# Connect Probability, Experiment Sizing, and Bootstrap Inference A product team wants a precise statistical readout for a billboard-style intervention. You must compute probabilities from a supplied discrete PMF, explain a p-value, size the study, interpret a dense slide deck, and use bootstrap resampling when an analytic interval is unreliable. ### Constraints & Assumptions - The PMF values and observed statistic are supplied during the interview. - The sampling unit may differ from the analysis unit, so dependence must be addressed. - The desired minimum detectable effect and significance level must be stated before sizing. - Bootstrap samples must preserve the dependence structure relevant to the estimand. ### Clarifying Questions to Ask - Is the estimand a mean, rate, quantile, or ratio? - What is randomized, and can one unit influence another? - Is the test one-sided or two-sided, and what power is required? ### Part 1 — PMF and p-value Show how to validate and use a discrete PMF, then define the p-value for a chosen test statistic without calling it the probability that the null is true. #### What This Part Should Cover - Normalization and support checks - A tail probability tied to a statistic - A distinction between statistical and practical significance ### Part 2 — Sample size Derive or outline the inputs to a sample-size calculation and explain how clustering, unequal allocation, or repeated looks change the requirement. #### What This Part Should Cover - Baseline variance or rate, effect size, alpha, and power - Design effect for clustered assignment - Multiple-testing or sequential-monitoring control ### Part 3 — Bootstrap implementation Design a bootstrap procedure for the final estimand, including resampling unit, number of replicates, confidence-interval method, and diagnostics. #### What This Part Should Cover - Resampling aligned with independence - Percentile versus basic or BCa interval trade-offs - Monte Carlo stability and failure checks ### What a Strong Answer Covers - Correct formulas with defined symbols - Design assumptions connected to the data-generating process - An interpretation a stakeholder could act on ```hint Name the estimand first The PMF calculation, test, power formula, slide interpretation, and bootstrap must all target the same population quantity. Write that quantity before choosing machinery. ``` ### Follow-up Questions - When would a permutation test be preferable? - How would you adjust if billboard exposure spills across regions?