You are shown a weekly dispute_rate time series (disputes/succeeded_payments) that rises sharply, then partially reverts. Diagnose whether the change is real vs noise and whether mix shifts explain it. - Significance: Using the counts below, compute the overall Week 35 vs Week 34 difference in proportions test (two-sided) and a 99% CI for the difference. Data: • Week 34 overall: disputes=800, succeeded=100000 (0.80%) • Week 35 overall: disputes=1400, succeeded=110000 (1.27%) - Stratification (Simpson’s paradox check): Per-country counts • Week 34 US: 400/50000; EU: 400/50000 • Week 35 US: 1200/60000; EU: 200/50000 Compute the per-country changes and the mix-adjusted overall change if Week 35 had Week 34’s country mix. Explain why overall increased while EU improved. - Change-point detection at scale: You must monitor 200 country×industry pairs weekly. Propose a multiple-testing procedure (e.g., Benjamini–Hochberg at q=0.10) and a practical effect-size floor. Describe how you’d combine statistical and practical significance. - Small denominators: When succeeded_payments < 5,000, propose a Bayesian smoothing approach (e.g., Beta-Binomial with informative prior) and how to report shrunken rates with intervals. Answer with formulas, numeric results for the provided counts, and clear decision rules.