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.