I have data from an experiment where I applied two different treatments in identical initial conditions, producing an integer between 0 and 500 in each case as the outcome. I want to use a paired t-test to determine whether the effects produced by the two treatments are significantly different. The outcomes for each treatment group are normally distributed, but the difference between each pair is not normally distributed (asymmetric + one long tail).
Can I use a paired t-test in this case, or is the assumption of normality violated, meaning I should use a non-parametric test of some kind?