Compute power and interpret uplift metrics

Two‑arm experiment on conversion. Baseline conversion p0 = 0.120. You want to detect an absolute uplift of Δ = +0.005 (two‑sided α = 0.05, power = 0.80). (1) Approximate the required per‑variant sample size using the normal approximation for a difference in proportions; show your formula and calculation. (2) You ran for a week and observed: Control nC = 150,000, xC = 18,000; Treatment nT = 150,000, xT = 18,900. Compute the point estimate Δ̂, a 95% confidence interval, and a p‑value; interpret business significance vs statistical significance. (3) With CUPED using a pre‑period covariate yielding R² = 0.30 on the outcome, estimate the new effective sample size or variance and the revised MDE; show your math. (4) You track 4 guardrails with unadjusted p‑values {0.03, 0.01, 0.20, 0.04}. Apply Holm–Bonferroni at familywise α = 0.05 and state which guardrails remain significant; show ordering and adjusted thresholds. (5) Explain how you would check and correct for overdispersion or miscalibration in conversion estimates when there is user clustering by geo.