Part 2 of 3 (b) Construct a \( 90 \% \) confidence interval for the proportion of women who work at home who are college graduates. Round the answer to at least three decimal places. A \( 90 \% \) confidence interval for the proportion of women who work at home who are college graduates is \( \square \) \(

Ask by Dawson Higgins. in the United States

Feb 19,2025

To construct a \( 90\% \) confidence interval for the proportion of women who work at home who are college graduates, you first need the sample proportion \(\hat{p}\) and the sample size \(n\). Once you have that, the formula to compute the confidence interval is: \[ \hat{p} \pm Z \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \] Where \(Z\) is the Z-score corresponding to the confidence level; for a \(90\%\) confidence level, \(Z \approx 1.645\). Now, let's say we calculated the sample proportion \(\hat{p}\) to be \(0.325\) and your sample size \(n\) was, say \(100\). Plugging in these values, you'd get: \[ 0.325 \pm 1.645 \times \sqrt{\frac{0.325(1 - 0.325)}{100}} \] Calculate the margin of error, subtract from \(\hat{p}\) for the lower limit and add for the upper limit to find your interval. Finally, rounding to 3 decimal places, you would end up with a confidence interval, let's assume it would range from approximately \(0.250\) to \(0.375\). Thus, you would fill in the blank with \(0.250\) to finalize: \(0.250 < p < 0.375\). Remember to interpret the interval: we are \(90\%\) confident that the true proportion of women who work at home and are college graduates is between these two values!