Question 5 (1 point)
Use the given degree of confidence and sample data to construct a confidence interval for the population proportion $p$.
\[
n=195, x=162 ; 95 \% \text { confidence }
\]

Step 1 ：Given the sample size (n) is 195 and the number of successes (x) is 162, we are asked to construct a 95% confidence interval for the population proportion (p).

Step 2 ：The sample proportion (p_hat) is calculated as the ratio of the number of successes to the sample size, i.e., \(\hat{p} = \frac{x}{n} = \frac{162}{195} = 0.8307692307692308\).

Step 3 ：The Z-score corresponding to a 95% confidence interval is 1.96.

Step 4 ：The standard error (se) is calculated using the formula \(se = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.8307692307692308(1-0.8307692307692308)}{195}} = 0.026851129563818747\).

Step 5 ：The lower and upper bounds of the confidence interval are calculated as \(\hat{p} \pm Z \times se\). Therefore, the lower bound is \(0.8307692307692308 - 1.96 \times 0.026851129563818747 = 0.778141016824146\) and the upper bound is \(0.8307692307692308 + 1.96 \times 0.026851129563818747 = 0.8833974447143156\).

Step 6 ：Final Answer: The 95% confidence interval for the population proportion p is \(\boxed{[0.778, 0.883]}\).