The FBI wants to determine the effectiveness of their 10 Most Wanted list. To do so, they need to find out the fraction of people who appear on the list that are actually caught. Step 2 of 2: Suppose a sample of 514 suspected criminals is drawn. Of these people, 164 were captured. Using the data, construct the $95 \%$ confidence interval for the population proportion of people who are captured after appearing on the 10 Most Wanted list. Round your answers to three decimal places. Answer How to enter your answer (opens in new window) Lower endpoint: Lower endpoint: Upper endpoint:

Step 1 ：Given a sample of 514 suspected criminals, 164 of them were captured. We are asked to construct the $95 \%$ confidence interval for the population proportion of people who are captured after appearing on the 10 Most Wanted list.

Step 2 ：First, we calculate the sample proportion ($\hat{p}$) which is the proportion of people who were captured. This is calculated as $\hat{p} = \frac{x}{n}$, where $x$ is the number of successes (people captured) and $n$ is the sample size. Substituting the given values, we get $\hat{p} = \frac{164}{514} = 0.319$.

Step 3 ：Next, we find the Z-score for the desired confidence level. For a $95 \%$ confidence level, the Z-score is approximately 1.96.

Step 4 ：We then calculate the standard error (SE) using the formula $SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$. Substituting the values we have, we get $SE = \sqrt{\frac{0.319(1-0.319)}{514}} = 0.021$.

Step 5 ：The margin of error (ME) is then calculated as $ME = Z \times SE = 1.96 \times 0.021 = 0.041$.

Step 6 ：Finally, we calculate the lower and upper endpoints of the confidence interval by subtracting and adding the margin of error from/to the sample proportion, respectively. The lower endpoint is $\hat{p} - ME = 0.319 - 0.041 = 0.278$ and the upper endpoint is $\hat{p} + ME = 0.319 + 0.041 = 0.360$.

Step 7 ：\(\boxed{\text{Final Answer: The $95 \%$ confidence interval for the population proportion of people who are captured after appearing on the 10 Most Wanted list is approximately $[0.278, 0.360]$.}}\)