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 $\alpha=0.01$ level of significance for the given sample data. (b) Construct a $95 \%$ confidence interval about $\mu_{1}-\mu_{2}$. \begin{tabular}{ccc} & Population 1 & Population 2 \\ \hline $\mathbf{n}$ & 24 & 18 \\ \hline$\overline{\mathbf{x}}$ & 47.3 & 42.9 \\ \hline $\mathbf{s}$ & 3.8 & 11.6 \end{tabular} (a) Identify the null and alternative hypotheses for this test. $\checkmark \mathrm{A}$ \[ \begin{array}{l} H_{0}: \mu_{1}=\mu_{2} \\ H_{1}: \mu_{1}>\mu_{2} \end{array} \] D. \[ \begin{array}{l} H_{0}: \mu_{1}=\mu_{2} \\ H_{1}: \mu_{1}<\mu_{2} \end{array} \] B. \[ \begin{array}{l} H_{0}: \mu_{1} \neq \mu_{2} \\ H_{1}: \mu_{1}=\mu_{2} \end{array} \] E. \[ \begin{array}{l} H_{0}: \mu_{1}<\mu_{2} \\ H_{1}: \mu_{1}=\mu_{2} \end{array} \] C. \[ \begin{array}{l} H_{0}: \mu_{1}=\mu_{2} \\ H_{1}: \mu_{1} \neq \mu_{2} \end{array} \] 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. 1.55 (Round to two decimal places as needed.) Determine the P-value for this hypothesis test. (Round to three decimal places as needed.)
Solution
Step 1 ：The null and alternative hypotheses for this test are: \[\begin{array}{l} H_{0}: \mu_{1}=\mu_{2} \\ H_{1}: \mu_{1}>\mu_{2} \end{array}\]
Step 2 ：The test statistic for this hypothesis test is calculated using the formula for the test statistic in a two-sample t-test, which is \((\bar{x}_{1} - \bar{x}_{2}) / \sqrt{(s_{1}^{2}/n_{1}) + (s_{2}^{2}/n_{2})}\), where \(\bar{x}_{1}\) and \(\bar{x}_{2}\) are the sample means, \(s_{1}\) and \(s_{2}\) are the sample standard deviations, and \(n_{1}\) and \(n_{2}\) are the sample sizes. Substituting the given values, we get a test statistic of approximately \(1.55\).
Step 3 ：The P-value for this hypothesis test is calculated using the test statistic and the degrees of freedom, which is \((n_{1} + n_{2} - 2)\) in a two-sample t-test. The calculated P-value is approximately \(0.065\).
Step 4 ：The final answer is: The null and alternative hypotheses for this test are: \[\begin{array}{l} H_{0}: \mu_{1}=\mu_{2} \\ H_{1}: \mu_{1}>\mu_{2} \end{array}\] The test statistic for this hypothesis test is \(\boxed{1.55}\). The P-value for this hypothesis test is \(\boxed{0.065}\).
