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: \[\boxed{H_{0}: \mu_{1}=\mu_{2}, H_{1}: \mu_{1}<\mu_{2}}\]