Compute p-values, probabilities, and regularization choices

Respond to every part.

A) By hand, calculate a two-sided p-value for the difference between two means using Welch's t-test. Sample A has $n_1=20$, mean $5.2$, and standard deviation $1.1$; Sample B has $n_2=24$, mean $4.6$, and standard deviation $1.5$. Compute the t statistic, the Satterthwaite degrees of freedom, and the p-value to three decimal places. Explain why Welch's version is appropriate rather than the pooled test.

B) Probability. A Poisson process has rate $\lambda=2$ events per hour.

1. Determine $P[N(1.5) \ge 3]$.
2. If $N(2)=1$, find the probability that the event happens within the first 0.5 hours. Name the property you relied on.

C) Regularization choice. With 200 standardized predictors, block correlations around 0.9, and $n=500$, select among LASSO, Ridge, and Elastic Net for prediction and interpretation. Discuss the trade-offs, describe how you would choose $\lambda$ (and $\alpha$) through nested cross-validation, and explain how correlated groups affect selection stability and prediction error.

D) Test selection. You have two related measurements per subject, unequal variances, and mild non-normality with $n=25$ pairs. Choose an appropriate test among paired t, Welch, or a nonparametric method, justify the assumptions, and report an effect size.

Overview: This question assesses inferential statistics and probability skills (hand calculation of Welch's t statistic, Satterthwaite degrees of freedom, and p-values; Poisson process probabilities), knowledge of model selection and regularization (LASSO, Ridge, Elastic Net, and tuning), and hypothesis test selection and effect size interpretation.