Introduction to Probability and Statistics for Engineers and Scientists

452 Chapter 10:Analysis of Variance

Hence, with 95 percent confidence, we can conclude that the mean grade point average
of first-year students from high school 3 is less than the mean average of students from
high school 1 or from high school 2 by an amount that is between .165 and .985, and
that the difference in grade point averages of students from high schools 1 and 2 is less
than .410. ■

10.3.2 One-Way Analysis of Variance with Unequal Sample Sizes

The model in the previous section supposed that there were an equal number of data points
in each sample. Whereas this is certainly a desirable situation (see the Remark at the end
of this section), it is not always possible to attain. So let us now suppose that we havem
normal samples of respective sizesn 1 ,n 2 ,...,nm. That is, the data consist of the

∑m
i= 1 n^1
independent random variablesXij,j=1,...,ni,i=1,...,m, where

Xij∼N(μi,σ^2 )

Again we are interested in testing the hypothesisH 0 that all means are equal.
To derive a test ofH 0 , we start with the fact that

∑m

i= 1

∑ni

j= 1

(Xij−E[Xij])^2 /σ^2 =

∑m

i= 1

∑ni

j= 1

(Xij−μi)^2 /σ^2

is a chi-square random variable with

∑m
i= 1 nidegrees of freedom. Hence, upon replacing
each meanμiby its estimatorXi., the average of the elements in samplei, we obtain

∑m

i= 1

∑ni

j= 1

(Xij−Xi.)^2 /σ^2

which is chi-square with

∑m

i= 1 ni−mdegrees of freedom. Therefore, letting

SSW=

∑m

i= 1

∑ni

j= 1

(Xij−Xi.)^2

it follows thatSSW/

(∑m

i= 1 ni−m

)
is an unbiased estimator ofσ^2.
Furthermore, ifH 0 is true andμis the common mean, then the random variables
Xi.,i=1,...,mwill be independent normal random variables with

E[Xi.]=μ, Var(Xi.)=σ^2 /ni

As a result, whenH 0 is true

∑m

i= 1

(Xi.−μ)^2

σ^2 /ni

=

∑m

i= 1

ni(Xi.−μ)^2 /σ^2