# Independent Samples - Quantum Column Proportions Test

Where $\displaystyle{ p_1 }$ and $\displaystyle{ p_2 }$ are the two proportions and $\displaystyle{ e_1 }$ and $\displaystyle{ e_2 }$ are the effective sample sizes and $\displaystyle{ \pi_1 }$ and $\displaystyle{ \pi_2 }$ are the weighted sample sizes:
$\displaystyle{ t=\frac{p_1-p_2}{\sqrt{p_{12}(1-p_{12})(e^{-1}_1+e^{-1}_2)(\frac{e_1+e_2}{e_1+e_2-1})}} }$, where $\displaystyle{ p_{12}=\frac{\pi_1 p_1 + \pi_2 p_2}{\pi_1+\pi_2} }$
where $\displaystyle{ p \approx 2 \Pr(t_{e_1+e_2-1} \ge |t|) }$.