UnofficialAP StatisticsUnit 6 of 9

AP Statistics Unit 6: Inference for Categorical Data: Proportions

Confidence intervals and significance tests for proportions.

Unit overview

Inference for proportions answers two kinds of question. A confidence interval estimates an unknown proportion: $\hat{p} \pm z^* \sqrt{\hat{p}(1-\hat{p})/n}$ . A significance test asks whether data give convincing evidence against a claimed value $p_0$ , using $z = (\hat{p} - p_0)/\sqrt{p_0(1-p_0)/n}$ .

Each procedure has the same four parts: state the parameter and hypotheses or confidence level, check the random, 10% and large counts conditions, do the calculation, and conclude in context. The same structure extends to a difference of two proportions, where the test uses a pooled proportion.

Key topics

Confidence intervals for $p$: Point estimate plus or minus margin of error; interpreting the interval and the confidence level.
Significance tests for $p$: Hypotheses, test statistic, P-value and conclusion at a significance level.
Errors and power: Type I error rejects a true $H_0$ ; Type II fails to reject a false one; power is $1 - P(\text{Type II})$ .
Two-proportion procedures: Intervals and tests for $p_1 - p_2$ , with pooling in the test.

Worked example

In a random sample of 400 voters, 220 support a measure. Find the 95% confidence interval's margin of error (use $z^* = 1.96$ ).

$\hat{p} = 220/400 = 0.55$ .
$SE = \sqrt{0.55 \cdot 0.45 / 400} \approx 0.02487$ .
$ME = 1.96 \cdot 0.02487 \approx 0.0488$ .

Answer: $ME \approx 0.0488$

Common mistakes

Writing hypotheses about $\hat{p}$ instead of the parameter $p$ .
Saying there is a 95% probability that the parameter is in this particular interval; the 95% describes the method.
Concluding that $H_0$ is true when the P-value is large; the correct phrase is that there is not convincing evidence for $H_a$ .

Exam tips

Use $p_0$ in the standard error for a test and $\hat{p}$ for an interval.
Link the conclusion to the P-value and $\alpha$ explicitly, and keep it in the language of the problem.

Descriptive Statistics Study Guide (study guide)
Statistics Formulas cheat sheet (one-page formulas)
Descriptive Statistics Quiz (quick quiz)
Statistics practice problems (with worked solutions)

See the AP Statistics exam format and every unit on the course page.