If $n=580$ and $\hat{p}$ (p-hat $)=0.7$, construct a $90 \%$ confidence interval.

Give your answers to three decimals

Step 1 ：To construct a \(90\%\) confidence interval for a proportion, we use the formula \(\hat{p} \pm z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\)

Step 2 ：For a \(90\%\) confidence interval, the z-score is approximately \(1.645\)

Step 3 ：Given \(n = 580\) and \(\hat{p} = 0.7\), we can calculate the margin of error as \(1.645 \times \sqrt{\frac{0.7(1-0.7)}{580}}\)

Step 4 ：The margin of error is approximately \(0.031\)

Step 5 ：The lower bound of the confidence interval is \(\hat{p} - \text{margin of error} = 0.7 - 0.031 = 0.669\)

Step 6 ：The upper bound of the confidence interval is \(\hat{p} + \text{margin of error} = 0.7 + 0.031 = 0.731\)

Step 7 ：\(\boxed{(0.669, 0.731)}\)