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 $H_{1}: \mu \neq 30,436$
Solution
Step 1 ：The question is asking for the appropriate null and alternative hypotheses for a test to determine whether chief justices live longer than the general population of males. The null hypothesis is usually a statement of no effect or no difference, while the alternative hypothesis is what we are trying to prove.
Step 2 ：In this case, the null hypothesis should be that the mean lifespan of chief justices is equal to the mean lifespan of the general population of males, which is given as 27,778 days.
Step 3 ：The alternative hypothesis should be that the mean lifespan of chief justices is greater than the mean lifespan of the general population of males.
Step 4 ：Therefore, the appropriate null and alternative hypotheses are: \(H_{0}: \mu=27,778\) versus \(H_{1}: \mu>27,778\).
Step 5 ：So, the answer should be option C.
Step 6 ：Final Answer: \(\boxed{C}\)
