Say we have a large sample size for each of the two groups, so that the central limit theorem can be applied and thus t-test to compare the two groups means can be justified.
Say group mean difference of the variable A was proved to be significantly different from zero in the t-test at alpha=0.05. This means the group means of variable A are significantly different between the two groups.
Is it possible that, when I draw the ROC curve of the variable A, the ROC curve of A will be close to or below the 45-degree line x=y? I wouldn't think this will happen since if the group means of variable A are significantly different between the two groups, that means A can be a good variable to use to distinguish between the two groups.
If such phenomenon is possible to take place, what can be the reason for it?