Introduction to Probability and Statistics for Engineers and Scientists

10.3One-Way Analysis of Variance 445

will be a chi-square random variable withm−1 degrees of freedom. That is, if we define
SSbby

SSb=n

∑m

i= 1

(Xi.−X..)^2

then it follows that

when H 0 is true,

SSb/σ^2 is chi-square withm−1 degrees of freedom

From the above we obtain that whenH 0 is true,

E[SSb]/σ^2 =m− 1

or, equivalently,

E[SSb/(m−1)]=σ^2 (10.3.4)

So, whenH 0 is true,SSb/(m−1) is also an estimator ofσ^2.

Definition

The statistic

SSb=n

∑m

i= 1

(Xi.−X..)^2

is called thebetween samples sum of squares. WhenH 0 is true,SSb/(m−1) is an estimator
ofσ^2.
Thus we have shown that

SSW/(nm−m) always estimatesσ^2

SSb/(m−1) estimatesσ^2 whenH 0 is true

Because* it can be shown thatSSb/(m−1) will tend to exceedσ^2 whenH 0 is not true, it
is reasonable to let the test statistic be given by

TS=

SSb/(m−1)

SSW/(nm−m)

and to rejectH 0 whenTSis sufficiently large.

* A proof is given at the end of this section.