Compare Normal vs Poisson; test dispersion and approximate tails

Q: Compare Normal vs Poisson; test dispersion and approximate tails

This is a Statistics & Math interview question from Apple for Data Scientist roles. View the full question and solution on PracHub.

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Question

You collect n=200 independent minute-level event counts with sample mean x̄=14.5 and sample variance s²=16.2.

Under a Poisson(λ) model, derive the MLE for λ and its asymptotic variance; construct a 95% Wald CI for λ.
Test H0: data ∼ Poisson(λ) (equidispersion) vs H1: overdispersed. Specify an appropriate dispersion test (e.g., variance-to-mean test or Pearson χ²/df test), its test statistic, reference distribution, and the decision at α=0.05 using the given x̄ and s². State any approximations you use.
For a single Poisson count with λ=120, approximate P(100 ≤ X ≤ 140) using the Normal approximation with continuity correction. Write the exact Normal integral you would evaluate and explain why the correction is needed.
Precisely state conditions under which a Normal approximation to a Poisson is acceptable, when it fails, and a principled alternative model when s² ≫ x̄ (include one diagnostic you would check).

Compare Normal vs Poisson; test dispersion and approximate tails

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