Answer all parts. A) Hand‑compute a two‑sided p‑value comparing two means using Welch’s t‑test. Sample A: n1=20, mean1=5.2, sd1=1.1. Sample B: n2=24, mean2=4.6, sd2=1.5. Compute t, Satterthwaite df, and the p‑value (3 decimals). Justify why Welch (not pooled) applies. B) Probability. For a Poisson process with rate λ=2 events/hour: 1) Find P[N(1.5) ≥ 3]. 2) Given N(2)=1, what is P(the event occurs within the first 0.5 hours)? State the property you used. C) Regularization choice. With 200 standardized predictors, block correlations ≈0.9, and n=500, choose LASSO, Ridge, or Elastic Net for prediction and interpretation. Explain trade‑offs, how you would select λ (and α) via nested CV, and how correlated groups affect selection stability and error. D) Test selection. Given two related measurements per subject with unequal variances and mild non‑normality at n=25 pairs, choose an appropriate test (paired t, Welch, or nonparametric), justify assumptions, and report an effect size.