Test conversion difference and adjust for clustering

Using aggregated results for the 7‑day window 2025‑08‑26..2025‑09‑01, evaluate statistical significance and power for conversion uplift, accounting for day‑level clustering: Given totals: Control (C): visits n_C=10,240, bookings x_C=308; Treatment (T): visits n_T=10,180, bookings x_T=351.

1. Point estimates: compute p_C, p_T, absolute lift (p_T − p_C, in percentage points) and relative lift.
2. Significance: perform a two‑sided test for difference in proportions (unpooled standard error). Report z, p‑value, and a 95% CI for (p_T − p_C). State any continuity correction you apply.
3. Clustering: adjust for day‑level clustering with ICC=0.01 and 7 days per variant. Use design effect DE = 1 + (\bar{m} − 1)·ICC where \bar{m} = n_variant / 7. Recompute effective sample sizes n_eff = n / DE and provide an adjusted p‑value/CI. Explain assumptions and limitations of this correction.
4. Power and sample size: What total visits per variant are required to detect a 0.30 percentage‑point absolute lift from a 3.00% baseline at 80% power and alpha=0.05 using an unpooled z‑test? Show the formula and final n per variant. Then recompute with the design effect from ICC=0.01 to give a clustered n per variant and the implied experiment duration if each variant receives 2,000,000 visits/day.
5. Robustness: briefly describe how you would check day‑to‑day heterogeneity (e.g., Q‑test or interaction with weekday) and how that influences the decision to launch.