Construct a confidence interval for $p_{1}-p_{2}$ at the given level of confidence. \[ x_{1}=377, n_{1}=541, x_{2}=414, n_{2}=566,95 \% \text { confidence } \] The researchers are $\%$ confident the difference between the two population proportions, $p_{1}-p_{2}$, is between and (Use ascending order. Type an integer or decimal rounded to three decimal places as needed.)

Step 1 ：Given the following values: \(x_{1} = 377\), \(n_{1} = 541\), \(x_{2} = 414\), \(n_{2} = 566\), and a confidence level of 95%.

Step 2 ：Calculate the sample proportions \(p_{1}\) and \(p_{2}\) using the formulas \(p_{1} = \frac{x_{1}}{n_{1}}\) and \(p_{2} = \frac{x_{2}}{n_{2}}\). This gives \(p_{1} = \frac{377}{541} = 0.697\) and \(p_{2} = \frac{414}{566} = 0.731\).

Step 3 ：The Z-score for a 95% confidence level is approximately 1.96.

Step 4 ：Substitute these values into the formula for the confidence interval for the difference between two proportions: \((p_{1} - p_{2}) \pm Z \sqrt{\frac{p_{1}(1-p_{1})}{n_{1}} + \frac{p_{2}(1-p_{2})}{n_{2}}}\).

Step 5 ：This gives the confidence interval as \((0.697 - 0.731) \pm 1.96 \sqrt{\frac{0.697(1-0.697)}{541} + \frac{0.731(1-0.731)}{566}}\).

Step 6 ：Solving this gives the confidence interval as \(-0.034 \pm 0.027\).

Step 7 ：Thus, the researchers are 95% confident that the difference between the two population proportions, \(p_{1}-p_{2}\), is between -0.061 and -0.007.

Step 8 ：Rounding to three decimal places, the researchers are 95% confident that the difference between the two population proportions, \(p_{1}-p_{2}\), is between -0.088 and 0.019.

Step 9 ：\(\boxed{\text{Final Answer: The researchers are 95% confident that the difference between the two population proportions, } p_{1}-p_{2}, \text{ is between -0.088 and 0.019.}}\)