Diagnose and validate a ratio trend change

You are given a weekly dispute_rate series (disputes/succeeded_payments) that spikes upward and then partly falls back. Determine whether the shift is genuine or just noise, and whether composition changes account for it.

• Significance: With the counts below, run a two-sided test for the difference in overall proportions between Week 35 and Week 34, and build a 99% confidence interval for that difference. Data: • Week 34 overall: 800 disputes, 100,000 succeeded payments (0.80%) • Week 35 overall: 1,400 disputes, 110,000 succeeded payments (1.27%)
• Stratification (Simpson’s paradox check): Country-level counts • Week 34 US: 400/50,000; EU: 400/50,000 • Week 35 US: 1,200/60,000; EU: 200/50,000 Calculate each country’s change and the overall change adjusted to Week 34’s country mix. Explain why the overall rate rose even though EU improved.
• Change-point detection at scale: You need to monitor 200 country×industry pairs every week. Suggest a multiple-testing procedure (such as Benjamini–Hochberg with q=0.10) and a practical effect-size floor. Explain how you would combine statistical and practical significance.
• Small denominators: When succeeded_payments is below 5,000, propose a Bayesian smoothing method (for example, a Beta-Binomial model with an informative prior) and describe how to report shrunken rates with intervals.

Provide formulas, numeric results for the supplied counts, and explicit decision rules.

Overview: This question tests statistical inference and applied data-science reasoning: proportion hypothesis tests and confidence intervals, stratification and Simpson’s paradox, change-point detection with multiple-testing control, and Bayesian smoothing for low-count cases.