Simplified Independent Complex Samples T-Test - Comparing Two Means

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Where [math]\displaystyle{ \bar{x} }[/math] and [math]\displaystyle{ \bar{y} }[/math] are the two means and [math]\displaystyle{ s_\bar{x} }[/math] and [math]\displaystyle{ s_\bar{y} }[/math] are their Standard Errors, the test statistic is:

[math]\displaystyle{ t=\frac{\bar{x}-\bar{y}}{\sqrt{s^2_\bar{x} + s^2_\bar{x}}} }[/math],


[math]\displaystyle{ p = 2\Pr(t_v \ge |t|) }[/math],
[math]\displaystyle{ 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} } }[/math]
the groups have sample sizes of [math]\displaystyle{ m }[/math] and [math]\displaystyle{ n }[/math] respectively.