Calculate power and test duration

Take the baseline rate of cold-food complaints to be $3.50\%$, and suppose every such order currently gets a full refund. The thermal-bag experiment is targeting a $10\%$ relative decrease in complaints, bringing the rate to $3.15\%$. With a two-sided z-test for two proportions at $\alpha=0.05$ and power $0.80$: (a) calculate the minimum sample size per arm under equal allocation, and show the formulas and numerical work. (b) If just $60\%$ of $200{,}000$ daily orders qualify and the traffic is divided evenly between arms, estimate the shortest test length in days, assuming no seasonal effects. (c) Repeat the calculation in (a) for a $6\%$ relative reduction target. List all assumptions explicitly, and round the final results to the nearest 100 orders and nearest 0.5 day.

Overview: This item assesses skill in statistical power analysis, sample-size determination for proportions, two-sided z-test hypothesis testing, and estimating practical A/B test duration.