Compute sample size and test duration correctly

For two experiments, calculate the required sample size and runtime, and defend the assumptions you make.

Scenario A: the baseline friend-accept rate is 9.0%; the target minimum detectable effect is +0.5 percentage points in absolute terms; use two-sided $\alpha = 0.05$, power $0.80$, and equal allocation; eligible traffic is 10 million users per day.

1. Derive the sample size per arm and the number of calendar days needed.
2. Show how the results change if the true standard deviation is 20% larger than the assumed value.

Scenario B has three arms with a shared control: detect a +0.3 percentage point effect for either treatment against control while keeping the familywise $\alpha = 0.05$; choose among Holm, Bonferroni, and Dunnett and justify your choice; recompute the per-arm $n$ and total duration when 20% of traffic goes to control and 40% goes to each of the two treatments.

For both scenarios: 3) discuss what happens if you peek daily and stop on a naive $p < 0.05$ rule; propose a sequential design such as O’Brien–Fleming and describe how it alters the stopping boundaries; 4) explain when you would move to user-level CUPED or cluster-robust standard errors (for example, feed-level clustering) and how that changes the required $n$.

Overview: This question tests experiment design and statistical power skills, covering sample size estimation, the effect of variance misspecification, multiple-comparison corrections, sequential monitoring boundaries, and covariate or clustering adjustments.