Step 1: Find the Critical Value (c)
The critical value c is the threshold for our decision. We find it using the significance level a , which is the probability of a Type I error (rejecting $H_0$ when it's true).
- Alpha $(a)=P\left(\right.$ Reject $H_0 \mid H_0$ is true $)=P(\bar{x}>c \mid \mu=0)=0.05$
To find $c$, we first need the standard deviation of the sample mean (the standard error):
- Population standard deviation, $\sigma=\sqrt{25}=5$
- Sample size, $n=100$
- Standard Error, $\sigma_{-} \bar{x}=0 / \sqrt{ } n=5 / \sqrt{ } 100=5 / 10=0.5$
Now, we standardize the expression for a to find the corresponding Z-score:
$$
\begin{gathered}
P\left(\frac{\bar{X}-\mu_0}{\sigma_X}>\frac{c-0}{0.5}\right)=0.05 \\
P\left(Z>\frac{c}{0.5}\right)=0.05
\end{gathered}
$$
From a Z -table, the critical value z that leaves 0.05 in the right tail is $\mathbf{1 . 6 4 5}$.
$$
\begin{gathered}
\frac{c}{0.5}=1.645 \\
c=1.645 \times 0.5=0.8225
\end{gathered}
$$
Calculate the Power of the Test
Now we calculate the probability of rejecting $H_0$ (i.e., $\bar{x}>0.8225$ ) given that the alternative hypothesis is true ( $\mu=1.5$ ).
- Power $=P\left(\right.$ Reject $H_0 \mid H_2$ is true $)=P(\bar{X}>0.8225 \mid \mu=1.5)$
We standardize this expression using the mean from the alternative hypothesis ( $\mu_2=1.5$ ). The standard error remains the same (0.5).
$$
\begin{gathered}
\text { Power }=P\left(\frac{\bar{X}-\mu_A}{\sigma_X}>\frac{0.8225-1.5}{0.5}\right) \\
\text { Power }=P\left(Z>\frac{-0.6775}{0.5}\right) \\
\text { Power }=P(Z>-1.355)
\end{gathered}
$$
This is the area under the standard normal curve to the right of $z=-1.355$. We can find this using a Z-table or calculator.
- $P(Z>-1.355) \approx 0.9122$
