Assume that both populations are normally distributed. a) Test whether $\mu_{1} \neq \mu_{2}$ at the $\alpha=0.05$ level of significance for the given sample data. b) Construct a $95 \%$ confidence interval about $\mu_{1}-\mu_{2}$. \begin{tabular}{ccc} & Sample 1 & Sample 2 \\ \hline $\mathbf{n}$ & 13 & 13 \\ \hline $\bar{x}$ & 15.3 & 17.2 \\ \hline $\mathbf{s}$ & 4.3 & 3.7 \end{tabular} Click the icon to kiem the Student t-distribution table. Determine the test statistic. $t=-1.21$ (Round to two decimal places as needed.) Determine the critical value(s). Select the correct choice below and fill in the answer box(es) within your choice. (Round to three decimal places as needed.) A. The critical value is B. The lower critical value is -2.066 . The upper critical value is 2.066 . Should the hypothesis be rejected? Do not reject the null hypothesis because the test statistic is not in the critical region. b) Construct a $95 \%$ confidence interval about $\mu_{1}-\mu_{2}$ : The confidence interval is the range from $\square$ to (Round to two decimal places as needed. Use ascending order.)

Step 1 ：Given the sample data for two populations, we are asked to perform a two-sample t-test to determine if the means of the two populations are equal. The null hypothesis is that the means are equal, and the alternative hypothesis is that the means are not equal.

Step 2 ：The test statistic for the t-test is calculated to be \(-1.21\).

Step 3 ：We then determine the critical value at the \(\alpha = 0.05\) level of significance. The critical values are found to be \(-2.066\) and \(2.066\).

Step 4 ：Comparing the test statistic with the critical values, we find that the test statistic is not in the critical region. Therefore, we do not reject the null hypothesis. This means that there is not enough evidence to conclude that the means of the two populations are different.

Step 5 ：We are also asked to construct a 95% confidence interval for the difference between the two means. Using the formula for the confidence interval for the difference between two means, we find the confidence interval to be \((-5.15, 1.35)\). This means that we are 95% confident that the true difference between the means of the two populations is within this interval.

Step 6 ：Final Answer: The test statistic is \(\boxed{-1.21}\). The critical values are \(\boxed{-2.066}\) and \(\boxed{2.066}\). We do not reject the null hypothesis. The 95% confidence interval for the difference between the two means is \(\boxed{(-5.15, 1.35)}\).