I have one sample with n=170 and two binary variables (A,B) that can take as a value 1 or 0, where 1 counts as a success and 0 counts as a failure. What I want to know is whether the means of these two variables are equal.
To find this out I generate a new variable that takes the difference between these two variables called C, so C = B-A. I then compute the p-value for the hypothesis that C is normally distributed with the Shapiro-Wilk test and I find a p-value of .96, so I choose not to reject this hypothesis. Apart from that the difference is normally distributed, I am not worried about the other assumptions required for a paired t-test.
Question: Can I use the paired t-test in this circumstance or is it a mistake to use the Shapiro-Wilk test for binary data to check for normality and should I use the Wilcoxon sign rank test instead?
I would much prefer to use the t-test, because I believe it has a higher power than the Wilcoxon sign rank test, but that higher power pretty much does not matter if the test used is the wrong one.