Suppose that $X_{1}, \ldots, X_{n}$ are independent normal random variables of unknown mean $\theta$ and variance 1 . It is desired to test the hypothesis $H_{0}: \theta \leq 0$ against the alternative $H_{1}: \theta>0$. Show that there is a uniformly most powerful test of size $\alpha=1 / 20$ and identify a critical region for such a test in the case $n=9$. If you appeal to any theoretical result from the course you should also prove it.

[The 95th percentile of the standard normal distribution is 1.65.]