\(\newcommand{\Cov}{\mathrm{Cov}}\) \(\newcommand{\Corr}{\mathrm{Corr}}\) \(\newcommand{\Sample}{X_{1},\dots,X_{n}}\)

For a Poission distribution with large \(n\), use \(Z=\frac{\bar{X}-\lambda}{\sqrt{\lambda/n}}\) (i.e. \(\lambda\) instead of \(S\)).

It is approximately normal. Use it to test \(H_{0}:\lambda=\lambda_{0}\)