EXAMPLE 24 Heights of Young Women

The distribution of heights of young adult women is approximately normal, with mean 64.5 inches and standard deviation 2.5 inches. This normal distribution describes the population of young women. It is also the probability model for choosing one woman at random from this population and measuring her height. For example, the 68-95-99.7 rule says that the probability is 0.95 that a randomly chosen woman is between 59.5 and 69.5 inches tall.

Now choose an SRS of 25 young women at random and take the mean of their heights. The mean varies in repeated samples; the pattern of variation is the sampling distribution of . The sampling distribution has the same center ( inches) as the population of young women. The standard deviation of the sampling distribution of is

The standard deviation describes the variation when we measure many individual women. The standard deviation of the distribution of describes the variation in the average heights of samples of women when we take many samples. The average height is less variable than individual heights.

Figure 8.24 compares the two distributions: Both are normal and both have the same mean, but the average height of 25 randomly chosen women has much less variability. For example, the 68-95-99.7 rule says that 95% of all averages lie between 63.5 and 65.5 inches because two of 's standard deviations make 1 inch. This 2-inch span is just one-fifth as wide as the 10-inch span that catches the middle 95% of heights for individual women.

382