Chapter 10.3, Problem 39E

Refer back to the previous exercise. The actual sample proportion observed in the study was $\hat{p}=0.33$. Based on this sample proportion, is there convincing evidence that more than 25% of law enforcement agencies review social media activity as part of background checks, or is this sample proportion consistent with what you would expect to see when the null hypothesis is true?

10.39

The article “Cops Get Screened for Digital Dirt” (USA TODAY, November 12, 2010) summarizes a report of law enforcement agencies regarding the use of social media to screen applicants for employment. The report was based on a survey of 728 law enforcement agencies. One question on the survey asked if the agency routinely reviewed applicants’ social media activity during background checks. For purposes of this exercise, suppose that the 728 agencies were selected at random, and that you want to use the survey data to decide if there is convincing evidence that more than 25% of law enforcement agencies review applicants’ social media activity as part of routine background checks.

1. a. The sampling distribution of $\hat{p}$ describes the behavior of $\hat{p}$ when random samples are selected from a particular population. Describe the shape, center, and spread of the sampling distribution of $\hat{p}$ for samples of size 728 if the null hypothesis H₀: p = 0.25 is true.
2. b. Would you be surprised to observe a sample proportion of $\hat{p}=0.27$ for a sample of size 728 if the null hypothesis H₀: p = 0.25 is true? Explain why or why not.
3. c. Would you be surprised to observe a sample proportion of $\hat{p}=0.31$ for a sample of size 728 if the null hypothesis H₀: p = 0.25 is true? Explain why or why not.