A newsgroup is interested in constructing a $90 \%$ confidence interval for the difference in the proportions of Texans and New Yorkers who favor a new Green initiative. Of the 598 randomly selected Texans surveyed, 443 were in favor of the initiative and of the 572 randomly selected New Yorkers surveyed, 454 were in favor of the initiative.
A. Create a $90 \%$ confidence interval to estimate the difference in the proportions of Texans and New Yorkers who favor a new Green initiative $\left(p_{1}-p_{2}\right.$, where population 1 is Texans \& population 2 is New Yorkers)

Enter your answer af an open-interval (i.e., parentheses) accurate to 3 decimal places. Example: $(3.24,9.21)$
\[
(0.741,0.794) \quad \times
\]

Step 1 ：First, calculate the sample proportions for Texans and New Yorkers. For Texans (population 1), the sample proportion (p1) is the number of Texans in favor of the initiative divided by the total number of Texans surveyed. \(p1 = \frac{443}{598} = 0.741\)

Step 2 ：For New Yorkers (population 2), the sample proportion (p2) is the number of New Yorkers in favor of the initiative divided by the total number of New Yorkers surveyed. \(p2 = \frac{454}{572} = 0.794\)

Step 3 ：The difference in the sample proportions (p1 - p2) is: \(p1 - p2 = 0.741 - 0.794 = -0.053\)

Step 4 ：Next, calculate the standard error of the difference in proportions. The formula for the standard error is: \(SE = \sqrt{(p1 * (1 - p1) / n1) + (p2 * (1 - p2) / n2)}\) where n1 and n2 are the sample sizes for population 1 and population 2, respectively. \(SE = \sqrt{(0.741 * (1 - 0.741) / 598) + (0.794 * (1 - 0.794) / 572)} = 0.022\)

Step 5 ：For a 90% confidence interval, the z-score is 1.645 (you can find this value in a standard z-table). The margin of error (ME) is the z-score times the standard error: \(ME = 1.645 * 0.022 = 0.036\)

Step 6 ：Finally, calculate the confidence interval by subtracting and adding the margin of error from the difference in sample proportions: Lower limit = p1 - p2 - ME = -0.053 - 0.036 = -0.089, Upper limit = p1 - p2 + ME = -0.053 + 0.036 = -0.017

Step 7 ：So, the 90% confidence interval for the difference in the proportions of Texans and New Yorkers who favor a new Green initiative is \(\boxed{(-0.089, -0.017)}\)