Population Comparison

Let and be the number of successes in variates taken from two populations. Define

$\displaystyle \hat p_1$	$\textstyle \equiv$	$\displaystyle {x_1\over n_1}$	(1)
$\displaystyle \hat p_2$	$\textstyle \equiv$	$\displaystyle {x_2\over n_2}.$	(2)

The Estimator of the difference is then $\hat p_1-\hat p_2$ . Doing a z-Transform,

$\begin{displaymath} z={(\hat p_1-\hat p_2)-(p_1-p_2)\over \sigma_{\hat p_1-\hat p_2}}, \end{displaymath}$

(3)

where

$\begin{displaymath} \sigma_{\hat p_1-\hat p_2}\equiv \sqrt{{\sigma_{\hat p_1}}^2-{\sigma_{\hat p_2}}^2}. \end{displaymath}$

(4)

$\displaystyle {\rm SE}_{\hat p_1-\hat p_2}$	$\textstyle =$	$\displaystyle \sqrt{{\hat p_1(1-\hat p_1)\over n_1}+{\hat p_2(1-\hat p_2)\over n_2}}$	(5)
$\displaystyle {\rm SE}_{\bar x_1-\bar x_2}$	$\textstyle =$	$\displaystyle \sqrt{{{s_1}^2\over n_1}+{{s_2}^2\over n_2}}$	(6)
$\displaystyle {s_{\rm pool}}^2$	$\textstyle =$	$\displaystyle {(n_1-1){s_1}^2+(n_2-1){s_2}^2\over n_1+n_2-2}.$	(7)

See also z-Transform