Chapter 22
Comparing Two
Proportions
Copyright © 2010 Pearson Education, Inc.
A county health department tries an experiment
using several hundred volunteers who were
planning to use a nicotine patch to help quit
smoking. The subjects were split into two groups.
Group 1 were given the patch and attended a
weekly discussion support group, Group 2 just got
the patch. After six months, 46 of 143 people in
Group 1 and 30 of 151 people in Group 2 has
successfully stopped smoking. Do these results
suggest that such support groups could be an
effective way to help people stop smoking?
Copyright © 2010 Pearson Education, Inc.
Slide 22 - 3
Comparing Two Proportions


Comparisons between two percentages are much
more common than questions about isolated
percentages. And they are more interesting.
We often want to know how two groups differ,
whether a treatment is better than a placebo
control, or whether this year’s results are better
than last year’s.
Copyright © 2010 Pearson Education, Inc.
Slide 22 - 4
Another Ruler


In order to examine the difference between two
proportions, we need another ruler—the standard
deviation of the sampling distribution model for
the difference between two proportions.
Recall that standard deviations don’t add, but
variances do. In fact, the variance of the sum or
difference of two independent random quantities
is the sum of their individual variances.
Copyright © 2010 Pearson Education, Inc.
Slide 22 - 5
The Standard Deviation of the Difference Between
Two Proportions


Proportions observed in independent random
samples are independent. Thus, we can add their
variances. So…
The standard deviation of the difference between
two sample proportions is
SD  pˆ1  pˆ 2  

p1q1 p2 q2

n1
n2
Thus, the standard error is
SE  pˆ1  pˆ 2  
Copyright © 2010 Pearson Education, Inc.
pˆ1qˆ1 pˆ 2 qˆ2

n1
n2
Slide 22 - 6


A survey of 886 randomly selected teenagers (1217) found that more than half of them have online
profiles. There appear to be differences between
boys and girls in their online behavior. Among
teens aged 15-17, 57% of the 248 boys had
online profiles, compared to 70% of the 256 girls.
If we want to estimate how large the difference
truly is, first calculate the standard error of the
sample proportions.
Copyright © 2010 Pearson Education, Inc.
Slide 22 - 7
Assumptions and Conditions

Independence Assumptions:
 Randomization Condition: The data in each
group should be drawn independently and at
random from a homogeneous population or
generated by a randomized comparative
experiment.
 The 10% Condition: If the data are sampled
without replacement, the sample should not
exceed 10% of the population.
 Independent Groups Assumption: The two
groups we’re comparing must be independent
of each other.
Copyright © 2010 Pearson Education, Inc.
Slide 22 - 8
Assumptions and Conditions (cont.)

Sample Size Condition:
 Each of the groups must be big enough…
 Success/Failure Condition: Both groups are big
enough that at least 10 successes and at least
10 failures have been observed in each.
Copyright © 2010 Pearson Education, Inc.
Slide 22 - 9



Among a random sample of teens aged 15-17,
57% of the 248 boys had online profiles,
compared to 70% of the 256 girls.
Can we use these results to make inferences
about all 15-17 year olds?
What are the assumptions and conditions?
Copyright © 2010 Pearson Education, Inc.
Slide 22 - 10
The Sampling Distribution


We already know that for large enough samples,
each of our proportions has an approximately
Normal sampling distribution.
The same is true of their difference.
Copyright © 2010 Pearson Education, Inc.
Slide 22 - 11
The Sampling Distribution (cont.)

Provided that the sampled values are
independent, the samples are independent, and
the samples sizes are large enough, the sampling
distribution of pˆ1  pˆ 2 is modeled by a Normal
model with
 Mean:
  p1  p2

Standard deviation:
SD  pˆ1  pˆ 2  
Copyright © 2010 Pearson Education, Inc.
p1q1 p2 q2

n1
n2
Slide 22 - 12
Two-Proportion z-Interval


When the conditions are met, we are ready to find
the confidence interval for the difference of two
proportions:
The confidence interval is
 pˆ1  pˆ 2   z

where
SE  pˆ1  pˆ 2  

 SE  pˆ1  pˆ 2 
pˆ1qˆ1 pˆ 2 qˆ2

n1
n2
The critical value z* depends on the particular
confidence level, C, that you specify.
Copyright © 2010 Pearson Education, Inc.
Slide 22 - 13



Among a random sample of teens aged 15-17,
57% of the 248 boys had online profiles,
compared to 70% of the 256 girls. We calculated
the SE for the difference in sample proportions to
be SE(𝑝girls - 𝑝boys ) = 0.0425 and found that the
assumptions and conditions have been met.
Construct a 95% confidence interval about the
difference in online behavior.
Explain your result in context.
Copyright © 2010 Pearson Education, Inc.
Slide 22 - 14

A Gallup poll asked whether the attribute
“intelligent” applied to men in general. The poll
revealed that 28% of 506 men thought it did, but
only 14% of 250 women agreed. We want to
estimate the true size of the gender gap by
creating a 95% confidence interval.
Copyright © 2010 Pearson Education, Inc.
Slide 22 - 15


A charity looking for donations runs a test to see if they
will be more effective soliciting donations by email or
regular mail. They send the same letter to two different
random groups of people and received donations 26% of
the time from the group that received an email, and 15%
from those who received the request by regular mail. A
90% confidence interval estimated the difference in
donation rates to be 11% ± 7%
Interpret this confidence interval in context.
Copyright © 2010 Pearson Education, Inc.
Slide 22 - 16
What Can Go Wrong?



Don’t use two-sample proportion methods when
the samples aren’t independent.
 These methods give wrong answers when the
independence assumption is violated.
Don’t apply inference methods when there was
no randomization.
 Our data must come from representative
random samples or from a properly
randomized experiment.
Don’t interpret a significant difference in
proportions causally.
 Be careful not to jump to conclusions about
causality.
Copyright © 2010 Pearson Education, Inc.
Slide 22 - 17
What have we learned?


We’ve now looked at inference for the difference
in two proportions.
Perhaps the most important thing to remember is
that the concepts and interpretations are
essentially the same—only the mechanics have
changed slightly.
Copyright © 2010 Pearson Education, Inc.
Slide 22 - 18