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I have a multiple choice problem with 4 options, A, B, C, and D. I know the number of people choosing each of the four options. How should I test the null hypothesis that the probability of choosing A is the same as the probability of choosing B? Thanks!

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To address the null hypothesis, you want to use the counts of responses for each of the options.

To assess if the distribution across all options is equal, typical omnibus tests would be a chi-square goodness-of-fit test or a multinomial test.

If you want to compare A and B specifically, you could use either a chi-square goodness-of-fit test or a binomial test.

To compare among the 4 options, you could use the tests mentioned above in a pairwise fashion. But it might make more sense to construct confidence intervals for each option. Be sure to use a method that is applicable for a multinomial situation, such as the Sison-Glaz method.

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