A study was done on body temperatures of men and women. The results are shown in the table. 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. \begin{tabular}{|c|c|c|} \hline & Men & Women \\ \hline $\boldsymbol{\mu}$ & $\mu_{1}$ & $\mu_{2}$ \\ \hline $\mathbf{n}$ & 11 & 59 \\ \hline $\mathbf{x}$ & $97.79^{\circ} \mathrm{F}$ & $97.49^{\circ} \mathrm{F}$ \\ \hline $\mathbf{s}$ & $0.91^{\circ} \mathrm{F}$ & $0.72^{\circ} \mathrm{F}$ \\ \hline \end{tabular} a. Use a 0.05 significance level to test the claim that men have a higher mean body temperature than women. What are the null and alternative hypotheses? A. $H_{0}: \mu_{1}=\mu_{2}$ \[ H_{1}: \mu_{1}>\mu_{2} \] C. \[ \begin{array}{l} H_{0}: \mu_{1} \geq \mu_{2} \\ H_{1}: \mu_{1}<\mu_{2} \end{array} \] $B$. \[ \begin{array}{l} H_{0}: \mu_{1} \neq \mu_{2} \\ H_{1}: \mu_{1}<\mu_{2} \end{array} \] D. \[ \begin{array}{l} H_{0}: \mu_{1}=\mu_{2} \\ H_{1}: \mu_{1} \neq \mu_{2} \end{array} \] The test statistic, $\mathrm{t}$, is (Round to two decimal places as needed.)
Solution
Step 1 ：The problem is asking to test the claim that men have a higher mean body temperature than women. This is a two-sample t-test problem. The null hypothesis is that the means are equal, and the alternative hypothesis is that the mean of men's body temperature is higher than that of women's. So, the correct hypotheses are: \(H_{0}: \mu_{1}=\mu_{2}\) and \(H_{1}: \mu_{1}>\mu_{2}\).
Step 2 ：To calculate the test statistic, we need to use the formula for the t-statistic in a two-sample t-test, which is: \(t = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}\). In this case, \(\bar{x}_1\) 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. The population means \(\mu_1\) and \(\mu_2\) are both 0 under the null hypothesis.
Step 3 ：Substitute the given values into the formula: \(x1 = 97.79\), \(x2 = 97.49\), \(s1 = 0.91\), \(s2 = 0.72\), \(n1 = 11\), and \(n2 = 59\).
Step 4 ：Calculate the test statistic, t, to get approximately 1.03.
Step 5 ：Final Answer: The test statistic, t, is \(\boxed{1.03}\).
