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Chapter 14
Comparing Groups: Analysis
of Variance Methods
Section 14.1
One-Way ANOVA: Comparing Several Means
Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Analysis of Variance
The analysis of variance method compares means of
several groups.
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
Let g denote the number of groups.

Each group has a corresponding population of
subjects.

The means of the response variable for the g
populations are denoted by m1 , m2 , ...m g .
Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Hypotheses and Assumptions for the
ANOVA Test Comparing Means
The analysis of variance is a significance test of the
null hypothesis of equal population means:
 H 0 : 1  2  ...   g
The alternative hypothesis is:

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Ha
: at least two of the population means are
unequal.
Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Hypotheses and Assumptions for the ANOVA
Test Comparing Means
The assumptions for the ANOVA test comparing
population means are as follows:
 The population distributions of the response
variable for the g groups are normal with the same
standard deviation for each group.
 Randomization (depends on data collection
method):
 In a survey sample, independent random
samples are selected from each of the g
populations.
 For an experiment, subjects are randomly
assigned separately to the g groups.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Example: Tolerance of Being on Hold?
An airline has a toll-free telephone number for
reservations. Often the call volume is heavy, and callers
are placed on hold until a reservation agent is free to
answer.
The airline hopes a caller remains on hold until the call
is answered, so as not to lose a potential customer.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Example: Tolerance of Being on Hold?
The airline recently conducted a randomized
experiment to analyze whether callers would remain on
hold longer, on the average, if they heard:



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An advertisement about the airline and its current
promotion
Muzak (“elevator music”)
Classical music
Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Example: Tolerance of Being on Hold?
The company randomly selected one out of every
1000 calls in a week.
For each call, they randomly selected one of the three
recorded messages.
They measured the number of minutes that the caller
stayed on hold before hanging up (these calls were
purposely not answered).
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Example: Tolerance of Being on Hold?
Table 14.1 Telephone Holding Times by Type of Recorded Message. Each
observation is the number of minutes a caller remained on hold before hanging up,
rounded to the nearest minute.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Example: Tolerance of Being on Hold?
Denote the holding time means for the populations that
these three random samples represent by:



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1 = mean for the advertisement
2 = mean for the Muzak
3 = mean for the classical music
Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Example: Tolerance of Being on Hold?
The hypotheses for the ANOVA test are:
 H 0 : 1  2  3

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Ha
: at least two of the population means are
different
Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Example: Tolerance of Being on Hold?
Here is a display of the sample means:
Figure 14.1 Sample Means of Telephone Holding Times for Callers Who Hear
One of Three Recorded Messages. Question: Since the sample means are quite
different, can we conclude that the population means differ?
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Example: Tolerance of Being on Hold?
As you can see from the output on the previous page,
the sample means are quite different. But even if the
population means are equal, we expect the sample
means to differ some because of sampling variability.
This alone is not sufficient evidence to enable us to
reject H 0 .
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Variability Between Groups and Within
Groups Is the Key to Significance
The ANOVA method is used to compare population
means.
It is called analysis of variance because it uses
evidence about two types of variability.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Variability Between Groups and Within
Groups Is the Key to Significance
Two examples of data sets with equal means but unequal variability:
Figure 14.2 Data from Table 14.1 in Figure 14.2a and Hypothetical Data in
Figure 14.2b That Have the Same Means but Less Variability Within Groups
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Variability Between Groups and Within
Groups Is the Key to Significance
Which case do you think gives stronger evidence against
H 0 : 1  2  3
?
What is the difference between the data in these two
cases?
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Variability Between Groups and Within
Groups Is the Key to Significance
In both cases the variability between pairs of means
is the same.
In ‘Case b’ the variability within each sample is
much smaller than in ‘Case a.’
The fact that ‘Case b’ has less variability within each
sample gives stronger evidence against H 0 .
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
ANOVA F-Test Statistic
The analysis of variance (ANOVA) F-test statistic is:
Between groups variability
F=
Within groups variability
The larger the variability between groups relative to
the variability within groups, the larger the F test statistic
tends to be.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
ANOVA F-Test Statistic
The test statistic for comparing means has the
F-distribution.
The larger the F-test statistic value, the stronger the
evidence against H 0 .
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
SUMMARY: ANOVA F-test for Comparing
Population Means of Several Groups
1. Assumptions:
 Independent random samples
 Normal population distributions with equal
standard deviations
2. Hypotheses:
 H 0 : 1  2  ...   g
 H a : at least two of the population means are
different
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
SUMMARY: ANOVA F-test for Comparing
Population Means of Several Groups
3. Test statistic:
Between groups variability
F=
Within groups variability
 F- sampling distribution has df1  g  1 ,
df 2  N  g  (total sample size – no. of groups)
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
SUMMARY: ANOVA F-test for Comparing
Population Means of Several Groups
4. P-value: Right-tail probability above the observed
F- value
5. Conclusion: If decision is needed, reject
P-value  significance level (such as 0.05)
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
H0
if
The Variance Estimates and the ANOVA Table
Let  denote the standard deviation for each of the g
population distributions
 One assumption for the ANOVA F-test is that each
population has the same standard deviation,  .

