Introduction to Probability and Statistics for Engineers and Scientists

362 Chapter 9: Regression

X

25

20

15

10

5

4 6 8 10 12 14 16 18 20

Y

FIGURE 9.4 Example 9.3a.

9.4.1 Inferences Concerningβ

An important hypothesis to consider regarding the simple linear regression model

Y=α+βx+e

is the hypothesis thatβ=0. Its importance derives from the fact that it is equivalent to
stating that the mean response does not depend on the input, or, equivalently, that there
is no regression on the input variable. To test

H 0 :β=0 versus H 1 :β= 0

note that, from Proposition 9.3.1,

B−β

√

σ^2 /Sxx

=

√

Sxx

(B−β)

σ

∼N(0, 1) (9.4.1)

and is independent of

SSR

σ^2

∼χn^2 − 2

Hence, from the definition of at-random variable it follows that

√

Sxx(B−β)/σ

√

SSR

σ^2 (n−2)

=

√

(n−2)Sxx

SSR

(B−β)∼tn− 2 (9.4.2)