9.4Statistical Inferences About the Regression Parameters 367
Father’s Height7570656055Son’s Height55 60 65 70 75FIGURE 9.6 Scatter diagram of son’s height versus father’s height.
EXAMPLE 9.4c To illustrate Galton’s thesis of regression to the mean, the British statistician
Karl Pearson plotted the heights of 10 randomly chosen sons versus that of their fathers.
The resulting data (in inches) were as follows.
Fathers’ height 60 62 64 65 66 67 68 70 72 74
Sons’ height 63.6 65.2 66 65.5 66.9 67.1 67.4 68.3 70.1 70A scatter diagram representing these data is presented in Figure 9.6.
Note that whereas the data appear to indicate that taller fathers tend to have taller
sons, it also appears to indicate that the sons of fathers that are either extremely short or
extremely tall tend to be more “average” than their fathers — that is, there is a “regression
toward the mean.”
We will determine whether the preceding data are strong enough to prove that there is
a regression toward the mean by taking this statement as the alternative hypothesis. That
is, we will use the above data to test
H 0 :β≥1 versus H 1 :β< 1which is equivalent to a test of
H 0 :β=1 versus H 1 :β< 1