I'm sure I've got this completely wrapped round my head, but I just can't figure it out.
The t-test compares two normal distributions using the Z distribution. That's why there's an assumption of normality in the DATA.
ANOVA is equivalent to linear regression with dummy variables, and uses sums of squares, just like OLS. That's why there's an assumption of normality of RESIDUALS.
It's taken me several years, but I think I've finally grasped those basic facts. So why is it that the t-test is equivalent to ANOVA with two groups? How can they be equivalent if they don't even assume the same things about the data?
Best Answer
The t-test with two groups assumes that each group is normally distributed with the same variance (although the means may differ under the alternative hypothesis). That is equivalent to a regression with a dummy variable as the regression allows the mean of each group to differ but not the variance. Hence the residuals (equal to the data with the group means subtracted) have the same distribution --- that is, they are normally distributed with zero mean.
A t-test with unequal variances is not equivalent to a one-way ANOVA.