6.10 Setting Up a Test for the Difference of Two Population Proportion

Part of Inference for Categorical Data: Proportions in AP® Statistics. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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Hypotheses for Two-Proportion z Test

Two-proportion z-test: Tests whether two population proportions are equal.

Hypotheses:

${H_0: p_1 = p_2}$ (or equivalently, ${p_1 - p_2 = 0}$ )
${H_a: p_1 \neq p_2}$ (two-sided, most common)
${H_a: p_1 > p_2}$ (right-tailed)
${H_a: p_1 < p_2}$ (left-tailed)

Conditions for the two-proportion z-test:

Random: Both samples are random
Independence: 10% condition for each group; groups are independent of each other
Large Counts (using ${\hat{p}_c}$ — the combined/pooled proportion):

$n_1 \hat{p}_c \geq 10, \quad n_1(1-\hat{p}_c) \geq 10, \quad n_2 \hat{p}_c \geq 10, \quad n_2(1-\hat{p}_c) \geq 10$

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