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.
Run the investigation
- 1Choose the skewed-right population and note its stated mean and standard deviation.
- 2Set the sample size to n = 5, press “Draw 500” a few times to build up the sampling distribution, then record the SD of x̄ it reports.
- 3Change the sample size to n = 25 and repeat, recording the SD of x̄ again.
- 4Repeat once more at n = 50, recording both the SD of x̄ and the mean of x̄.
Booting the lab…
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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