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.)