Apply Bayes theorem with conjugate priors
Overview
This question evaluates a candidate's ability to apply Bayesian inference with a conjugate Beta prior to a binomial likelihood, derive the posterior distribution, compute posterior summaries (mean, credible interval, tail probabilities), and contrast Bayesian credible intervals with frequentist Wald and Clopper–Pearson intervals.
A website’s true daily purchase conversion p is unknown. Prior: p ~ Beta(2, 8). Day 1: 10 purchases out of 120 visits; Day 2: 18 purchases out of 150 visits. (a) Derive the posterior after both days. (b) Compute the posterior mean and a 95% equal-tailed credible interval. (c) Compute P(p > 0.12 | data). (d) Compare your Bayesian 95% interval to a frequentist Wald interval and an exact Clopper–Pearson interval; explain differences and when each is appropriate.