Question 8 of 9, Step 3 of 3 $13 / 20$ Correct Consider two independent random samples with the following results: \[ \begin{array}{ll} n_{1}=676 & n_{2}=742 \\ x_{1}=63 & x_{2}=316 \end{array} \] Use this data to find the $90 \%$ confidence interval for the true difference between the population proportions. Step 3 of 3 : Construct the $90 \%$ confidence interval. Round your answers to three decimal places. Answer How to enter your answer (opens in new window) Lower endpoint: Upper endpoint:

Step 1 ：Given two independent random samples with the following results: \(n_{1}=676, n_{2}=742, x_{1}=63, x_{2}=316\).

Step 2 ：Calculate the sample proportions: \(p1 = \frac{x1}{n1} = \frac{63}{676} = 0.093\), \(p2 = \frac{x2}{n2} = \frac{316}{742} = 0.426\).

Step 3 ：Calculate the difference of sample proportions: \(d = p1 - p2 = 0.093 - 0.426 = -0.333\).

Step 4 ：Calculate the standard error of the difference: \(SE = \sqrt{\frac{p1 * (1 - p1)}{n1} + \frac{p2 * (1 - p2)}{n2}} = \sqrt{\frac{0.093 * (1 - 0.093)}{676} + \frac{0.426 * (1 - 0.426)}{742}} = 0.021\).

Step 5 ：Use the Z-score for a 90% confidence interval, which is 1.645.

Step 6 ：Calculate the margin of error: \(ME = Z * SE = 1.645 * 0.021 = 0.035\).

Step 7 ：Calculate the lower and upper endpoints of the confidence interval: lower = d - ME = -0.333 - 0.035 = -0.368, upper = d + ME = -0.333 + 0.035 = -0.298.

Step 8 ：The $90 \%$ confidence interval for the true difference between the population proportions is \(\boxed{[-0.368, -0.298]}\).