Question about two sample t-test (unpaired)

[chever]

I have read that the statistic computed for the unpaired two sample t-test is:

$t = \frac{\bar{x} - \bar{y}}{\sqrt{SEM_x + SEM_y}}$

where:

$SEM_x = \frac{\sigma^2_x}{n_x}$

(and likewise for y).

Part of this makes sense: it is satisfactorily proven to me that that $Var(\bar{x} - \bar{y}) = Var(\bar{x}) + Var(\bar{y})$ when the two variables are independent. Then the denominator is the standard deviation of the term $\bar{x} - \bar{y})$. What doesn't make sense is that the numerator isn't normalized. In the one sample t-test, one computes:

$t = \frac{\bar{x} - \mu_x}{\sqrt{SEM_x}}$

so, here, $\bar{x}$ is normalized with $\mu_x$. I don't see why this shouldn't also apply to the two-sample case. Can someone enlighten me?

 

[pmsrw3]

In the two-sample t-test, you're testing whether the two samples are different from each other; that is, whether the mean of the difference between them is zero. You can think of the numerator as having an implicit 0 subtracted from it, if you like.

 

[chever]

In the two-sample t-test, you're testing whether the two samples are different from each other; that is, whether the mean of the difference between them is zero. You can think of the numerator as having an implicit 0 subtracted from it, if you like.

That clarifies matters a bit. Thank you.