Chapter 10 Analysis of Variance
10.1 Introduction
A large company is considering purchasing, in quantity, one of four different computer
packages designed to teach a new programming language. Some influential people within
this company have claimed that these packages are basically interchangeable in that the one
chosen will have little effect on the final competence of its user. To test this hypothesis the
company has decided to choose 160 of its engineers, and divide them into 4 groups of size
- Each member in groupiwill then be given teaching packagei,i= 1, 2, 3, 4, to learn
the new language. When all the engineers complete their study, a comprehensive exam
will be given. The company then wants to use the results of this examination to determine
whether the computer teaching packages are really interchangeable or not. How can they
do this?
Before answering this question, let us note that we clearly desire to be able to conclude
that the teaching packages are indeed interchangeable when the average test scores in all
the groups are similar and to conclude that the packages are essentially different when
there is a large variation among these average test scores. However, to be able to reach
such a conclusion, we should note that the method of division of the 160 engineers
into 4 groups is of vital importance. For example, suppose that the members of the first
group score significantly higher than those of the other groups. What can we conclude
from this? Specifically, is this result due to teaching package 1 being a superior teaching
package, or is it due to the fact that the engineers in group 1 are just better learners? To be
able to conclude the former, it is essential that we divide the 160 engineers into the 4 groups
in such a way so as to make it extremely unlikely that one of these groups is inherently
superior. The time-tested method for doing this is to divide the engineers into 4 groups
in a completely random fashion. That is, we should do it in such a way so that all possible
divisions are equally likely; for in this case, it would be very unlikely that any one group
would be significantly superior to any other group. So let us suppose that the division of the
engineers was indeed done “at random.” (Whereas it is not at all obvious how this can be
accomplished, oneefficientprocedureistostartbyarbitrarilynumberingthe160engineers.
Then generate a random permutation of the integers 1, 2,..., 160 and put the engineers
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