Listed below are the numbers of years that archbishops and monarchs in a certain country lived after their election or coronation. Assume that the two samples are independent simple random samples selected from normally distributed populations. Do not assume that the population standard deviations are equal. Use a 0.05 significance level to test the claim that the mean longevity for archbishops is less than the mean for monarchs after coronation. All measurements are in years.
Click the icon to view the table of longevities of archbishops and monarchs.
What are the null and alternative hypotheses? Assume that population 1 consists of the longevity of archbishops and population 2 consists of the longevity of monarchs.
A. $\mathrm{H}_{0}: \mu_{1} \neq \mu_{2}$
$H_{1}: \mu_{1}> \mu_{2}$
C.
\[
\begin{array}{l}
H_{0}: \mu_{1}=\mu_{2} \\
H_{1}: \mu_{1} \neq \mu_{2}
\end{array}
\]
B.
\[
\begin{array}{l}
H_{0}: \mu_{1}=\mu_{2} \\
H_{1}: \mu_{1}< \mu_{2}
\end{array}
\]
D.
\[
\begin{array}{l}
H_{0}: \mu_{1} \leq \mu_{2} \\
H_{1}: \mu_{1}> \mu_{2}
\end{array}
\]
Answer
Final Answer: \[\boxed{H_{0}: \mu_{1}=\mu_{2}, H_{1}: \mu_{1}<\mu_{2}}\]
Step 1 ：The null hypothesis is always a statement of no effect or no difference. It is the hypothesis that we are trying to find evidence against in our hypothesis test. The alternative hypothesis is what we would believe if we found evidence against the null hypothesis.
Step 2 ：In this case, we are testing the claim that the mean longevity for archbishops is less than the mean for monarchs after coronation. This means that our null hypothesis should be that the means are equal, and our alternative hypothesis should be that the mean longevity for archbishops is less than the mean for monarchs.
Step 3 ：Therefore, the correct null and alternative hypotheses are: \[H_{0}: \mu_{1}=\mu_{2}\] and \[H_{1}: \mu_{1}<\mu_{2}\]
Step 4 ：Final Answer: \[\boxed{H_{0}: \mu_{1}=\mu_{2}, H_{1}: \mu_{1}<\mu_{2}}\]
