Let $X_1,\dots, X_n$ be a random sample from $N(\theta,\sigma^2)$, and we would like to test $H_0:\theta \le \theta_0 \text{ vs } H_1:\theta>\theta_0$. Assume $\sigma^2$ is known and derive a likelihood ratio test (LRT).
Obviously the question should be solved via UMP test, however if you are requested to perform the above hypothesis testing via LRT, how would you do it?
