wo Means: Question 5, 9.2.11-T HW Score: $64.64 \%, 9.05$ of 14 points aples (2-Samp T-Test) Part 5 of 6 Points: 0.25 of 1 Save Given in the table are the BMI statistics for random samples of men and women. Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Complete parts (a) and (b) below. Use a 0.01 significance level for both parts. \begin{tabular}{|c|c|c|} \hline & Male BMI & Female BMI \\ \hline $\boldsymbol{\mu}$ & $\mu_{1}$ & $\mu_{2}$ \\ \hline $\mathbf{n}$ & 42 & 42 \\ \hline$\overline{\mathbf{x}}$ & 27.0382 & 24.5775 \\ \hline $\mathbf{s}$ & 7.169285 & 5.428095 \\ \hline \end{tabular} The test statistic, $\mathrm{t}$, is 1.77 . (Round to two decimal places as needed.) The P-value is 0.080 . (Round to three decimal places as needed.) State the conclusion for the test. A. Fail to reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that men and women have the same mean BMI. B. Reject the null hypothesis. There is not sufficient evidence to warrant rejection of the claim that men and women have the same mean BMI. C. Fail to reject the null hypothesis. There is not sufficient evidence to warrant rejection of the claim that men and women have the same mean BMI. D. Reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that men and women have the same mean BMI. b. Construct a confidence interval suitable for testing the claim that males art females have the same mean BMI. \[ \square<\mu_{1}-\mu_{2}<\square \] (Round to three decimal places as needed.)
Solution
Step 1 ：Given that the P-value is 0.080 and the significance level is 0.01. Since the P-value is greater than the significance level, we fail to reject the null hypothesis. This means there is not sufficient evidence to warrant rejection of the claim that men and women have the same mean BMI. So the answer to the first part is C.
Step 2 ：For the second part, we need to use the formula for the confidence interval for the difference in means. The formula is: \[ (\overline{x}_1 - \overline{x}_2) \pm t_{\alpha/2} \cdot \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \]
Step 3 ：Substitute the given values into the formula: n1 = 42, n2 = 42, x_bar1 = 27.0382, x_bar2 = 24.5775, s1 = 7.169285, s2 = 5.428095, alpha = 0.01, t_alpha_2 = 2.701181303578512, se = 1.3875540550624463, ci_lower = -1.2873350712392302, ci_upper = 6.208735071239229
Step 4 ：The confidence interval for the difference in mean BMI between males and females is \(\boxed{(-1.287, 6.209)}\).
