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 0.05 significance level for both parts.
\begin{tabular}{|c|c|c|}
\hline & Male BMI & Female BMI \\
\hline $\boldsymbol{\mu}$ & $\mu_{1}$ & $\mu_{2}$ \\
\hline $\mathbf{n}$ & 47 & 47 \\
\hline$\overline{\mathbf{x}}$ & 28.3834 & 26.6054 \\
\hline $\mathbf{s}$ & 8.746558 & 4.280232 \\
\hline
\end{tabular}
a. Test the claim that males and females have the same mean body mass index (BMI).

What are the null and alternative hypotheses?
A.
\[
\begin{array}{l}
H_{0}: \mu_{1} \geq \mu_{2} \\
H_{1}: \mu_{1}< \mu_{2}
\end{array}
\]
c.
\[
\begin{array}{l}
H_{0}: \mu_{1}=\mu_{2} \\
H_{1}: \mu_{1}> \mu_{2}
\end{array}
\]
B.
\[
\begin{array}{l}
H_{0}: \mu_{1} \neq \mu_{2} \\
H_{1}: \mu_{1}< \mu_{2}
\end{array}
\]
D.
\[
\begin{array}{l}
H_{0}: \mu_{1}=\mu_{2} \\
H_{1}: \mu_{1} \neq \mu_{2}
\end{array}
\]

The test statistic, $t$, is $\square$. (Round to two decimal places as needed.)
The P-value is $\square$. (Round to three decimal places as needed.)
State the conclusion for the test.
A. 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.

Step 1 ：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. Use a 0.05 significance level for both parts.

Step 2 ：First, we need to set up our null and alternative hypotheses. The null hypothesis is that the mean BMI for men and women is the same, while the alternative hypothesis is that the mean BMI for men and women is not the same. In mathematical terms, this can be written as: \n\[\begin{array}{l}H_{0}: \mu_{1}=\mu_{2} \H_{1}: \mu_{1} \neq \mu_{2}\end{array}\]

Step 3 ：Next, we need to calculate the test statistic. The formula for the test statistic in a two-sample t-test is: \n\[t = \frac{{\overline{x}_1 - \overline{x}_2}}{{\sqrt{{\frac{{s_1^2}}{{n_1}} + \frac{{s_2^2}}{{n_2}}}}}}\] where: \n- $\overline{x}_1$ and $\overline{x}_2$ are the sample means, \n- $s_1^2$ and $s_2^2$ are the sample variances, \n- $n_1$ and $n_2$ are the sample sizes.

Step 4 ：Using the given values, we find that the test statistic, $t$, is approximately 1.25.

Step 5 ：After calculating the test statistic, we can use a t-distribution table or a statistical software to find the p-value. The p-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true.

Step 6 ：Using the calculated test statistic and degrees of freedom, we find that the p-value is approximately 0.214.

Step 7 ：Since the p-value is greater than the significance level of 0.05, we 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.

Step 8 ：Final Answer: \nThe null and alternative hypotheses are: \n\[\begin{array}{l}H_{0}: \mu_{1}=\mu_{2} \H_{1}: \mu_{1} \neq \mu_{2}\end{array}\] \nThe test statistic, $t$, is $\boxed{1.25}$. \nThe P-value is $\boxed{0.214}$. \nWe 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.