$\hat{p}_c = \frac{x_1 + x_2}{n_1 + n_2}$, where $x_1$ and $x_2$ are the number of successes, and $n_1$ and $n_2$ are the sample sizes.

$z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}_c(1-\hat{p}_c)(\frac{1}{n_1} + \frac{1}{n_2})}}$

Calculate: $n_1\hat{p}_c$, $n_1(1-\hat{p}_c)$, $n_2\hat{p}_c$, and $n_2(1-\hat{p}_c)$.

All must be greater than or equal to 10 to satisfy the Normal condition.

$z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$

Statement of no difference between population proportions: $H_0: p_1 = p_2$

Statement that there is a difference between population proportions ($H_a: p_1 > p_2$, $H_a: p_1 < p_2$, or $H_a: p_1 \neq p_2$).

$p_1$ and $p_2$ represent the population proportions of two different groups being compared.

A weighted average of the sample proportions, used when the null hypothesis assumes equal population proportions.

A procedure for measuring the strength of evidence against a null hypothesis.

Null Hypothesis: Assumes no difference between population proportions. | Alternative Hypothesis: Claims there is a significant difference between population proportions.

Type I error: Rejecting a true null hypothesis. | Type II error: Failing to reject a false null hypothesis.

One-sided: Tests for difference in a specific direction ($p_1 > p_2$ or $p_1 < p_2$). | Two-sided: Tests for any difference ($p_1 \neq p_2$).

Random Sampling: Selecting a sample randomly from a population. | Random Assignment: Assigning participants to treatment groups randomly in an experiment.

Z-test: Used when the population standard deviation is known or sample size is large. | T-test: Used when the population standard deviation is unknown and sample size is small.