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

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Answer

$(0.669, 0.731)$

Steps

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 $\hat{p} - margin of error = 0.7 - 0.031 = 0.669$

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

Step 7 ： $(0.669, 0.731)$