Let $X_{1}, X_{2}, ldots, X_{n}$ be iid $Nleft(mu, sigma^{2} ight)$. a. Show that the power function of

Let $X_{1}, X_{2}, \ldots, X_{n}$ be iid $N\left(\mu, \sigma^{2}\right)$.

a. Show that the power function of the test $H_{0}: \mu=0$ versus $H_{1}: \mu>0$ at $\mu=1$ is

\[1-\Phi\left(z_{\frac{\alpha}{2}}-\frac{\sqrt{n}}{\sigma}\right)\]

where $z_{\frac{\alpha}{2}}$ is the $100\left(1-\frac{\alpha}{2}\right)^{t h}$ percentile of the $N(0,1)$.

b. Use $\mathrm{R}$ to calculate the power when $n=10, \alpha=0.01$, and $\sigma=1$.