey of Exercise IV.10 provide good evidence that fewer than one-fourth of all American adults smoked in the week prior to the survey?

IV.13. When shall we call you? Suppose that we know that 57% of all calls made by sample surveys on weekday mornings are answered. We make 2454 calls to randomly chosen numbers during weekday evenings. Of these, 1840 are answered. Is this good evidence that the proportion of answered calls is higher in the evening?

IV.14. Not significant. The study cited in Exercise IV.9 looked at the factors that may affect whether doctors bend medical plan rules. Perhaps doctors who fear being prosecuted will bend the rules less often. The study report said, “Notably, greater worry about prosecution for fraud did not affect physicians’ use of these tactics (P = .34).’’ Explain why the result (P = 0.34) supports the conclusion that doctors’ fears about potential prosecution did not affect behavior.

IV.15. Going to church. Opinion polls show that about 40% of Americans say they attended religious services in the last week. This result has stayed stable for decades. Studies of what people actually do, as opposed to what they say they do, suggest that actual church attendance is much lower. One study calculated 95% confidence intervals based on what a sample of Catholics said and then based on a sample of actual behavior. In Chicago, for example, the 95% confidence interval from the opinion poll said that between 45.7% and 51.3% of Catholics attended mass weekly. The 95% confidence interval from actual counts said that between 25.7% and 28.9% attended mass weekly.

The following exercises are based on the optional sections of Chapters 21 and 22.

IV.16. We’ve been hacked. An October 2014 Gallup Poll asked a random sample of 1017 adults if they or another household member had information from a credit card used at a store stolen by computer hackers in the previous year. Of these adults, 275 said Yes. Assume that the sample was an SRS. Give 90% and 99% confidence intervals for the proportion of all adults who had information from a credit card used at a store by themselves or another household member stolen by computer hackers in the year prior to October 2014. Explain briefly what important fact about confidence intervals is illustrated by comparing these two intervals and the 95% confidence interval from Exercise IV.1.

IV.17. Do you drink? In a July 2014 Gallup Poll a random sample of 1013 adults were asked whether they drank alcoholic beverages (liquor, wine, or beer) or completely abstained from drinking any alcohol. Among respondents, 365 said that they completely abstain from drinking alcohol. Assume that the sample was an SRS. Is this good evidence that more than one-third of American adults completely abstain from drinking alcoholic beverages? Show the five steps of the test (hypotheses, sampling distribution, data, P-value, conclusion) clearly. Use a significance level of 0.05. (Hint: See pages 522–531.)

The following exercises are based on the optional material in Chapters 21, 22, and 24.

IV.18. Honesty in the media. A Gallup Poll conducted from December 5 to 8, 2013, asked a random sample of 1031 adults to rate the honesty and ethical standards of people in a variety of professions. Among the respondents, 206 rated the honesty and ethical standards of TV reporters as very high or high. Assume that the sample was an SRS. Give a 90% confidence interval for the proportion of all adults who rate the honesty and ethical standards of TV reporters as very high or high. For what purpose might a 90% confidence interval be less useful than a 95% confidence interval? For what purpose might a 90% interval be more useful?

IV.19. A poll of voters. You are the polling consultant to a member of Congress. An SRS of 500 registered voters finds that 37% name “economic problems’’ as the most important issue facing the nation. Give a 90% confidence interval for the proportion of all voters who hold this opinion. Then explain carefully to the member of Congress what your conclusion reveals about voters’ opinions. (Hint: See pages 502–505.)

IV.20. Smoking in the United States. Carry out the significance test called for in Exercise IV.12 in all detail. Show the five steps of the test (hypotheses, sampling distribution, data, P-value, conclusion) clearly.

IV.21. When shall we call you? Carry out the significance test called for in Exercise IV.13 in all detail. Show the five steps of the test (hypotheses, sampling distribution, data, P-value, conclusion) clearly. (Hint: See pages 528–530.)

IV.22. CEO pay. A study of 104 corporations found that the pay of their chief executive officers had increased an average of $\overline{x}=6.9$ % per year in real terms. The standard deviation of the percentage increases was s = 17.4%.

IV.23. Water quality. An environmentalist group collects a liter of water from each of 45 random locations along a stream and measures the amount of dissolved oxygen in each specimen. The mean is 4.62 milligrams (mg) and the standard deviation is 0.92 mg. Is this strong evidence that the stream has a mean oxygen content of less than 5 mg per liter? (Hint: See pages 531–535.)

IV.24. Pleasant smells. Do pleasant odors help work go faster? Twenty-one subjects worked a paper-and-pencil maze wearing a mask that was either unscented or carried the smell of flowers. Each subject worked the maze three times with each mask, in random order. (This is a matched pairs design.) Here are the differences in their average times (in seconds), unscented minus scented. If the floral smell speeds work, the difference will be positive because the time with the scent will be lower.

IV.25. Sharks. Great white sharks are big and hungry. Here are the lengths in feet of 44 great whites:

IV.26. Pleasant smells. Return to the data in Exercise IV.24. Give a 95% confidence interval for the mean improvement in time to solve a maze when wearing a mask with a floral scent. Are you confident that the scent does improve mean working time?

IV.27. Sharks. Return to the data in Exercise IV.25. Is there good evidence that the mean length of sharks in the population that these sharks represent is greater than 15 feet? (Hint: See pages 531–535.)

IV.28. Simpson’s paradox. If we compare average 2015 SAT mathematics scores, we find that female college-bound seniors in Connecticut do better than female college-bound seniors in New York. But if we look only at white students, New York does better. If we look only at minority students, New York again does better. That’s Simpson’s paradox: the comparison reverses when we lump all students together. Explain carefully why this makes sense, using the fact that a much higher percentage of Connecticut female college-bound seniors are white.

IV.29. Unhappy HMO patients. A study of complaints by HMO members compared those who filed complaints about medical treatment and those who filed nonmedical complaints with an SRS of members who did not complain that year. Here are the data on the number who stayed and the number who voluntarily left the HMO:

IV.30. Treating ulcers. Gastric freezing was once a recommended treatment for stomach ulcers. Use of gastric freezing stopped after experiments showed it had no effect. One randomized comparative experiment found that 28 of the 82 gastric-freezing patients improved, while 30 of the 78 patients in the placebo group improved.

IV.31. When shall we call you? In Exercise IV.11, we learned of a study that dialed telephone numbers at random during two periods of the day. Of 2304 numbers called on weekday mornings, 1313 answered. Of 2454 calls on weekday evenings, 1840 were answered.