Significance Tests for Proportions

One-Sample z-Test for $p$

Hypotheses:

• $H_0: p = p_0$
• $H_a: p \neq p_0$ (or $>$ or $<$)

Test Statistic: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

Note: We use $p_0$ (not $\hat{p}$) in the standard error because we assume $H_0$ is true.

Conditions:

1. Random sample or randomized experiment
2. 10% Condition: $n < 0.10N$
3. Large Counts: $np_0 \geq 10$ and $n(1-p_0) \geq 10$

Two-Sample z-Test for $p_1 - p_2$

Hypotheses:

• $H_0: p_1 = p_2$ (equivalently, $p_1 - p_2 = 0$)
• $H_a: p_1 \neq p_2$ (or $>$ or $<$)

Pooled Proportion (used because $H_0$ assumes $p_1 = p_2$): $\hat{p}_c = \frac{x_1 + x_2}{n_1 + n_2}$

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

Conditions (check for BOTH samples):

1. Random: Both samples are random/independent
2. 10%: $n_1 < 0.10N_1$ and $n_2 < 0.10N_2$
3. Large Counts: $n_1\hat{p}_c \geq 10$, $n_1(1-\hat{p}_c) \geq 10$, $n_2\hat{p}_c \geq 10$, $n_2(1-\hat{p}_c) \geq 10$

Finding the P-Value

• Right-tailed ($H_a: p > p_0$): $P(Z > z)$
• Left-tailed ($H_a: p < p_0$): $P(Z < z)$
• Two-tailed ($H_a: p \neq p_0$): $2 \cdot P(Z > |z|)$

Two-Sample z-Interval for $p_1 - p_2$

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

Note: For the CI, we do NOT pool — we use each sample's $\hat{p}$.

AP Tip: For tests, use the pooled proportion $\hat{p}_c$. For confidence intervals, use the individual sample proportions. This is a common source of errors.