Say I had two distributions and I wanted to determine whether the mechanism behind these two distributions was statistically different.
We know that Student's T-test is pooled, meaning BOTH the distributions must have the same variance. Welch's T-Test is unpooled, meaning these two distributions do not have to be the same.
It seems that Welch's test would be more applicable in the real-world. In almost all cases I can think of, the variance would be different! When does it ever make sense to apply a pooled t-test like Student's t-test?
Should we always check if the variance of the two normal distributions is the same?