Introduction to Probability and Statistics for Engineers and Scientists

368 Chapter 9: Regression

It now follows from Equation 9.4.2 that whenβ=1, the test statistic

TS=

√

8 Sxx/SSR(B−1)

has at-distribution with 8 degrees of freedom. The significance levelαtest will rejectH 0
when the value ofTSis sufficiently small (since this will occur whenB, the estimator of
β, is sufficiently smaller than 1). Specifically, the test is to

reject H 0 if

√

8 Sxx/SSR(B−1)<−tα,8

Program 9.2 gives that
√
8 Sxx/SSR(B−1)=30.2794(.4646−1)=−16.21

Sincet.01,8=2.896, we see that

TS<−t.01,8

and so the null hypothesis thatβ≥1 is rejected at the 1 percent level of significance. In
fact, thep-value is

p-value=P{T 8 ≤−16.213}≈ 0

and so the null hypothesis thatβ≥1 is rejected at almost any significance level, thus
establishing a regression toward the mean (see Figure 9.7).
A modern biological explanation for the regression to the mean phenomenon would
roughly go along the lines of noting that as an offspring obtains a random selection of

Father’s Height

72

70

68

66

64

62

60

Son’s Height

60 62 64 66 68 70 72 74

x

y

FIGURE 9.7 Example 9.4c for x small, y>x. For x large, y<x.