Question

The article "Cops Get Screened for Digital Dirt" (USA Today, Nov. 12,2010 ) summarizes a report on law enforcement agency 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. a. Describe the shape, center, and spread of the sampling distribution of \(\hat{p}\) for samples of size 728 if the null hypothesis \(H_{0}: p=0.25\) is true. b. Would you be surprised to observe a sample proportion as large as \(\hat{p}=0.27\) for a sample of size 728 if the null hypothesis \(H_{0}: p=0.25\) were true? Explain why or why not. c. Would you be surprised to observe a sample proportion as large as \(\hat{p}=0.31\) for a sample of size 728 if the null hypothesis \(H_{0}: p=0.25\) were true? Explain why or why not. d. 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?

Short answer

In conclusion, the actual sample proportion of 0.33 would be very surprising if the null hypothesis of \(p=0.25\) were true. There is convincing evidence supporting that more than 25% of law enforcement agencies review social media activity as part of background checks.

Step-by-step solution

Define the Null Hypothesis

The null hypothesis \(H_{0}\) is that the proportion \(p\) of law enforcement agencies that review social media activity as part of their background checks is 0.25.

Describe the Shape, Center, and Spread of the Sampling Distribution

The shape of the sampling distribution of \(\hat{p}\) is approximately normally distributed by the Central Limit Theorem (CLT) as the sample size is large (\(n = 728\)). The center of this distribution is the same as the population proportion \(p = 0.25\). The standard deviation (spread) of the sampling distribution can be calculated using the formula \(\sqrt{{p(1-p)}/{n}} = \sqrt{{0.25(0.75)}/{728}} \approx 0.0176

Calculation and Interpretation of the Sample Proportion \(\hat{p}=0.27\)

We calculate the Z-score for \(\hat{p}=0.27\) using \(Z = (\hat{p} - p) / \sqrt{(p(1-p)/n)} = (0.27 - 0.25) / 0.0176 =1.13\). Near the center of our normal distribution (within 1 standard deviation), we would not be surprised to observe this sample proportion given the null hypothesis is true.

Calculation and Interpretation of the Sample Proportion \(\hat{p}=0.31\)

We calculate the Z-score for \(\hat{p}=0.31\) using \(Z = (\hat{p} - p) / \sqrt{(p(1-p)/n)} = (0.31 - 0.25) / 0.0176 = 3.41\). This is more than 3 standard deviations away from the center, which is a rare event under the standard normal distribution. So we would be pretty surprised to observe this sample proportion given the null hypothesis is true.

Decision Based on the Actual Sample Proportion \(\hat{p}=0.33\)

The Z-score for \(\hat{p}=0.33\) is \(Z = (\hat{p} - p) / \sqrt{(p(1-p)/n)} = (0.33 - 0.25) / 0.0176 = 4.54\). This value is far in the tail of the standard normal distribution, a very unlikely occurrence if the null hypothesis is true. Hence, we have convincing evidence that the proportion of law enforcement agencies that review social media activity as part of background checks is more than \(25\% \).