Simplified Independent Complex Samples T-Test - Comparing Two Means

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

$t=\frac{\overline{x}-\overline{y}}{\sqrt{s_{\overline{x}}^2+s_{\overline{x}}^2}}$,

where:

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

$v=\frac{(s_{\overline{x}}^2+s_{\overline{x}}^2{)}^2}{\frac{(s_{\overline{x}}^2{)}^2}{n-1}+\frac{s_{\overline{y}}^2{)}^2}{m-1}}$

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