# Incorrect implementation of a t-test

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?

• Thanks Dave! So to clarify - the differences between pre- and post-workout levels came from same subjects too. So let me see if I understand it correctly - it was wrong from me to be using the $\mu$ of post-workout data to compare against pre-workout distribution because datasets are related, but if we use differences, even though they came from same subjects, it is fine since we'd be comparing them to $\mu=0$, which is independent of either dataset? Also, was my approach completely incorrect or was the second approach simply better? Aug 17 '20 at 2:50