Difference of Successes

If and are the observed proportions from standard Normally Distributed samples with proportion of success $\theta$ , then the probability that

$\begin{displaymath} w\equiv {x_1\over n_1}-{x_2\over n_2} \end{displaymath}$

(1)

will be as great as observed is

$\begin{displaymath} P_\delta = 1-2\int_0^{\vert\delta\vert} \phi(t)\,dt, \end{displaymath}$

(2)

where

$\displaystyle \delta$	$\textstyle \equiv$	$\displaystyle {w\over \sigma_w}$	(3)
$\displaystyle \sigma_w$	$\textstyle \equiv$	$\displaystyle \sqrt{\hat\theta (1-\hat\theta )\left({{1\over n_1}+{1\over n_2}}\right)}$	(4)
$\displaystyle \hat\theta$	$\textstyle \equiv$	$\displaystyle {x_1+x_2\over n_1+n_2}.$	(5)

Here, $\hat\theta$ is the Unbiased Estimator. The Skewness and Kurtosis of this distribution are

$\displaystyle {\gamma_1}^2$	$\textstyle =$	$\displaystyle {(n_1-n_2)^2\over n_1n_2(n_1+n_2)} {1-4\hat\theta (1-\hat\theta )\over \hat\theta (1-\hat\theta )}$	(6)
$\displaystyle {\gamma_2}$	$\textstyle =$	$\displaystyle {{n_1}^2-n_1n_2+{n_2}^2\over n_1n_2(n_1+n_2)} {1-6\hat\theta(1-\hat\theta)\over \hat\theta (1-\hat\theta)}.$	(7)