I'm having trouble understanding how the 95% confidence interval equation for population proportions gets simplified.
Here is the initial equation: $$ Pr(\overline{X} - 2\hat{SE}(\overline{X}) ≤ p ≤ \overline{X} + 2\hat{SE}(\overline{X})) $$
Here is what I should get after simplifying (z-score in the middle):
$$ -2 ≤ {\overline{X} - p \over \hat{SE}(\overline{X})} ≤ 2 $$
My issue is that when I subtract $\overline{X}$ in the three parts of the equation and then divide by $\hat{SE}(\overline{X})$ again in all parts of the equation, I obtain $ p - \overline{X} \over \hat{SE}(\overline{X})$, which is not the formula for the z-score.
You can go from $$-2 ≤ {p-\overline{X} \over \hat{SE}(\overline{X})} ≤ 2$$ by multiplying everything by $-1$ (changing the direction of inequalities) to get $$2 \ge {\overline{X} - p \over \hat{SE}(\overline{X})} \ge -2$$ and then reordering to arrive at $$-2 ≤ {\overline{X} - p \over \hat{SE}(\overline{X})} ≤ 2$$