Calculate A/B sample size, CI, decision rules

Plan and evaluate an A/B test for a signup funnel. Setup: Baseline conversion $p_0 = 0.040$. Minimum detectable effect (absolute) is $0.004$ (so target $p_1 = 0.044$). Two-sided $\alpha = 0.05$, power $= 0.80$.

1. Sample size: Calculate the required sample size per arm for a two-proportion z-test. Provide the formula you use and the numeric result.
2. CI and p-value: Suppose after launch you observe Control with $n_C = 50{,}000$, $x_C = 2{,}150$ (4.30%) and Treatment with $n_T = 49{,}500$, $x_T = 2{,}090$ (4.22%). Compute the p-value for the difference in proportions (both pooled and unpooled variants) and a 95% confidence interval for $(p_T - p_C)$. Interpret the results practically.
3. Design choices: Specify primary versus guardrail metrics and how you would adjust for multiple comparisons if you track cancellation rate and latency as guardrails.
4. Power/variance reduction: Explain how you would use CUPED or stratification to reduce variance and how that affects the required sample size.
5. Operational risks: Discuss sequential peeking and how to control type-I error (for example, alpha-spending). What do you do if traffic is heterogeneous and the global average obscures segment-level reversals? Overview: This question assesses skills in experimental design and statistical inference, including sample size calculation, hypothesis testing, confidence interval interpretation, multiple comparison control, variance reduction techniques, and operational considerations for A/B testing.