d) We are performing a hypothesis test where $\mathrm{H}_{1}: \mu> 50$. If the test is done at the $5 \%$ significance level and the truth is that $\mu=50$, then what percentage of the time should we expect to Reject the null hypothesis in the long run?
\[
0 \%
\]
\[
1 \% \quad 5 \%
\]
\[
10 \%
\]
\[
100 \%
\]
Answer
\(\boxed{5 \%}\)
Step 1 ：The null hypothesis in this case is that \(\mu \leq 50\).
Step 2 ：The alternative hypothesis is that \(\mu > 50\).
Step 3 ：We are performing the test at a 5% significance level, which means that we are willing to accept a 5% chance of incorrectly rejecting the null hypothesis when it is true.
Step 4 ：However, in this case, the truth is that \(\mu = 50\), which means the null hypothesis is true.
Step 5 ：Therefore, we should expect to incorrectly reject the null hypothesis 5% of the time in the long run.
Step 6 ：\(\boxed{5 \%}\)
