I conducted a training session and for each of my users, I measured pre and post score (before and after training session). Each of my users (n=28) is measured twice, so my data is paired:
Data:
user pre-score post-score
1 10 30
....
Hypothesis: $$ H_0: \mu_{Post-Pre} \leq 0\\ H_1: \mu_{Post-Pre} > 0 $$
In order to check the hypothesis, whether the post-score is significantly higher than the pre-score (whether the training had a positive effect on users' score), I would conduct a paired t-test (t.test(Post, Pre, mu=0, conf.level=0.95, alt="greater")
in R).
Problem: The post data is not normally distributed. That's why I would go with a non-parametric test, i.e. Wilcoxon signed-rank, instead of the paired t-test.
However, if a calculate the difference between the pairs (Post-Pre), I end up with one sample which is normally distributed. So I would have to test the following hypothesis in a one sample t-test (t.test(Post-Pre, mu=0, conf.level=0.95, alt="greater")
in R):
$$
H_0: \mu_{Diff.} \leq 0\\
H_1: \mu_{Diff.} > 0\\
$$
Question: Which method is appropriate? Should I use the non-parametric paired Wilcoxon text because my post data is not normally distributed? Or can I work with the difference of the pairs (one sample) which is normally distributed and use a one sample t-test?