 Solution

The Neyman-Pearson lemma concerns testing two simple hypotheses with densities $f_0$ and $f_1$ relative to a common measure. A test that rejects when $f_1(x)>k{f}_0(x)$, accepts when $f_1(x)<k{f}_0(x)$, and randomizes on equality if necessary to achieve size $\alpha$, is most powerful among all tests with size at most $\alpha$:Here $\phi$ is the described likelihood-ratio test and test functions take values in $[0,1]$. In the continuous normal setting below the boundary has probability zero, so randomization is unnecessary.