Simplified Independent Complex Samples T-Test - Comparing Two Means

Where $\bar{x}$ and $\bar{y}$ are the two means and $s_\bar{x}$ and $s_\bar{y}$ are their Standard Errors, the test statistic is:

$t=\frac{\bar{x}-\bar{y}}{\sqrt{s^2_\bar{x} + s^2_\bar{x}}}$,

where:

$p = 2\Pr(t_v \ge |t|)$,

$v = \frac{(s^2_\bar{x} +s^2_\bar{x})^2}{\frac{(s^2_\bar{x})^2}{n-1}+\frac{s^2_\bar{y})^2}{m-1} }$

the groups have sample sizes of $m$ and $n$ respectively.