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.

Ensures the samples are representative of the populations, avoiding bias and allowing for generalization of results.

Ensures that the observations within each sample are independent of one another. Often checked using the 10% condition.

To ensure that the sampling distribution of the difference in sample proportions is approximately normal, allowing us to use the z-distribution for inference.

The probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true.

Rejecting the null hypothesis means there is sufficient evidence to support the alternative hypothesis. The observed difference is statistically significant.

$\hat{p}_c = \frac{x_1 + x_2}{n_1 + n_2}$, 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.

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