The Central Limit Theorem
Statisticians need to understand the central limit theorem, how to use it, when to use it, and when it’s not needed. The central limit theorem is used only in certain situations. Solve the following problems that involve the central limit theorem.
Suppose that a researcher draws random samples of size 20 from an unknown distribution. Can the researcher claim that the sampling distribution of sample means is at least approximately normal?
Answer: No, because the sample sizes are too small to use the central limit theorem.
In this case, the original population distribution is unknown, so you can’t assume that you have a normal distribution. The central limit theorem can’t be invoked because the sample sizes are too small (less than 30).
As a general rule, approximately what is the smallest sample size that can be safely drawn from a non-normal distribution of observations if someone wants to produce a normal sampling distribution of sample means?
Answer: n = 30
According to the central limit theorem, if you repeatedly take sufficiently large samples, the distribution of the means from those samples will be approximately normal. For most non-normal populations, you can choose sample sizes of at least 30 from the distribution, which usually leads to a normal sampling distribution of sample means no matter what the underlying distribution of scores is.
In fact, if the underlying distribution of values approximates a normal distribution, it may be possible to achieve a normal sampling distribution of sample means with smaller samples.
For populations with several peaks, wild variation, and/or extreme outliers, you may need larger sample sizes.
A researcher draws 150 samples of 10 apiece from a normally distributed population of individual observations. What can the researcher conclude in this case?
Answer: The sampling distribution of sample means is normal.
The sampling distribution of sample means is normal whenever the observations come from a normally distributed population of scores, which is true in this case.
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