The following distinguishes TWO cases for large samples confidence interval for difference in means:
where Sp^2 is the pooled estimate of the common variance, n1 is the sample size from the first population, n2 is the sample size from the second population, and z_alpha/2 is 100(1-alpha/2) th percentile of the standard normal.
It seems to me that case 1 is a special case of case 2 with the population variances being equal. If this is the case, the formula for case 2 should reduce to the formula for case 1 when the population variances are equal. However, I have no way of seeing it being the case.
[aside: I am trying to cut down on the number of formulas that I have to memorize. Instead of two different formulas, if case 2 contains case 1, then I only have to memorize the general case 2 formula which is nice.]
Could somebody please show me how I can reduce case 2 to case 1?
Any help would be appreciated!