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Analysis of Variance  Introduction
We have just studied hypothesis tests about the differences of means of two independent
populations.
For example,
0
1
2
a
1
2
H :
H :
μ = μ
μ μ
Our decision about the null hypothesis is based on knowing about the probability distribution
of a sample statistic  a single number  computed using
1
2
x
x

, the difference of the
means of random samples from the two populations. Recall that in the case of small sample size,
it is important to be able to assume that the sampled populations are normal.
nBut we may wish to compare the means of
more than two
populations. Clearly, now
things are
more complicated  a single such difference cannot suffice. But we still do base the decision on
the value of a test statistic  just
not simply z or t.
Since ANOVA involves
somewhat forbidding
nomenclature, we'll first try to get
the basic idea without all the jargon.
For example, we
might be interested in the following complementary hypotheses:
0
1
2
3
a
H :
H : These 3 means are not all the same.
μ = μ = μ
Notice that the null hypothesis is not one equation but three, and that the alternative
hypothesis includes more than one possibility. How do we proceed?
nFirst of all, we make some not very surprising assumptions. By now you are well aware that
in this crapshoot called statistics, we usually have to do this to say anything significant at all.
1. All 3 populations are assumed to be normal. Not shocking  what does "normal"
mean?
2. All 3 populations have the same variance,
2
2
2
1
2
3
σ = σ = σ
, so just call them
all
2
σ
. Recall
that it is not shocking that different populations can have
different means, but the same amount of variability.
3. All the observations of the variable involved are independent.
And for the purposes of this introduction, we make the simplifying assumption that
the
3 samples taken from the 3 populations are all the same size n. We will abandon this assumption
when we develop the full ANOVA terminology.
nNow, to start the analysis, suppose that the null hypothesis is true. But
then
1
2
3
μ = μ = μ
and
2
2
2
1
2
3
σ = σ = σ
actually means that there is really just
one
population with mean
μ
and variance
2
σ
.
The 3 populations
we are interested in can just be thought of as manifestations of this one
population. In this case, the 3 sample means,
1
2
3
x ,x ,x ,
should be close together, i.e., have small
variability. In fact we could say that small variability of these means favors the null hypothesis,
while large variability contradicts it. Of course, we need more precision than just this.
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This note was uploaded on 05/07/2008 for the course QMBE 2786 taught by Professor Easly during the Spring '08 term at University of New Orleans.
 Spring '08
 Easly

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