440 Chapter 10:Analysis of Variance
whose numbers are among the first 40 of the permutation into group 1, those whose num-
bers are among the 41st through the 80th of the permutation into group 2, and so on.)
It is now probably reasonable to suppose that the test score of a given individual
should be approximately a normal random variable having parameters that depend on
the package from which he was taught. Also, it is probably reasonable to suppose that
whereas the average test score of an engineer will depend on the teaching package she
was exposed to, the variability in the test score will result from the inherent varia-
tion of 160 different people and not from the particular package used. Thus, if we let
Xij,i=1,...,4,j =1,..., 40, denote the test score of thejth engineer in groupi,
a reasonable model might be to suppose that theXijare independent random variables
withXijhaving a normal distribution with unknown meanμiand unknown varianceσ^2.
The hypothesis that the teaching packages are interchangeable is then equivalent to the
hypothesis thatμ 1 =μ 2 =μ 3 =μ 4.
In this chapter, we present a technique that can be used to test such a hypothesis. This
technique, which is rather general and can be used to make inferences about a multitude
of parameters relating to population means, is known as theanalysis of variance.
10.2 An Overview
Whereas hypothesis tests concerning two population means were studied in Chapter 8,
tests concerning multiple population means will be considered in the present chapter. In
Section 10.3, we suppose that we have been provided samples of sizenfrommdistinct
populations and that we want to use these data to test the hypothesis that thempopulation
means are equal. Since the mean of a random variable depends only on a single factor,
namely, the sample the variable is from, this scenario is said to constitute aone-way
analysis of variance. A procedure for testing the hypothesis is presented. In addition, in
Section 10.3.1 we show how to obtain multiple comparisons of the
(m
2)
differences between
the pairs of population means; and in Section 10.3.2 we show how the equal means
hypothesis can be tested when themsample sizes are not all equal.
In Sections 10.4 and 10.5, we consider models that assume that there are two factors
that determine the mean value of a variable. In these models, the variables can be thought
of as being arranged in a rectangular array, with the mean value of a specified variable
depending both on the row and on the column in which it is located. Such a model is
called atwo-way analysis of variance. In these sections we suppose that the mean value of a
variable depends on its row and column in an additive fashion; specifically, that the mean
of the variable in rowi, columnjcan be written asμ+αi+βj. In Section 10.4, we
show how to estimate these parameters, and in Section 10.5 how to test hypotheses to the
effect that a given factor — either the row or the column in which a variable is located —
does not affect the mean. In Section 10.6, we consider the situation where the mean of
a variable is allowed to depend on its row and column in a nonlinear fashion, thus allowing
for a possibleinteractionbetween the two factors. We show how to test the hypothesis that
there is no interaction, as well as ones concerning the lack of a row effect and the lack of
a column effect on the mean value of a variable.