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The F-test statistic is the ratio of two estimates of
the population variance for each group.
Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
,
2
The Variance Estimates and the ANOVA
Table
24

The estimate of  in the denominator of the F-test
statistic uses the variability within each group.

The estimate of  in the numerator of the F-test
statistic uses the variability between each sample
mean and the overall mean for all the data.
2
2
Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
The Variance Estimates and the ANOVA
Table
25


Computer software displays the two estimates of
in the ANOVA table similar to tables displayed in
regression.

The MS column contains the two estimates, which
are called mean squares.

The ratio of the two mean squares is the F- test
statistic.

This F- statistic has a P-value.
Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
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Example: Telephone Holding Times
This example is a continuation of a previous
example in which an airline conducted a randomized
experiment to analyze whether callers would remain on
hold longer, on the average, if they heard:



26
An advertisement about the airline and its current
promotion
Muzak (“elevator music”)
Classical music
Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Example: Telephone Holding Times
Denote the holding time means for the populations that
these three random samples represent by:



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1 = mean for the advertisement
2 = mean for the Muzak
3 = mean for the classical music
Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Example: Telephone Holding Times
The hypotheses for the ANOVA test are:

H 0 : 1  2  3

H a : at least two of the population means are
different
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Example: Telephone Holding Times
Table 14.2 ANOVA Table for F Test Using Data From Table 14.1
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Example: Telephone Holding Times
Since P-value < 0.05, there is sufficient evidence to
reject H 0 : 1  2  3
.
We conclude that a difference exists among the three
types of messages in the population mean amount of
time that customers are willing to remain on hold.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Assumptions and the Effects of Violating Them
1. Population distributions are normal
 Moderate violations of the normality assumption are
not serious.
2. These distributions have the same standard deviation
 Moderate violations are also not serious.
3. The data resulted from randomization.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Assumptions and the Effects of Violating
Them
You can construct box plots or dot plots for the sample
data distributions to check for extreme violations of
normality.
Misleading results may occur with the F-test if the
distributions are highly skewed and the sample size N is
small.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Assumptions and the Effects of Violating
Them
 Misleading results may also occur with the F-test if
there are relatively large differences among the
standard deviations (the largest sample standard
deviation being more than double the smallest one).
 The ANOVA methods presented here are for
independent samples. For dependent methods, other
techniques must be used.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Using One F Test or Several t Tests to
Compare the Means
Why Not Use Multiple t-tests?
 When there are several groups, using the F test
instead of multiple t tests allows us to control the
probability of a type I error.
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
If separate t tests are used, the significance level
applies to each individual comparison, not the overall
type I error rate for all the comparisons.

However, the F test does not tell us which groups
differ or how different they are.
Copyright © 2013, 2009, and 2007, Pearson Education, Inc.