Connecting Sample Means and Sampling Distributions
Sampling distributions and the central limit theorem are difficult topics. Everything builds on the understanding of what a sampling distribution is — work until you get it. Solve the following problems about sample means and sampling distributions.
A researcher draws a series of samples of exam scores from a population of scores that has a normal distribution. What required sample size (if any) is necessary for the distribution of the sample means to also have a normal distribution?
Answer: No specific requirement for sample size is needed.
Because you know that the individual scores come from a normal distribution, the distribution of the sample means will also have a normal distribution, regardless of the sample size.
The daily productivity of honey of individual bees has a normal distribution. If random samples of ten bees are taken, what is the shape of the sampling distribution of the sample means of honey production?
Answer: It is exactly normal.
Because the individual data points are normally distributed, the sampling distribution of sample means is also normal, no matter what the size of each sample is. (You don’t need the central limit theorem and the requirement of having a sample size of at least 30 if you start with a normal distribution.)
If the population distribution is __________ and the sample size is ___________, you are required to apply the central limit theorem to assume that the sampling distribution of the sample means is normal.
(A) normal, 10
(B) normal, 50
(C) right-skewed, 60
(D) Choices (B) and (C)
(E) Choices (A), (B), and (C)
Answer: right-skewed, 60
If the population’s distribution is normal, the sampling distribution of the sample means is also normal, so the central limit theorem is required only for non-normal (in this case, right-skewed) populations.
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