Comparing mean difference between groups where raw data follows a long tail distribution I am trying to determine if there is a mean difference between two test groups. The raw data (continuous scores) is distributed according to a long tail distribution.
Each test group has more than 10K observations.
After a lot of reading I concluded that I can probably use the Welch t-test to calculate a 95% confidence interval for the mean difference. 
Is this correct or am I violating any important assumptions by doing that?
Cheers,
Marcus
 A: As with other T-tests, Welch's T-test hinges on the use of the central limit theorem to ensure that the sample means of the two samples are (approximately) normally distributed.  This is what gets us to the (approximate) T-distribution for the test statistic.  Now, it is not necessary for your samples to be normally distributed, but they must meet the requirements of one of the central limit theorems so that the sample means are normally distributed.  (See further discussion on this issue in comments below.)
To meet the requirements of the classical CLT (the Lindberg-Lévy theorem) the two populations must each have finite variance, and this would exclude some long-tailed distributions.  Finite variance requires that the tails of the distribution each decrease faster than cubic decay.  To check if your distributions have tails that decrease faster than cubic decay (and therefore have finite variance) you should construct a tail plot for each sample, and compare this to a line showing cubic decay; see e.g., this related answer.  If you are unable to establish the conditions for the CLT, and it looks like they might not hold, then the requirements of the T-test may not be valid, and the test may give poor results.
A: This video shows a clear explanation and simple explanation of comparing two means. Three assumptions are made for this tool:


*

*Homogeneity of variance

*Normal distribution of populations (according to you, this is your case)

*Each value is sampled independently
If you can assume those three:yes, you can use Welch's t-test or student's t-test
