# CLT: Question 6

Imagine that you select a sample of N = 25 women from Ourtown and find that the mean height of the sample is 5 feet, 7 inches (67 inches). The population mean is known to be 5 feet, 4 inches (64 inches) with SD = 3 inches. Would a sample mean this large be more likely or less likely if N = 5? Why?

#### Equally likely. Sample means are randomly distributed.

You answered that a mean as large as 67 inches is equally likely when N = 25 as when N = 5. Consider the standard error for sample means in each case and the corresponding z-score for a sample mean that is 3 inches larger than the population mean.

#### More likely. The sampling distribution of the mean would be wider if the sample were smaller, so sample means would be more likely to be farther from the population mean.

Yes, excellent!

The shape of the sampling distribution is narrower with larger sample sizes. You can observe that the curve gets thinner as the sample size increases. As a result, a sample mean score 3 inches larger than the population mean is farther in the tail of the sampling distribution (and less likely) when the sample size is larger.
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