Question: In a special double issue of Time magazine, the cover story featured Pope John Paul II as "man of the year" (26 December 1994-2 January 1995, pp. 74-76). As part of the story, Time reported on the results of a survey of 507 adult American Catholics, taken by telephone on December 7-8. It was also reported that "sampling error is 4.4%."
a. One question asked was, "Do you favor allowing women to be priests?" to which 59% of the respondents answered yes. Using the reported margin of error of 4.4%, calculate a 95% confidence interval for the response to this question. Write a sentence interpreting the interval that could be understood by someone who knows nothing about statistics. Be careful about specifying the correct population.
b. Calculate a 95% confidence interval for the question in part a, using the formula in this chapter rather than the reported margin of error. Compare your answer to the answer in part a.
c. Another question in the survey was, "Is it possible to disagree with the Pope and still be a good Catholic?" to which 89% of respondents said yes. Using the formula in this chapter, compute a 95% confidence interval for the true proportion who would answer yes to the question. Now compute a 95% confidence interval using the reported margin of error of 4.4%. Compare your two intervals.
d. If you computed your intervals correctly, you would have found that the two intervals in parts a and b were quite similar to each other, whereas the two intervals in part c were not. In part c, the interval computed using the reported margin of error was wider than the one computed using the formula. Explain why the two methods for computing the intervals agreed more closely for the survey question in parts a and b than for the survey question in part c.