2006-68.md

course	course_year	question_number	tags	title	year
Statistics	IB	68	IB 2006 Statistics	2.II.19C	2006

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.]