P.Mean: Normality assumptions for the paired t-test (created 2008-10-14).
I am confused about which data have to be normally distributed on a paired t-test for testing that two data sets differ significantly. Everitt-Hothorn "A handbook of statistical analyses using R", page 33 says that the differences between the data should be normally distributed without implying anything about if the original data should be normally distributed, while Wiki t-test and Field "Discovering statistics using SPSS" page 287 imply that both of the original data should be normally distributed? Considering that I am a beginner in statistics, I am confused. can you give me any clues please?
Everitt-Hotthorn is right. In practice, it is unusual for the differences to be normal without the original variables not being normal, but it could happen. Your other two sources are too conservative. Their conditions would insure acceptability of the assumptions for the paired t-test, but they might rule out a paired t-test in situations where it is actually quite fine.
It's rather hard to visualize, but suppose you added a skewed random variable to a normally distributed random variable. That would almost certainly produce a skewed sum (though the skewness might be a bit less pronounced). If it is possible for normal plus skewed to produce skewed, then would it also be possible for a skewed distribution minus another skewed distribution to produce a normal distribution? Probably so, but I'd have a hard time proving this rigorously.
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