91Ó°ÊÓ

An urban economist wishes to estimate the proportion of Americans who own their homes. What size sample should be obtained if he wishes the estimate to be within 0.02 with \(90 \%\) confidence if (a) he uses a 2010 estimate of 0.669 obtained from the U.S. Census Bureau? (b) he does not use any prior estimates?

A researcher for the U.S. Department of the Treasury wishes to estimate the percentage of Americans who support abolishing the penny. What size sample should be obtained if he wishes the estimate to be within 2 percentage points with \(98 \%\) confidence if (a) he uses a 2006 estimate of \(15 \%\) obtained from a Coinstar National Currency Poll? (b) he does not use any prior estimate?

A school administrator is concerned about the amount of credit-card debt that college students have. She wishes to conduct a poll to estimate the percentage of full-time college students who have credit-card debt of \(\$ 2000\) or more. What size sample should be obtained if she wishes the estimate to be within 2.5 percentage points with \(94 \%\) confidence if (a) a pilot study indicates that the percentage is \(34 \% ?\) (b) no prior estimates are used?

A sociologist wishes to conduct a poll to estimate the percentage of Americans who favor affirmative action programs for women and minorities for admission to colleges and universities. What sample size should be obtained if she wishes the estimate to be within 4 percentage points with \(90 \%\) confidence if (a) she uses a 2003 estimate of \(55 \%\) obtained from a Gallup Youth Survey? (b) she does not use any prior estimates? (c) Why are the results from parts (a) and (b) so close?

A recent Gallup poll asked Americans to disclose the number of books they read during the previous year. Initial survey results indicate that \(s=16.6\) books. (a) How many subjects are needed to estimate the number of books Americans read the previous year within four books with \(95 \%\) confidence? (b) How many subjects are needed to estimate the number of books Americans read the previous year within two books with \(95 \%\) confidence? (c) What effect does doubling the required accuracy have on the sample size? (d) How many subjects are needed to estimate the number of books Americans read the previous year within four books with \(99 \%\) confidence? Compare this result to part (a). How does increasing the level of confidence in the estimate affect sample size? Why is this reasonable?

Jane wants to estimate the proportion of students on her campus who eat cauliflower. After surveying 20 students, she finds 2 who eat cauliflower. Obtain and interpret a \(95 \%\) confidence interval for the proportion of students who eat cauliflower on Jane's campus using Agresti and Coull's method.

Explain what "95\% confidence" means in a \(95 \%\) confidence interval.

Explain why quadrupling the sample size causes the margin of error to be cut in half.