Questions: 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. Male BMI Female BMI --------- **μ** μ₁ μ₂ **n** 48 48 **x̄** 27.4282 26.0503 **s** 8.538108 5.185315 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. B. 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. C. 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 and females have the same mean BMI. <μ₁-μ₂<

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. Male BMI Female BMI --------- **μ** μ₁ μ₂ **n** 48 48 **x̄** 27.4282 26.0503 **s** 8.538108 5.185315 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. B. 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. C. 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 and females have the same mean BMI. <μ₁-μ₂<
Transcript text: 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 $\boldsymbol{n}$ & 48 & 48 \\ \hline$\overline{\boldsymbol{x}}$ & 27.4282 & 26.0503 \\ \hline $\mathbf{s}$ & 8.538108 & 5.185315 \\ \hline \end{tabular} 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. B. 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. C. 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 and females have the same mean BMI. $\square$ $\square$ $<\mu_{1}-\mu_{2}<$
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Solution

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Solution Steps

Step 1: Calculate the Standard Error

The Standard Error \( (SE) \) is calculated using the formula:

\[ SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} = \sqrt{\frac{8.5381^2}{48} + \frac{5.1853^2}{48}} \approx 1.0201 \]

Step 2: Calculate the Test Statistic

The test statistic \( (t) \) is calculated as follows:

\[ t = \frac{\bar{x}_1 - \bar{x}_2}{SE} = \frac{27.4282 - 26.0503}{1.0201} \approx 1.3470 \]

Step 3: Calculate the Degrees of Freedom

The degrees of freedom \( (df) \) for Welch's t-test is calculated using the formula:

\[ df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1}} \approx 47.0 \]

Step 4: Calculate the p-value

The p-value is calculated as:

\[ P = 2(1 - T(|t|)) \approx 0.0 \]

Step 5: Determine the Critical Value

The critical value for a two-tailed test at a significance level of \( \alpha = 0.05 \) with \( df \approx 47 \) is:

\[ \text{Critical value} \approx 2.0117 \]

Step 6: Conclusion of the Hypothesis Test

Since the p-value \( (0.0) \) is less than the significance level \( (0.05) \), we reject the null hypothesis. Thus, there is sufficient evidence to warrant rejection of the claim that men and women have the same mean BMI.

Step 7: Construct the Confidence Interval

The confidence interval for the difference in means is calculated as follows:

\[ (\bar{x}_1 - \bar{x}_2) \pm z \cdot SE = (27.4282 - 26.0503) \pm 1.96 \cdot 1.0201 \]

Calculating the interval gives:

\[ (-1.448, 4.204) \]

Final Answer

The conclusion for the test is: \\(\boxed{D}\\).
The confidence interval for the difference in means is: \\(\boxed{(-1.448, 4.204)}\\).

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