15. A survey of 852 voters in one state reveals that 478 favor approval of an issue before the legislature. Construct the $95 \%$ confidence interval for the true proportion of all voters in the state who favor approval. (10 poinți)
Answer
\( \boxed{(0.5277, 0.5944)} \)
Step 1 ：Calculate the sample proportion, \( \hat{p} = \frac{478}{852} \)
Step 2 ：Find the z-score for a 95\% confidence level, which is approximately 1.96
Step 3 ：Calculate the standard error, \( \text{SE} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \)
Step 4 ：Construct the confidence interval, \( \text{CI} = \hat{p} \pm z \times \text{SE} \)
Step 5 ：Calculate the lower bound of the confidence interval, \( \text{LB} = \hat{p} - z \times \text{SE} \)
Step 6 ：Calculate the upper bound of the confidence interval, \( \text{UB} = \hat{p} + z \times \text{SE} \)
Step 7 ：\( \boxed{(0.5277, 0.5944)} \)
