Chapter 11 – Analysis of Variance
Objective: To test the hypothesis that three or more population means are equal.
Section 11.1:

F Distribution properties
1) The F distribution is not symmetric; it is skewed to the right.
2)
3)
The values of F can be zero (0) or positive (+), but they cannot be negative.
There is a different F distribution for each pair of degrees of freedom for the numerator and denominator.
11.2: One-Way ANOVA

In the past, we used the two-sample t procedures to compare the means of two populations or the mean
responses to two treatments in an experiment.

What about problems involving more than two populations?
ANOVA: Analysis of Variance – a statistical technique for comparing several means

The Problem of Multiple Regressions
 How do we compare multiple means?
o
Use the two-sample t test several times? No.
 For example, we might compare three means. The weakness of doing three tests is that we
get three P-values, one for each test alone. That doesn’t tell us how likely it is that three
sample means are spread apart as far as these are.
o
We can’t safely compare many parameters by doing tests or confidence intervals for two
parameters at a time. What would we conclude?
 This is the problem of multiple comparisons.

Statistical methods for dealing with multiple comparisons usually have two steps:
1. An overall test to see if there is good evidence of any differences among the parameters
2. A detailed follow-up analysis to decide which of the parameters differ and to estimate how large the
differences are.
The Analysis of Variance F-test
 Let’s say we want to test the null hypothesis that there are no differences among the 3 means:

The alternative hypothesis is that there is some difference:

The alternative hypothesis is no longer one-sided or two-sided. It is “many-sided,” because it allows any
relationship other than “all three equal.”

The test of H0 against HA is called the analysis of variance F test. (Don’t confuse the ANOVA F, which compares
several means, with the F statistic.)

Comparing several means is the simplest form of ANOVA, called one-way ANOVA, and is the only form we will
study.
o Example: We have three different groups (smokers, nonsmokers, and exposed to smoke) and we
measure the nicotine.
Why can’t we just test two hypothesis samples at a time?
Null Hypothesis:
Alternative Hypothesis:

F-Statistic:

Degrees of Freedom:
o

Example: Find the degrees of freedom for
Conditions:

The ANOVA F statistic has the form:


The measures of variation in the numerator and denominator of F are called mean squares.
You can find 𝑥̅ from the I sample means by

Mean Squares for Groups:

Mean Square for Error:

Because MSE is an average of the individual sample variances, it is also called the pooled sample variance,
written as .
The square root of MSE is the pooled standard deviation, sp.
The confidence interval for µi is:


One-Way ANOVA Examples: Check the appropriate assumptions and conditions and carry out the ANOVA test.
1. A small study was run to compare head injuries of car sizes.
Subcompact
Compact
Midsize
Full Size
Source
Group
Error
Total
n=5
n=5
n=5
n=5
DF
x = 668.8
x = 555.8
x = 486.8
x = 537.8
Sum of Square
88,425
475,323
s =242
s = 91
s =167.7
s = 154.6
Mean Square
F
P-Value
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2. The percentage of students taking state assessment tests was compared for samples of three different levels:
elementary, middle, high school.
Level
n
Mean
S.D
Elem
Middle
HS
49
17
12
70.98
62.72
46.33
18.25
18.44
14.09
Source
Group
Error
Total
DF
Sum of Square
5986
23604
Mean Square
F
P-Value
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3. Based on the preceding results, what do you conclude about the claim that the three age-group populations
have the same mean body temperature?
Source
Group
Error
Total
18-20
21-29
30 and older
98.0
98.4
97.7
98.5
97.1
99.6
98.2
99.0
98.2
97.9
98.6
98.6
97.0
97.5
97.3
DF
Sum of Square
Mean Square
F
P-Value
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