You are given daily counts of comments per active user. (1) Define and compute mean, median, and 95th percentile (P95) for a day; write 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; specify required conditions. (3) If you repeatedly sample 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) Show how the plots of daily mean, median, and P95 change as n increases, and why. (5) If comments/user increased week-over-week, list at least three plausible statistical or product reasons and how you would test each.

This question evaluates proficiency in statistical inference, descriptive statistics (mean, median, percentiles), sampling theory and the Central Limit Theorem within the Statistics & Math domain for data scientist interviews.

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This is a medium difficulty Statistics & Math question, commonly asked during Onsite rounds at Meta.

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This question is commonly asked for Data Scientist candidates at Meta during technical interviews.

Analyze daily comments distribution and sampling

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:

Define and compute the daily mean, median, and 95th percentile (P95). Provide explicit formulas, including how to compute P95 from a sorted empirical distribution.
Explain the Central Limit Theorem (CLT) and why the sampling distribution of the sample mean tends toward normality. State the required conditions.
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?
Describe how the plots of the sampling distributions of the daily mean, median, and P95 evolve as n increases, and explain why.
If comments per user increased week-over-week, list at least three plausible statistical or product reasons and how you would test each.

Daily Comments per Active User: Sampling and Inference

Define and compute the daily mean, median, and 95th percentile (P95). Provide explicit formulas, including how to compute P95 from a sorted empirical distribution.

Explain the Central Limit Theorem (CLT) and why the sampling distribution of the sample mean tends toward normality. State the required conditions.

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?

Describe how the plots of the sampling distributions of the daily mean, median, and P95 evolve as n increases, and explain why.

If comments per user increased week-over-week, list at least three plausible statistical or product reasons and how you would test each.

Analyze daily comments distribution and sampling

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Daily Comments per Active User: Sampling and Inference

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Analyze daily comments distribution and sampling

Quick Overview

Daily Comments per Active User: Sampling and Inference

Solution

Submit Your Answer