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Population Comparison

Let $x_1$ and $x_2$ 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,
z={(\hat p_1-\hat p_2)-(p_1-p_2)\over \sigma_{\hat p_1-\hat p_2}},
\end{displaymath} (3)

\sigma_{\hat p_1-\hat p_2}\equiv \sqrt{{\sigma_{\hat p_1}}^2-{\sigma_{\hat p_2}}^2}.
\end{displaymath} (4)

The Standard Error is
$\displaystyle {\rm SE}_{\hat p_1-\hat p_2}$ $\textstyle =$ $\displaystyle \sqrt{{\hat p_1(1-\hat p_1)\over n_1}+{\hat p_2(1-\hat p_2)\over n_2}}$ (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

© 1996-9 Eric W. Weisstein