Example 4.30

EXAMPLE 4.30

Heights of young women. The distribution of the heights of all young women is close to the Normal distribution with mean 64.5 inches and standard deviation 2.5 inches. Suppose that μ = 64.5 were exactly true.

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Figure 4.14 The law of large numbers in action, Example 4.30. As we take more observations, the sample mean always approaches the mean of the population.

Figure 4.14 shows the behavior of the mean height of n women chosen at random from a population whose heights follow the N(64.5, 2.5) distribution. The graph plots the values of as we add women to our sample. The first woman drawn had height 64.21 inches, so the line starts there. The second had height 64.35 inches, so for n = 2 the mean is

This is the second point on the line in the graph.

At first, the graph shows that the mean of the sample changes as we take more observations. Eventually, however, the mean of the observations gets close to the population mean μ = 64.5 and settles down at that value. The law of large numbers says that this always happens.