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A lecture note on Single Factor Analysis of Variance (ANOVA) by Yuan Huang. It covers the introduction, definition, construction, and applications of ANOVA for comparing means of multiple populations. It also discusses the assumptions, calculations, and interpretation of ANOVA results.
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Yuan Huang
April 5, 2013
Chapter 9 : hypothesis testing for one population mean.
Chapter 10: hypothesis testing for two population means.
What comes next?
Examples of multiple population comparisons:
studies in crop variation)).
Example : An experiment to study the effect of sunshine and water
in growing corns. Suppose the sunshine can be: intense, weak,
bare; amount of water can be: much, little.
and little for water.
When the response is numerical and factor is categorical (finite
number of levels), we could apply the analysis of variance ANOVA.
Depending on how many factors you have in your study (to label
your population), comparisons can be classified as:
In this lecture, we are introducing One-way ANOVA.
There are three assumptions:
balanced.
2
1
=... = σ
2
I
Note: To verify equality of variances, there is a formal test called
the Levene test. A rule of thumb that one can use is that the
largest standard deviation is not larger than two times the smaller.
Let me first give some numbers to help understand the notations
will be introduced.
and I am giving them a test. I want to test if the true
averages on the test for all classes are equal. I am selecting a
sample of 5 students from each class.
Variance
Sample variance:
2
i
J ∑
j=
ij
i.
2
Hypothesis:
0
: μ 1
=... = μ I
vs H 1
: at least two μ
′
i
s differ.
sample means ¯xi. to be as close to the grand mean ¯x.. as
possible.
Sum of Squares
Definition: error sum of squares SSE:
I ∑
i=
J ∑
j=
ij
i.
2
= (J − 1)
I ∑
i=
2
i
σ
2
∼ χ
2
I (J−1)
Sum of Squares
Definition: treatment sum of squares SSTr:
SSTr = J
I ∑
i=
i.
..
2
SSTr
σ
2
∼ χ
2
I − 1