Analyze daily comments distribution and sampling

You have the daily number of comments for each active user.

1. For a single day, define and calculate the mean, median, and 95th percentile (P95); give the explicit formulas, and explain how to obtain P95 from the sorted empirical distribution.
2. Describe the Central Limit Theorem (CLT), why the sampling distribution of the sample mean becomes approximately normal, and the conditions required for it to hold.
3. If you draw 200 independent simple random samples, each of size n, from that same day’s user-level comment counts, what value of n is enough for the sample mean to be roughly normal, and how does the variance of its sampling distribution depend on n?
4. Show how the plotted daily mean, median, and P95 change as n increases, and explain why.
5. If comments per user rose week over week, list at least three plausible statistical or product explanations and describe how you would test each one.

Overview: The question tests skills in statistical inference, descriptive statistics (mean, median, percentiles), sampling theory, and the Central Limit Theorem, within the Statistics & Math area for data scientist interviews.