Choose the right test for proportions

You run an A/B test on delivery completion rate (delivered_bool). Results: - Control: 1,000 orders, 920 delivered, 80 not delivered. - Variant: 1,000 orders, 880 delivered, 120 not delivered. 1) Select the most appropriate significance test and justify why it’s preferred over a t-test on proportions. Derive the test statistic formula starting from Bernoulli → Binomial → Normal approximations and state the conditions under which the approximation is valid (expected cell counts, continuity corrections). 2) Compute the p-value to 3 significant digits and a 95% confidence interval for the difference in proportions; show intermediate steps. 3) Now suppose a small-market rollout yields: Control 20 delivered / 25 total; Variant 12 delivered / 18 total. Re-evaluate test choice (Chi-square vs. Fisher’s exact vs. z-test). Compute the exact p-value or explain precisely how you would obtain it. 4) Explain when a two-sample t-test gives approximately the same result as a z-test for proportions, and when it fails. Include at least two concrete failure modes (e.g., low counts, unpooled variance misuse).