6.8 Confidence Intervals for the Difference of Two Proportions

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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Two-Proportion z Confidence Interval

Parameter: ${p_1 - p_2}$ (difference in two population proportions)

Conditions:

Random: Both samples are random (or randomized experiment)
Independence: Each ${n_i < 10\%}$ of respective population (10% condition for each group)
Large Counts: All four counts ${\geq 10}$ :

$n_1 \hat{p}_1 \geq 10, \quad n_1(1-\hat{p}_1) \geq 10, \quad n_2 \hat{p}_2 \geq 10, \quad n_2(1-\hat{p}_2) \geq 10$

Formula:

(\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: Use ${\hat{p}_1}$ and ${\hat{p}_2}$ (not a pooled estimate) in the standard error for confidence intervals.

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