91Ó°ÊÓ

Suppose that a particular candidate for public office is favored by \(48 \%\) of all registered voters in the district. A polling organization will take a random sample of 500 of these voters and will use \(\hat{p},\) the sample proportion, to estimate \(p\). a. Show that \(\sigma_{\hat{p}}\), the standard deviation of \(\hat{p},\) is equal to 0.022 . b. If for a different sample size, \(\sigma_{j}=0.071,\) would you expect more or less sample-to-sample variability in the sample proportions than when \(n=500 ?\) c. Is the sample size that resulted in \(\sigma_{\hat{p}}=0.071\) larger than 500 or smaller than \(500 ?\) Explain your reasoning.

Some colleges now allow students to rent textbooks for a semester. Suppose that \(38 \%\) of all students enrolled at a particular college would rent textbooks if that option were available to them. If the campus bookstore uses a random sample of size 100 to estimate the proportion of students at the college who would rent textbooks, is it likely that this estimate would be within 0.05 of the actual population proportion? Use what you know about the sampling distribution of \(\hat{p}\) to support your answer.

In a study of pet owners, it was reported that 24\% celebrate their pet's birthday (Pet Statistics, Bissell Homecare, Inc., 2010). Suppose that this estimate was based on a random sample of 200 pet owners. Is it reasonable to conclude that the proportion of all pet owners who celebrate their pet's birthday is less than \(0.25 ?\) Use what you know about the sampling distribution of \(\hat{p}\) to support your answer.