Technical Screen: P-values and Robust Two-Sample/Paired Tests

Context: You are a data scientist evaluating healthcare interventions. Answer in clear, interview-ready explanations that a stakeholder could understand and that a peer could reproduce.

Part A — Plain-language p-value

Explain a p-value to a non-scientist without using the phrase "probability the null is true." Use a concrete everyday analogy and clarify common misinterpretations, including:

• p is not an effect size.
• p depends on the chosen test, analysis plan, and assumptions.
• A non-significant p does not prove no effect, and a significant p does not prove practical importance.

Part B — Wilcoxon vs t-test

For each scenario, select and justify the appropriate test, the assumptions, how you would check them, and an effect size measure. Be explicit about handling ties/zeros for rank tests and how you would report confidence intervals.

1. Paired data (n = 12): Systolic BP measured pre/post a low-sodium diet. The distribution of paired differences is skewed with outliers. Choose between a paired t-test and a Wilcoxon signed-rank test. State:
   • Assumptions and how you'd check them.
   • How ties/zero differences are handled.
   • An appropriate effect size (e.g., Cohen's dz vs rank-biserial or matched-pairs r) and CI.
2. Independent samples (n1 = 18, n2 = 25): Compare length of stay between two clinics with unequal variances and heavy-tailed, non-normal distributions. Choose between Welch's t-test and Wilcoxon rank-sum (Mann–Whitney). Discuss:
   • What Mann–Whitney estimates (probability of superiority) vs Welch's mean-difference focus.
   • When each is preferable given goals and data features.
   • How you'd complement with confidence intervals and a robust effect size (e.g., Hedges' g with HC3 SEs or Cliff's delta).