 Solution

A possibly randomized statistical test has rejection probability $\phi (x)\in [0,1]$ after observing $x$. Its statistical power at parameter $\theta$ is ${\mathbb{E}}_{\theta }\phi$, and its test size is the supremum of that statistical power over the null parameter set.

For simple hypotheses with densities $f_0,f_1$, the Neyman-Pearson lemma states that a level-$\alpha$ test is most powerful if it rejects where $f_1>k{f}_0$, accepts where $f_1<k{f}_0$, and randomizes on equality to obtain null rejection probability $\alpha$. Here $k\ge 0$ is chosen to reach that test size. To prove optimality, let ${\phi }_{\ast }$ be this test and $\phi$ any test of level at most $\alpha$. Pointwise,Integrating and using $\int ({\phi }_{\ast }-\phi )f_0\ge 0$ gives ${\mathbb{E}}_1{\phi }_{\ast }-{\mathbb{E}}_1\phi \ge 0$. This proves maximal statistical power. For $k>0$, equality also forces any equally powerful test to use all its allowed test size and agree with the indicated decisions away from the likelihood-ratio tie set, up to null sets.

For the Poisson sample, $S={\sum }_i{X}_i\sim \mathrm{Poisson}(n\mu )$ and, for every fixed ${\mu }_1>{\mu }_0>0$, the likelihood ratio isIt increases strictly with $S$. Choose an integer $k$ so that ${\mathbb{P}}_{{\mu }_0}(S>k)\le \alpha \le {\mathbb{P}}_{{\mu }_0}(S\ge k)$, and reject for $S>k$, with boundary rejection probabilityThis has exact test size $\alpha$. Its rejection ordering and calibration do not depend on which ${\mu }_1>{\mu }_0$ is selected, so the lemma proves it uniformly most powerful for the whole one-sided alternative. Randomization is generally needed for an exact arbitrary test size in a discrete model.

For the approximate design, the central limit theorem under ${\mu }_0=1$ gives critical sum $n+1.645\sqrt{n}$. Under ${\mu }_1$, the sum is approximately normal with mean $n{\mu }_1$ and variance $n{\mu }_1$. Target statistical power $0.9$ therefore givesor $\sqrt{n}({\mu }_1-1)\approx 1.645+1.282\sqrt{{\mu }_1}$. Thus the requested design expression isRound upward to obtain an integer design. This is a normal approximation to the requested statistical power, rather than an exact discrete sample-size guarantee; the exact Poisson test above provides its exact calibration.