Confidence Interval for proportions

For a Single Population Proportion:

Type	Symbol	Description
Population Proportion	$p$	The true proportion you're estimating
Sample Proportion	$\hat{p}$	Number of successes / sample size

\hat{p} = \frac{number of successes}{n}

Find in standard error, but for proportion:

S E = \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}

\hat{p} \pm z^{*} \times S E

"We are [confidence level]% confident that the true population proportion is between [lower bound] and [upper bound]."

Condition	Requirement	Notes
Random Samples	Both groups randomly sampled	Essential for inference
Independence	Groups must be independent	No overlap between groups
Success/Failure	Check for both groups	All must be $\geq 10$

Find in standard error, but for difference in proportions:

S E = \sqrt{\frac{{\hat{p}}_{1} (1 - {\hat{p}}_{1})}{n_{1}} + \frac{{\hat{p}}_{2} (1 - {\hat{p}}_{2})}{n_{2}}}

({\hat{p}}_{1} - {\hat{p}}_{2}) \pm z^{*} \times S E

Result	Interpretation
Positive Interval	$p_{1}$ likely larger than $p_{2}$
Negative Interval	$p_{1}$ likely smaller than $p_{2}$
Contains Zero	No significant difference detected

Component	Single Proportion	Difference in Proportions
Point Estimate	$\hat{p}$	${\hat{p}}_{1} - {\hat{p}}_{2}$
Standard Error	$\sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}$	$\sqrt{\frac{{\hat{p}}_{1} (1 - {\hat{p}}_{1})}{n_{1}} + \frac{{\hat{p}}_{2} (1 - {\hat{p}}_{2})}{n_{2}}}$
Critical Value	$z^{*}$	$z^{*}$
Conditions	$n \hat{p} \geq 10$ and $n (1 - \hat{p}) \geq 10$	Check both groups separately