Compute power and cost-aware thresholds

You are evaluating a new ACH velocity+shared-device block rule. Assumptions - Volume: 1,000,000 ACH credits/month. - Baseline fraud prevalence: 0.15% of credits return as fraud within 5 business days. - Mean loss per fraudulent credit: $900; SD ≈ $600. - Proposed rule: Recall 70% of fraudulent credits; FPR 0.05% on legitimate credits. - Cost per false positive (ops + churn): $25. Tasks A) Expected value: Compute monthly gross fraud loss without the rule; with the rule; then net savings after false-positive costs. Show all formulas and units. B) Sample size: Using “loss per 1,000 credits” as the primary metric, design a two-arm online A/B test to detect a 15% relative reduction at α=0.05 (two-sided) and 80% power. State distributional assumptions (e.g., approximate as a Poisson rate or two-part model), derive the per-arm sample size, and justify your choice. C) Interval estimation: Control arm observes 240,000 credits with 0.12% fraud prevalence. Compute a 95% Wilson interval for the prevalence and interpret it for decision-making. D) Sequential monitoring: If results are reviewed daily for 14 days, propose a valid sequential testing plan (e.g., alpha-spending or group-sequential boundaries) and stopping rules that control Type I error. Explain how early stopping interacts with the chosen metric.