444 Chapter 10:Analysis of Variance
Definition
The statistic
SSW=
∑m
i= 1
∑n
j= 1
(Xij−Xi.)^2
is called thewithin samples sum of squaresbecause it is obtained by substituting the sample
population means for the population means in expression 10.3. The statistic
SSW/(nm−m)
is an estimator ofσ^2.
Our second estimator ofσ^2 will only be a valid estimator when the null hypothesis is
true. So let us assume thatH 0 is true and so all the population meansμiare equal, say,
μi=μfor alli. Under this condition it follows that themsample meansX 1 .,X 2 .,...,Xm.
will all be normally distributed with the same meanμand the same varianceσ^2 /n. Hence,
the sum of squares of themstandardized variables
Xi.−μ
√
σ^2 /n
=
√
n(Xi.−μ)/σ
will be a chi-square random variable withmdegrees of freedom. That is, whenH 0 is true,
n
∑m
i= 1
(Xi.−μ)^2 /σ^2 ∼χm^2 (10.3.3)
Now, when all the population means are equal toμ, then the estimator ofμis the average
of all thenmdata values. That is, the estimator ofμisX.., given by
X..=
∑m
i= 1
∑n
j= 1
Xij
nm
=
∑m
i= 1
Xi.
m
If we now substituteX.. for the unknown parameterμin expression 10.5, it follows,
whenH 0 is true, that the resulting quantity
n
∑m
i= 1
(Xi.−X..)^2 /σ^2