= Normal sample-size design for an upper-tail Poisson test
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{title2=$n\approx[(z_{1-\alpha}+z_{1-\beta}\sqrt{\mu_1})/(\mu_1-1)]^2$}
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.
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