Difference of Two Population Proportions (Setup) - AP Statistics

Thresholds for deciding whether to reject $H_0$. Z-scores that define the rejection region boundaries.

Random samples, independent groups, large sample size. Required conditions for valid statistical inference.

$\hat{p}_1$ and $\hat{p}_2$. Observed proportions from each sample group.

Both $n_1\hat{p}_1$, $n_1(1-\hat{p}_1)$, $n_2\hat{p}_2$, $n_2(1-\hat{p}_2) > 5$. Ensures sufficient data for normal approximation validity.

Samples are not related or paired. No connection between observations in different groups.

Use the Z formula with pooled standard error. Apply the standardized test statistic formula.

1.96. Boundary value for 95% confidence in two-tailed tests.

Reject $H_0$. P-value is less than significance level.

Fail to reject $H_0$. P-value exceeds the significance level.

Fail to reject $H_0$. Zero difference is plausible, so no significant difference.

$\hat{p} = \frac{70}{250}$. Total successes divided by total sample size.

Area in tails of distribution where $H_0$ is rejected. Region where test statistic leads to $H_0$ rejection.

Larger samples lead to more reliable results. Larger samples reduce sampling variability and increase precision.

Strong evidence against $H_0$. Statistically significant result supporting the alternative hypothesis.

Decreases the standard error. Standard error decreases as sample size increases.

Insufficient evidence to reject $H_0$. Not enough evidence to conclude a significant difference.

Probability of rejecting $H_0$ when $H_a$ is true. Ability to detect a true difference when it exists.

Power increases. Larger samples improve the test's ability to detect differences.

Reducing one increases the other, given a fixed sample size. Trade-off relationship between the two types of errors.

Increases Type I error probability. Higher $\alpha$ means greater chance of false positive.

$H_0: p_1 - p_2 = 0$. Assumes no difference between the two population proportions.

$H_a: p_1 - p_2 \neq 0$. Tests if proportions differ in either direction (two-sided test).

$H_a: p_1 - p_2 > 0$. Tests if the first proportion is greater than the second.

$H_a: p_1 - p_2 < 0$. Tests if the first proportion is less than the second.

$SE = \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}$. Measures variability of the difference between sample proportions.

$\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$. Combines both samples to estimate common proportion under $H_0$.

Both $n_1$ and $n_2$ should be sufficiently large. Ensures normal approximation is valid.

$\alpha = 0.05$. Standard threshold for statistical significance.

To determine the threshold for rejecting $H_0$. Sets the cutoff for statistical significance decisions.