question about two sample t-test (unpaired)

chever

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

[itex]t = \frac{\bar{x} - \bar{y}}{\sqrt{SEM_x + SEM_y}}[/itex]

where:

[itex]SEM_x = \frac{\sigma^2_x}{n_x}[/itex]

(and likewise for y).

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

[itex]t = \frac{\bar{x} - \mu_x}{\sqrt{SEM_x}}[/itex]

so, here, [itex]\bar{x}[/itex] is normalized with [itex]\mu_x[/itex]. 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

That clarifies matters a bit. Thank you.