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Difference of Successes

If $x_1/n_1$ and $x_2/n_2$ are the observed proportions from standard Normally Distributed samples with proportion of success $\theta$, then the probability that

w\equiv {x_1\over n_1}-{x_2\over n_2}
\end{displaymath} (1)

will be as great as observed is
P_\delta = 1-2\int_0^{\vert\delta\vert} \phi(t)\,dt,
\end{displaymath} (2)

$\displaystyle \delta$ $\textstyle \equiv$ $\displaystyle {w\over \sigma_w}$ (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)

© 1996-9 Eric W. Weisstein