The life expectancy of a male during the course of the past 100 years is approximately 27,778 days. Use the table to the right to conduct a test using $\alpha=0.05$ to determine whether the evidence suggests that chief justices live longer than the general population of males. Suggest a reason why the conclusion drawn may be flawed.
\begin{tabular}{|c|c|c|c|}
\hline \begin{tabular}{c}
Chief \\
Justice
\end{tabular} & \begin{tabular}{c}
Life \\
Span (Days)
\end{tabular} & \begin{tabular}{c}
Chief \\
Justice
\end{tabular} & \begin{tabular}{c}
Life \\
Span (Days)
\end{tabular} \\
\hline A & 32,084 & I & 30,668 \\
\hline B & 29,722 & J & 32,800 \\
\hline C & 30,457 & K & 27,836 \\
\hline D & 27,456 & L & 30,915 \\
\hline E & 32,947 & M & 31,082 \\
\hline F & 28,416 & N & 31,833 \\
\hline G & 26,682 & O & 32,556 \\
\hline H & 32,815 & P & 28,700 \\
\hline
\end{tabular}

State the appropriate null and alternative hypotheses.
A. $H_{0}: \mu=30,436$ versus $H_{1}: \mu< 30,436$
B. $H_{0}: \mu=27,778$ versus $H_{1}: \mu \neq 27,778$
C. $H_{0}: \mu=27,778$ versus $H_{1}: \mu> 27,778$
D. $H_{0}: \mu=27,778$ versus $H_{1}: \mu< 27,778$
E. $H_{0}: \mu=30,436$ versus $H_{1}: \mu> 30,436$
F. $H_{0}: \mu=30,436$ versus $H_{1}: \mu \neq 30,436$

Use the P-value approach at the $\alpha=0.05$ level of significance to test the hypotheses.
P-value $=.000$ (Round to three decimal places as needed.)
State the conclusion for the test. Choose the correct answer below.
A. Reject the null hypothesis. There is sufficient evidence to conclude that the mean life span of males is longer than 27,778 days. Thus, there is sufficient evidence to indicate that chief justices live longer than the general population of males.
B. Do not reject the null hypothesis. There is sufficient evidence to conclude that the mean life span of males is longer than 30,436 days. Thus, there is sufficient evidence to indicate that chief justices live longer than the general population of males.
c. Reject the null hypothesis. There is not sufficient evidence to conclude that the mean life span of males is longer than 27,778 days. Thus, there is not sufficient evidence to indicate that chief justices live longer than the general population of males.
D. Do not reject the null hypothesis. There is not sufficient evidence to conclude that the mean life span of males is longer than 27,778 days. Thus, there is not sufficient evidence to indicate that chief justices live longer than the general population of males.

Step 1 ：Identify the null and alternative hypotheses. In this case, we are testing whether the mean lifespan of chief justices is greater than the mean lifespan of the general male population, which is given as 27,778 days. Therefore, the null hypothesis is that the mean lifespan of chief justices is equal to 27,778 days, and the alternative hypothesis is that the mean lifespan of chief justices is greater than 27,778 days. This corresponds to option C: \(H_{0}: \mu=27,778\) versus \(H_{1}: \mu>27,778\).

Step 2 ：Conduct a hypothesis test using the P-value approach at the \(\alpha=0.05\) level of significance. To do this, calculate the sample mean and standard deviation of the lifespans of the chief justices, and then use these values to calculate the test statistic and P-value.

Step 3 ：Compare the P-value to the significance level to determine whether to reject the null hypothesis. If the P-value is less than \(\alpha\), reject the null hypothesis and conclude that there is sufficient evidence to suggest that chief justices live longer than the general population of males. If the P-value is greater than \(\alpha\), do not reject the null hypothesis and conclude that there is not sufficient evidence to suggest that chief justices live longer than the general population of males.

Step 4 ：The lifespans of the chief justices are [32084, 29722, 30457, 27456, 32947, 28416, 26682, 32815, 30668, 32800, 27836, 30915, 31082, 31833, 32556, 28700]. The sample mean is 30435.5625, the standard deviation is 2077.7070364145825, the sample size is 16, the test statistic is 5.116337295725872, the degrees of freedom is 15, and the P-value is 6.325553965391695e-05.

Step 5 ：The P-value is less than the significance level of 0.05, so reject the null hypothesis. This suggests that there is sufficient evidence to conclude that the mean life span of chief justices is longer than the mean life span of the general population of males. Therefore, the answer to the second part of the question is A: Reject the null hypothesis. There is sufficient evidence to conclude that the mean life span of males is longer than 27,778 days. Thus, there is sufficient evidence to indicate that chief justices live longer than the general population of males.

Step 6 ：The appropriate null and alternative hypotheses are \(H_{0}: \mu=27,778\) versus \(H_{1}: \mu>27,778\). The conclusion for the test is to reject the null hypothesis. There is sufficient evidence to conclude that the mean life span of males is longer than 27,778 days. Thus, there is sufficient evidence to indicate that chief justices live longer than the general population of males. \(\boxed{C, A}\).