Z formula

Sigma (standard deviation) is the square root of variance so you could restate the first formula as:

z = (x1-x2)/sqrt(var1/n1)

This looks a lot like the second formula. For two processes, variances are additive, standard deviations are not. What that second formula does is show you how to add the variances to get the overall variance.

Let me see if I understand.
If var1 not equal var 2,the formula sums the two variances.
The new t distribution then is a dist where sigma = sqrt (sum of 2 var.)
This kinda Z test solves for P (Z < Z test) where var 1 not = var 2;
while regular Z test solves for P( Z < Z test where var 1 DOES = var 2.
Am I close?
Thanks;
joe b.

Yes, pretty close. If Var1=Var2, then you essentially have one big distribution. The second formula allows you to merge two distributions with different variances.

This s sorta a weighted average, sqrt (var 1/n1 + var 2 / n2), but why not

sqrt((var1*n1 + var2*n2)/n1+n2) ? Maybe it's me, it ain't clicking yet.
Thanks;
joe b.