Test the central limit theorem against a skewed population
Three steps, the way the exam actually works: work through the lab, write down your own measurements, then answer a 6-point free response. What you recorded goes to the grader with your writing, so a conclusion that does not follow from your own numbers will cost you the point — exactly as it would with a real reader.
Predict before you look
- The central limit theorem concerns the distribution of sample MEANS, not of individual observations. Which of the two does the population's skew describe?
- Standard error is the population standard deviation divided by the square root of the sample size. What happens to it as n grows?
Nothing to submit here — these are to think through, so the prediction below is an informed one rather than a guess.
Commit to an answer now. It is not graded and being wrong costs nothing — the point is to have something specific to reconcile against once you have the data.
Answer every prediction to unlock the lab. A sentence is enough.
Run the investigation
Predictions first
The procedure and the simulation unlock once you have committed above. Observing before predicting is how a wrong intuition survives a lab intact.
Record what you measured
These are your numbers, not ours. The grader sees them, so your conclusions have to follow from what you actually recorded.
| Population standard deviation σ | |
|---|---|
| Standard deviation of sample means at n = 5 | |
| Standard deviation of sample means at n = 25 | |
| Standard deviation of sample means at n = 50 | |
| Mean of the sample means at n = 50 |
Answer the free response
You repeatedly sampled from a strongly right-skewed population. (a) Describe what happened to the CENTER of the distribution of sample means as n increased, and state what value it estimates. (b) Describe what happened to the SPREAD, and test your recorded standard deviations against the theoretical formula for the standard deviation of the sampling distribution. Show one calculation. (c) The population is visibly skewed, yet the distribution of sample means became bell-shaped. Name the theorem responsible and state precisely what it does and does not claim. (d) A classmate concludes that because of this theorem, it is always safe to use a normal model for a sample mean. Explain the flaw in that reasoning.
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