Daily Comments per Active User: Sampling and Inference

You have, for a given day d, the count of comments made by each active user. Let there be m active users on day d, and let X_i be the number of comments for user i (i = 1, 2, …, m). Answer the following:

1. Define and compute the daily mean, median, and 95th percentile (P95). Provide explicit formulas, including how to compute P95 from a sorted empirical distribution.
2. Explain the Central Limit Theorem (CLT) and why the sampling distribution of the sample mean tends toward normality. State the required conditions.
3. Suppose you repeatedly take 200 independent simple random samples of size n from the same day’s user-level comments. What n is sufficient for the sample mean to be approximately normal, and how does the sampling distribution’s variance change with n?
4. Describe how the plots of the sampling distributions of the daily mean, median, and P95 evolve as n increases, and explain why.
5. If comments per user increased week-over-week, list at least three plausible statistical or product reasons and how you would test each.