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A New York Times/CBS News Poll asked a random sample of U.S. adults the question, "Do you favor an amendment to the Constitution that would permit organized prayer in public schools?" Based on this poll, the \(95 \%\) confidence interval for the population proportion who favor such an amendment is (0.63,0.69) (a) Interpret the confidence interval. (b) What is the point estimate that was used to create the interval? What is the margin of error? (c) Based on this poll, a reporter claims that more than two-thirds of U.S. adults favor such an amendment. Use the confidence interval to evaluate this claim.

Losing weight A Gallup Poll asked a random sample of U.S. adults, "Would you like to lose weight?" Based on this poll, the \(95 \%\) confidence interval for the population proportion who want to lose weight is (0.56,0.62) (a) Interpret the confidence interval. (b) What is the point estimate that was used to create the interval? What is the margin of error? (c) Based on this poll, Gallup claims that more than half of U.S. adults want to lose weight. Use the confidence interval to evaluate this claim.

The news article goes on to say: "The theoretical errors do not take into account \(\cdots\) additional error resulting from the various practical difficulties in taking any survey of public opinion." List some of the "practical difficulties" that may cause errors which are not included in the ±3 percentage point margin of error.

Statistics class in Exercise 1 also asked an SRS of 20 boys at their school how many pairs of shoes they have. A \(95 \%\) confidence interval for the difference in the population means (girls - boys) is 10.9 to \(26.5 .\) Interpret the confidence interval and the confidence level.

Check whether each of the conditions is met for calculating a confidence interval for the population proportion \(\bar{p}\). Latoya wants to estimate what proportion of the seniors at her boarding high school like the cafeteria food. She interviews an SRS of 50 of the 175 seniors living in the dormitory. She finds that 14 think the cafeteria food is good.

In a recent National Survey of Drug Use and Health, 2312 of 5914 randomly selected full-time U.S. college students were classified as binge drinkers. \({ }^{13}\) (a) Calculate and interpret a \(99 \%\) confidence interval for the population proportion \(p\) that are binge drinkers. (b) A newspaper article claims that \(45 \%\) of full-time U.S. college students are binge drinkers. Use your result from part (a) to comment on this claim.

Have efforts to promote equality for women gone far enough in the United States? A poll on this issue by the cable network MSNBC contacted 1019 adults. A newspaper article about the poll said, "Results have a margin of sampling error of plus or minus 3 percentage points." (a) The news article said that \(65 \%\) of men, but only \(43 \%\) of women, think that efforts to promote equality have gone far enough. Explain why we do not have enough information to give confidence intervals for men and women separately. (b) Would a \(95 \%\) confidence interval for women alone have a margin of error less than \(0.03,\) about equal to \(0.03,\) or greater than 0.03 ? Why? (You see that the news article's statement about the margin of error for poll results is a bit misleading.)

PTC is a substance that has a strong bitter taste for some people and is tasteless for others. The ability to taste \(\mathrm{PTC}\) is inherited. About \(75 \%\) of Italians can taste \(\mathrm{PTC}\), for example. You want to estimate the proportion of Americans who have at least one Italian grandparent and who can taste PTC. (a) How large a sample must you test to estimate the proportion of PTC tasters within 0.04 with \(90 \%\) confidence? Answer this question using the \(75 \%\) estimate as the guessed value for \(\hat{p}\). (b) Answer the question in part (a) again, but this time use the conservative guess \(\hat{p}=0.5 .\) By how much do the two sample sizes differ?

A Gallup Poll found that only \(28 \%\) of American adults expect to inherit money or valuable possessions from a relative. The poll's margin of error was ±3 percentage points at a \(95 \%\) confidence level. This means that (a) the poll used a method that gets an answer within \(3 \%\) of the truth about the population \(95 \%\) of the time. (b) the percent of all adults who expect an inheritance is between \(25 \%\) and \(31 \%\) (c) if Gallup takes another poll on this issue, the results of the second poll will lie between \(25 \%\) and \(31 \%\) (d) there's a \(95 \%\) chance that the percent of all adults who expect an inheritance is between \(25 \%\) and \(31 \%\). (e) Gallup can be \(95 \%\) confident that between \(25 \%\) and \(31 \%\) of the sample expect an inheritance.

Most people can roll their tongues, but many can't. The ability to roll the tongue is genetically determined. Suppose we are interested in determining what proportion of students can roll their tongues. We test a simple random sample of 400 students and find that 317 can roll their tongues. The margin of error for a \(95 \%\) confidence interval for the true proportion of tongue rollers among students is closest to (a) 0.0008 . (c) 0.03 . (c) 0.05 . (b) 0.02 (d) 0.04