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The Distribution of Sample Means The
Sampling Distribution of the Mean (SDM)
is a distribution composed of the means (X) for all the possible random samples
(of size n) from a population. The
mean of the SDM is called the expected
value of the mean or
The Central Limit Theorem (CLT) tells us that a SDM has the following characteristics: (1) SDM is normally distributed if the population is normally distributed or if n ³ 30. (2)
(3)
Because the SDM is normally distributed, we can use z-scores to answer probability questions about samples.
Example: For a normally distributed population with m = 60 and s = 12, what is the probability of selecting a random sample of n=36 scores with a sample mean (X) > 64? (What proportion of the SDM is > 64?) (1) Express the problem in a probability statement: p(X > 64) = ?? (2) Sketch the distribution, label, and shade the area you wish to find.
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(3) Compute z-score.
(4) Use the unit normal table to determine the probability associated with this z-score. p(X > 64) = p(z > 2) = .0228. (5) With a population of m = 60 and s = 12, only 2.28% of samples of n=36 are expected to have means > 64. Therefore drawing a sample of n=36 with a mean > 64 is very unlikely.
Don’t let word problems fool you! The average age of residents at a large retirement community is m = 60 years. Of course, there is some variability in age with s = 12 years. If you randomly select 36 individuals from this community, what is the probability that the average age of your sample will be X > 64? This problem is the same as the one above.
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