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Chapter 10
Comparing Two Groups
Section 10.3
Other Ways of Comparing Means and
Comparing Proportions
Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Alternative Method for Comparing Means
An alternative t-method can be used when, under the null
hypothesis, it is reasonable to expect the variability as well
as the mean to be the same.
This method requires the assumption that the population
standard deviations be equal.
Figure 10.7 Two Groups With Equal Population Standard Deviations. Question:
What would a graph of the sample data look like to make you doubt this assumption?
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
The Pooled Standard Deviation
This alternative method estimates the common value
1
and  1 by:
(
n

1
)
s
(
n

1
)
s
s

n

n

2
2
1
1
1
4
2
2
2
Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
2

of
SUMMARY: Comparing Population Means,
Assuming Equal Population Standard
Deviations
Using the pooled standard deviation estimate, a 95% CI
for ( 1  2 ) is:
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(
x

x
)
ts 
nn
1
2
.025
1
This method has
5
2
df  n1  n2  2
Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
SUMMARY: Comparing Population Means,
Assuming Equal Population Standard
Deviations
The test statistic for H 0 : 1  2 is:
(x  x )
t
1 1
s

n n
1
2
1
This method has
6
2
df  n1  n2  2
Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
SUMMARY: Comparing Population Means,
Assuming Equal Population Standard
Deviations
These methods assume:
 Independent random samples from the two groups.
 An approximately normal population distribution for each
group. (This is mainly important for small sample sizes,
and even then, the CI and the two-sided test are usually
robust to violations of this assumption).
1   2 (In practice, this type of inference is not usually
relied on if one sample standard deviation is more than
double the other one).
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Example: Arthroscopic Surgery
Calculate the P-Value and determine if there is a statistical difference
between Arthroscopic surgery and Placebo at 5% level of significance.
Table 10.7 Summary of Knee Pain Scores The descriptive statistics compare
lavage and debridement arthroscopic surgery to a placebo (fake surgery) treatment.
With a P-value of 0.63, we should not reject the null that there is
no difference between placebo and Arthroscopic surgery.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Example: Arthroscopic Surgery
Calculate a 95% Confidence Interval.
Table 10.8 MINITAB Output for Comparing the Mean Knee Pain for Placebo and
Arthroscopic Surgery Groups.
We are 95% Confident that the difference between the placebo and
surgery is in this range -10.6 to 6.4.
Notice that 0 is within this range. Thus, we should not reject the null
hypothesis at the 5% significance level that there is no difference
between the two treatment groups.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
The Ratio of Proportions: The Relative Risk
The ratio of proportions for two groups is:
pˆ
1
pˆ
2
In medical applications for which the proportion refers to a
category that is an undesirable outcome, such as death or
having a heart attack, this ratio is called the relative risk.
The ratio describes the sizes of the proportions relative to
each other.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
The Ratio of Proportions: The Relative
Risk
Recall Physician’s Health Study:
pˆ 1  0.030 for placebo
pˆ 2  0.023 for aspirin
sample relative risk = pˆ 1 pˆ 2  0.030 0.023  1.30
This means that the proportion of the placebo group who had a
cancer death was 1.30 times the proportion of the aspirin group
who had a cancer death.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.