Compare Normal vs Poisson; test dispersion and approximate tails

You have $n=200$ independent minute-level event counts, with sample mean $\bar{x}=14.5$ and sample variance $s^2=16.2$.

1. Assuming a Poisson($\lambda$) model, derive the MLE of $\lambda$ and its asymptotic variance, then construct a 95% Wald confidence interval for $\lambda$.

2. Test $H_0$: the data follow a Poisson($\lambda$) distribution (equidispersion) against $H_1$: overdispersion. Choose an appropriate dispersion test (such as a variance-to-mean test or a Pearson $\chi^2/df$ test), give its test statistic and reference distribution, and state the decision at $\alpha=0.05$ using the supplied $\bar{x}$ and $s^2$. Mention any approximations you rely on.

3. For a single Poisson count with $\lambda=120$, approximate $P(100 \le X \le 140)$ using a Normal approximation with continuity correction. Write the precise Normal integral you would evaluate, and explain why the correction is necessary.

4. State precisely the conditions under which a Normal approximation to a Poisson is reasonable, when it breaks down, and a principled alternative model when $s^2 \gg \bar{x}$ (include one diagnostic you would check).

Overview: This item assesses a data scientist's ability to perform statistical inference for count data, including MLE and asymptotic variance estimation, Wald confidence intervals, equidispersion-versus-overdispersion tests, Normal approximation with continuity correction for Poisson tail probabilities, and model-selection diagnostics.