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Is there a test I can use to compare two proportions from the same sample that are not mutually exclusive?

For example, I would like see if the proportion of people that answered "yes" to a particular question is significantly different from the proportion of people that answered "yes" to a different question.

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The within-subjects test of equality of proportions is McNemar's test. I discuss it fairly thoroughly here: What is the difference between McNemar's test and the chi-squared test, and how do you know when to use each? The answer also provides an example demo in R, should you use that, but it should be easy to do in any software.

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You could state as your null hypothesis the possibility that both answer statistics are drawn from the same binomial distribution, for the binomial that maximizes the chance that they are drawn from the same binomial.

I think @gung provides the better answer here. Note that the McNemar's test wikipedia article describes the use of the exact binomial sign test for small sample size (assuming equal population for both questions).

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McNemar's test is equivalent to a binomial test, but a very specific one. The details are spelled out in my linked answer. – gung Mar 17 '14 at 19:29

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