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Stat 250 Gunderson Lecture Notes
Learning about the Difference in Population Proportions
Part 3: Testing about a Difference in Population Proportions
Chapter 12: Section 3 HT Module 2
12.3 HT Module 2: Testing Hypotheses about the
Difference in Two Population Proportions
 We have two populations or groups from which independent samples are available,
(or one population for which two groups can be formed using a categorical variable).
 The response variable is also categorical and we are interested in comparing the
proportions for the two populations.
Let p1 be the population proportion for the first population.
Let p2 be the population proportion for the second population.
Parameter: the difference in the population proportions p1 – p2.
Sample estimate: the difference in the sample proportions pˆ 1  pˆ 2 .
p1 (1  p1 ) p 2 (1  p 2 )

n1
n2
Standard deviation of pˆ 1  pˆ 2 : s.d.( pˆ 1  pˆ 2 ) 
Recall from Chapter 11 that the multiplier in the confidence interval was a z* value. So we will
be computing a Z test statistic for performing a significance test.
The standard error used in constructing the confidence interval for the difference between two
population proportions is not the same as that used for the standardized z test statistic.
We will need to construct the null standard error, the standard error for the statistic when the
null hypothesis is true. Let’s start with what the hypotheses will look like.
Possible null and alternative hypotheses.
1. H0:
versus Ha:
2. H0:
versus Ha:
3. H0:
versus Ha:
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Next we need to determine the test statistic and understand the conditions required for the
test to be valid. The general form of the test statistic is:
Test statistic = Sample statistic – Null value
Standard error
In the case of two population proportions, if the null hypothesis is true, we have p1 - p2 = 0 or
that the two population proportions are the same, p1 = p2 = p. What is a reasonable way to
estimate the common population proportion p?
The general standard error for pˆ 1  pˆ 2 is given by: s.e.( pˆ1  pˆ 2 ) 
pˆ1 (1  pˆ1 ) pˆ 2 (1  pˆ 2 )

n1
n2
but if the null hypothesis is true, then p̂ is the best estimate for each population proportion
and should be used in the standard error.
So, the null standard error for pˆ 1  pˆ 2 is given by:
And the corresponding test statistic is:
If the null hypothesis is true, this z-statistic will have a _____________ distribution.
This distribution is used to find the p-value for the test.
Conditions: This test requires that the sample proportions are based on independent random
samples from the two populations. Also, all of the quantities n1 pˆ , n1(1  pˆ ) , n2 pˆ , and n2 (1  pˆ )
be preferably at least 10. Note these are checked with the estimate of the common population
proportion p̂ .
From the Stat 250 Formula Card we have the following summary:
Two Population Proportions
Parameter
p1  p 2
Statistic
Large-Sample z-Test
pˆ 1  pˆ 2
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z
pˆ1  pˆ 2
1 1
pˆ (1  pˆ )  
 n1 n2 
where pˆ 
n1 pˆ1  n2 pˆ 2
n1  n2
Try It! Returning Lost Money $$
A random sample of college students was asked if they would return the money if they found a
wallet on the street. They were also asked to report their gender so a comparison could be
made between the two populations of male versus female college students.
Of the 93 women, 83 said they would, and of the 75 men, 53 said they would. Assume these
students represent all college students.
Test the hypothesis that equal proportions of college men and women would say they would
return the money versus the alternative hypothesis that a higher proportion of women would
do so. Use a 5% significance level and show all steps.
105
Additional Notes
A place to … jot down questions you may have and ask during office hours, take a few extra notes, write
out an extra practice problem or summary completed in lecture, create your own short summary about
this chapter.
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