I've just received a result to an assessment on statistical methods I did a few days ago and for one of the questions I got the result that a t-test was incorrectly implemented. The feedback is quite vague and I just want to try and understand what I did wrong.
So the problem was: I have a dataset of random selection of people with $\beta$-Endorphin levels measured before workout, after and also the difference between pre- and post-workout levels is provided. I needed to test the hypothesis that $\beta$-Endorphin levels increase with exercise with 95% confidence level.
What I did: For my t-test I decided to go with the data on post-workout levels. My idea was to calculate the mean of post-workout levels and then perform a t-test to compare it with the distribution of the pre-workout data. My thinking was that if I find that it is highly unlikely (within the confidence level) for the post-workout mean to have come from the same distribution as the pre-workout data, then I can conclude that there is indeed a change. Also, since the question was to test whether $\beta$-Endorphin levels increase with exercise, I decided to perform a 1-sided t-test with the alternative hypothesis being that pre-workout mean is less than the post-workout.
What the marker commented: He or she wrote that I had to instead do the test using the data on the difference between pre- and post-workout levels and do a t-test using a $\mu=0$.
I've actually thought of doing it the way the marker considers correct, but in my head I thought that what both approaches are very similar. Can anyone please explain what I did wrong here?