Analyze skewed comments and sampling effects
Overview
This question evaluates understanding of descriptive statistics for right‑skewed count data and inferential concepts such as sampling distributions, standard error, percentiles, and the Central Limit Theorem, and is commonly asked to probe how skewness and sample size affect central tendency measures and uncertainty in estimates.
Right‑Skewed Daily Comments: Location Stats and Sampling Distributions
You’re analyzing daily user comments per user, which are right‑skewed count data. Answer the following:
1) Position statistics on a right‑skewed distribution
Given a 20‑user sample of daily comments: [0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 6, 7, 10, 15, 20, 40]
- Compute the mean, median, and 95th percentile (p95).
- Place mean, median, and p95 on a number line and explain why, for right‑skewed count data, mean > median > mode typically holds.
- Briefly note why the median and p95 can be integers.
2) Sampling distribution of group averages
Assume the true individual‑level population has mean μ = 3.0 comments and standard deviation σ = 4.0 comments. If you repeatedly sample n = 50 users and compute each group’s average comments:
- Specify the approximate distribution (by the CLT) of that average.
- State its mean, standard error, and p95.
- Explain qualitatively how this p95 compares to the individual‑level p95 from part (1).
3) Effect of larger samples
If n increases to 400, recompute the standard error and describe how the locations of the mean, median, and p95 of the sampling distribution change relative to one another as n grows.
Solution
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