Normal sample-size design for an upper-tail Poisson test ($n\approx [(z_{1-\alpha }+z_{1-\beta }\sqrt{{\mu }_1})/({\mu }_1-1){]}^2$)

For a Poisson sample tested against mean ${\mu }_0=1$, the exact uniformly most powerful one-sided test thresholds the sample sum and randomizes on its boundary if necessary. A normal approximation gives critical sum $n+z_{1-\alpha }\sqrt{n}$. Equating its standardized alternative threshold to $-z_{1-\beta }$ gives the displayed design for approximate power $1-\beta$ at mean ${\mu }_1>1$. At size $0.05$ and target power $0.9$, the two quantiles are $1.645$ and $1.282$. Rounding up supplies an integer design; exact Poisson power still requires discrete evaluation.