A/B Test Sizing: Reducing Cold-Food Complaint Rate

You are running an A/B test of thermal delivery bags aiming to reduce the cold-food complaint rate. Baseline complaint rate is 3.50% and all such orders currently receive full refunds (this cost context motivates why even small deltas matter, though it doesn't change the stats math). The experiment targets a 10% relative reduction to 3.15%. Use a two-sided z-test for two proportions with α = 0.05 and power = 0.80.

Assume equal allocation between control and treatment and no seasonality.

Tasks:

(a) Compute the minimum sample size per arm for equal allocation to detect a 10% relative reduction (3.50% → 3.15%). Show formulas and numeric steps.

(b) If only 60% of 200,000 daily orders are eligible and traffic is split 50/50, estimate the minimum test duration in days.

(c) Recompute (a) for a 6% relative reduction target (3.50% → 3.29%).

Round final answers to the nearest 100 orders and 0.5 day, and state assumptions clearly.