Paired Z-Test of Proportions

Where $g_1$ and $g_2$ are the two proportions and $s_{g_1-g_2}$ is an estimate of the standard error of the difference between the proportions:

$z=\frac{g_1-g_2}{{\sigma }_{g_1-g_2}}$

where:

$p\approx 2(1-\mathrm{\Phi }(|z|))$ if $g_1\ne g_2$ and NaN otherwise,

$s_{g_1-g_2}$ is computed as the Standard Error of $d$,

the value for the $i$th observation is computed as $d_i=x_1-x_2$,

$x_1,x_2\in \{0,1\}$ are the observed values on the two variables.