Compare Normal vs Poisson; test dispersion and approximate tails
Company: Apple
Role: Data Scientist
Category: Statistics & Math
Difficulty: Medium
Interview Round: Technical Screen
You collect n=200 independent minute-level event counts with sample mean x̄=14.5 and sample variance s²=16.2.
1) Under a Poisson(λ) model, derive the MLE for λ and its asymptotic variance; construct a 95% Wald CI for λ.
2) 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.
3) 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.
4) 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).
Quick Answer: This question evaluates a data scientist's competency in statistical inference for count data, covering MLE and asymptotic variance estimation, Wald confidence intervals, tests for equidispersion versus overdispersion, Normal approximation with continuity correction for Poisson tails, and model selection diagnostics.