Introduction to Probability and Statistics for Engineers and Scientists

10.6Two-Way Analysis of Variance with Interaction 467

Namely,

reject H 0 int if

SSint/(n−1)(m−1)

SSe/nm(l−1)

>F(n−1)(m−1),nm(l−1),α

do not reject H 0 int otherwise

Alternatively, we can compute thep-value. IfFint=v, then thep-value of the test of the
null hypothesis that all interactions equal 0 is

p-value=P{F(n−1)(m−1),nm(l−1)>v}

If we want to test the null hypothesis

H 0 r:αi=0,i=1,...,m

then we use the fact that whenH 0 ris true,Xi..is the average ofnlindependent normal
random variables, each with meanμand varianceσ^2. Hence, underH 0 r,

E[Xi..]=μ, Var(Xi..)=σ^2 /nl

and so

∑m

i= 1

nl

(Xi..−μ)^2

σ^2

is chi-square withmdegrees of freedom. Thus, if we let

SSr=

∑m

i= 1

nl(Xi..−ˆμ)^2 =

∑m

i= 1

nl(Xi..−X..)^2

then, whenH 0 ris true,

SSr

σ^2

is chi-square withm−1 degrees of freedom

and so

SSr

m− 1

is an unbiased estimator ofσ^2

Because it can be shown that, underH 0 r,SSeandSSrare independent, it follows that
whenH 0 ris true

SSr/(m−1)

SSe/nm(l−1)

is anFm− 1 ,nm(l−1) random variable