Quote Originally Posted by jessica01 View Post
Does the Central Limit Theorem apply? If so, the data is normally distributed. You then verify this by checking the data. So either do not do any testing at all, or do it only after you think about whether CLT applies.
I think we had this one pretty well wrapped up already but I'll comment on a few things you bring up. The CLT almost definitely applies with a sample size this large as long as the data isn't distributed as a Cauchy.

The CLT says nothing about the data being normally distributed. It's a common misconception that we need the data to be normally distributed to use a t-test. That's not what we need and it's not even what the CLT gives us. The assumption of the t-test is that the sampling distribution of the mean is normally distributed which is what the CLT gives us under some fairly loose conditions.

But if we didn't have that then the nonparametric tests could still apply so having data that isn't normally distributed isn't a reason to not do testing.