Use the given statistics to complete parts (a) and (b). Assume that the populations are normally distributed. (a) Test whether $\mu_{1}>\mu_{2}$ at the $\alpha=0.10$ level of significance for the given sample data. (b) Construct a $90 \%$ confidence interval about $\mu_{1}-\mu_{2}$. \begin{tabular}{ccc} & Population 1 & Population 2 \\ \hline $\mathrm{n}$ & 21 & 22 \\ \hline$\overline{\mathrm{x}}$ & 48.6 & 40.5 \\ \hline $\mathrm{5}$ & 4.8 & 12.7 \end{tabular} (a) Identify the null and alternative hypotheses for this test. A. \[ \begin{array}{l} H_{0}: \mu_{1}=\mu_{2} \\ H_{1}: \mu_{1}<\mu_{2} \end{array} \] D. \[ \begin{array}{l} H_{0}: \mu_{1}=\mu_{2} \\ H_{1} \cdot \mu_{1} \neq \mu_{2} \end{array} \] B. \[ \begin{array}{l} H_{0}: \mu_{1}<\mu_{2} \\ H_{1}: \mu_{1}=\mu_{2} \end{array} \] E. $H_{0}: \mu_{1} \neq \mu_{2}$ $H_{1} \cdot \mu_{1}=\mu_{2}$ C. $\mathrm{H}_{0}: \mu_{1}>\mu_{2}$ \[ H_{1}: \mu_{1}=\mu_{2} \] F. \[ \begin{array}{l} H_{0}: \mu_{1}=\mu_{2} \\ H_{1}: \mu_{1}>\mu_{2} \end{array} \] Find the test statistic for this hypothesis test. (Round to two decimal places as needed)

Step 1 ：Identify the null and alternative hypotheses for this test. The null hypothesis is that the means of the two populations are equal, and the alternative hypothesis is that the mean of population 1 is greater than the mean of population 2. The hypotheses are as follows: \[H_{0}: \mu_{1}=\mu_{2}\] \[H_{1}: \mu_{1}>\mu_{2}\]

Step 2 ：Use the formula for the test statistic in a two-sample t-test, which is: \[t = \frac{(\overline{x}_1 - \overline{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}\] where \(\overline{x}_1\) and \(\overline{x}_2\) are the sample means, \(\mu_1\) and \(\mu_2\) are the population means (which are both 0 under the null hypothesis), \(s_1^2\) and \(s_2^2\) are the sample variances, and \(n_1\) and \(n_2\) are the sample sizes.

Step 3 ：Substitute the given sample data into the formula: \[n1 = 21\] \[n2 = 22\] \[\overline{x}_1 = 48.6\] \[\overline{x}_2 = 40.5\] \[s_1 = 4.8\] \[s_2 = 12.7\]

Step 4 ：Calculate the test statistic: \[t = 2.790035350960466\]

Step 5 ：The test statistic calculated from the given sample data is approximately 2.79. This value will be used to determine whether we reject or fail to reject the null hypothesis.

Step 6 ：Final Answer: The test statistic for this hypothesis test is approximately \(\boxed{2.79}\).