EXAMPLE 7.10

Heights of 10-year-old girls and boys. A fourth-grade class has 12 girls and 8 boys. The children’s heights are recorded on their 10th birthdays. What is the chance that the girls are taller than the boys? Of course, it is very unlikely that all the girls are taller than all the boys. We translate the question into the following: what is the probability that the mean height of the girls is greater than the mean height of the boys?

Based on information from the National Health and Nutrition Examination Survey, we assume that the heights (in inches) of 10-year-old girls are N(56.9, 2.8) and the heights of 10-year-old boys are N(56.0, 3.5).²² The heights of the students in our class are assumed to be random samples from these populations. The two distributions are shown in Figure 7.11(a).

The difference between the female and male mean heights varies in different random samples. The sampling distribution has mean

μ₁ −μ₂ = 56.9 − 56.0 = 0.9 inches

and variance

= 2.18

The standard deviation of the difference in sample means is, therefore, inches.

If the heights vary Normally, the difference in sample means is also Normally distributed. The distribution of the difference in heights is shown in Figure 7.11(b). We standardize by subtracting its mean (0.9) and dividing by its standard deviation (1.48). Therefore, the probability that the girls, on average, are taller than the boys is

= P(Z > −0.61) = 0.7291