Introduction to Probability and Statistics for Engineers and Scientists

10.3One-Way Analysis of Variance 453

is chi-square withmdegrees of freedom; therefore, replacingμin the preceding by its
estimatorX.., the average of all theXij, results in the statistic

∑m

i= 1

ni(Xi.−X..)^2 /σ^2

which is chi-square withm−1 degrees of freedom. Thus, letting

SSb=

∑m

i= 1

ni(Xi.−X..)^2

it follows, whenH 0 is true, thatSSb/(m−1) is also an unbiased estimator ofσ^2. Because it
can be shown that whenH 0 is true the quantitiesSSbandSSWare independent, it follows
under this condition that the statistic

SSb/(m−1)

SSW

/(∑m

i= 1

ni−m

)

is anF-random variable withm−1 numerator and

∑m
i= 1 ni−mdenominator degrees of
freedom. From this we can conclude that a significance levelαtest of the null hypothesis

H 0 :μ 1 = ··· =μm

is to

reject H 0 if

SSb/(m−1)

SSW

/(∑m

i= 1

ni−m

)>Fm−1,N,α

(

N=

∑

i

ni−m

)

not reject H 0 otherwise

REMARK

When the samples are of different sizes we say that we are in theunbalancedcase. Whenever
possible it is advantageous to choose a balanced design over an unbalanced one. For one
thing, the test statistic in a balanced design is relatively insensitive to slight departures from
the assumption of equal population variances. (That is, the balanced design is more robust
than the unbalanced one.)