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:

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

where:

$p=2\Pr(t_{v}\geq |t|)$,

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

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