Paired Z-Test of Proportions

Where [math]\displaystyle{ g_1 }[/math] and [math]\displaystyle{ g_2 }[/math] are the two proportions and [math]\displaystyle{ s_{g_1 - g_2} }[/math] is an estimate of the standard error of the difference between the proportions:

[math]\displaystyle{ z=\frac{g_1-g_2}{\sigma_{g_1-g_2}} }[/math]

where:

[math]\displaystyle{ p \approx 2(1-\Phi(|z|)) }[/math] if [math]\displaystyle{ g_1 \ne g_2 }[/math] and NaN otherwise,

[math]\displaystyle{ s_{g_1 - g_2} }[/math] is computed as the Standard Error of [math]\displaystyle{ d }[/math],

the value for the [math]\displaystyle{ i }[/math]th observation is computed as [math]\displaystyle{ d_i = x_1 - x_2 }[/math],

[math]\displaystyle{ x_1,x_2 \in {\{0,1\}} }[/math] are the observed values on the two variables.