In random, independent samples of 275 adults and 350 teenagers who watched a certain television show, 148 adults and 150 teens indicated that they liked the show. Let $p_{1}$ be the proportion of all adults watching the show who liked it, and let $p_{2}$ be the proportion of all teens watching the show who liked it. Find a $90 \%$ confidence interval for $p_{1}-p_{2}$. Then find the lower limit and upper limit of the $90 \%$ confidence interval. Carry your intermediate computations to at least three decimal places. Round your responses to at least three decimal places. (If necessary, consult a list of formulas.) Lower limit: Upper limit:

Step 1 ：We are given that in random, independent samples of 275 adults and 350 teenagers who watched a certain television show, 148 adults and 150 teens indicated that they liked the show. Let \(p_{1}\) be the proportion of all adults watching the show who liked it, and let \(p_{2}\) be the proportion of all teens watching the show who liked it. We are asked to find a 90% confidence interval for \(p_{1}-p_{2}\).

Step 2 ：The formula for a confidence interval for the difference in proportions is given by: \[(p_1 - p_2) \pm z \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}\] where \(p_1\) and \(p_2\) are the sample proportions, \(n_1\) and \(n_2\) are the sample sizes, and \(z\) is the z-score corresponding to the desired level of confidence.

Step 3 ：In this case, \(p_1 = \frac{148}{275}\), \(p_2 = \frac{150}{350}\), \(n_1 = 275\), \(n_2 = 350\), and \(z\) is the z-score for a 90% confidence interval, which is approximately 1.645.

Step 4 ：We can plug these values into the formula to find the lower and upper limits of the confidence interval.

Step 5 ：Calculating the standard error, we get \(se = 0.040043702209788784\).

Step 6 ：Substituting the values into the formula, we get the lower limit as \(lower\_limit = 0.0437384994752871\) and the upper limit as \(upper\_limit = 0.17548227974549221\).

Step 7 ：Rounding to three decimal places, we get the lower limit as approximately 0.044 and the upper limit as approximately 0.176.

Step 8 ：Final Answer: The lower limit of the 90% confidence interval for \(p_{1}-p_{2}\) is approximately \(\boxed{0.044}\) and the upper limit is approximately \(\boxed{0.176}\).