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Introduction
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When we have a categorical variable of interest measured in two populations, it is quite often that we are interested in comparing the proportions of a certain category for the two populations.

Let’s consider the following example.

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Application

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$100 Received by Mistake
Section* *

Males and females were asked about what they would do if they received a $100 bill by mail, addressed to their neighbor, but wrongly delivered to them. Would they return it to their neighbor? Of the 69 males sampled, 52 said "yes" and of the 131 females sampled, 120 said "yes."

Does the data indicate that the proportions that said "yes" are different for male and female? How do we begin to answer this question?

If the proportion of males who said “yes, they would return it” is denoted as \(p_1\) and the proportion of females who said “yes, they would return it” is denoted as \(p_2\), then the following equations indicate that \(p_1\) is equal to \(p_2\).

\(p_1-p_2=0\) or \(\dfrac{p_1}{p_2}=1\)

We would need to develop a confidence interval or perform a hypothesis test for one of these expressions.

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Moving forward
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There may be other ways of setting up these equations such that the proportions are equal. We choose the difference due to the theory discussed in the last section. Under certain conditions, the sampling distribution of \(\hat{p}_1\), for example, is approximately normal and centered around \(p_1\). Similarly, the sampling distribution of \(\hat{p}_2\) is approximately normal and centered around \(p_2\). Their difference, \(\hat{p}_1-\hat{p}_2\), will then be approximately normal and centered around \(p_1-p_2\), which we can use to determine if there is a difference.

In the next subsections, we explain how to use this idea to develop a confidence interval and hypothesis tests for \(p_1-p_2\).