Inference for Proportions

Name: AP Statistics Unit 6: Inference for Proportions
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12–15% of the exam4 lessons · 58 min20 terms

What this unit covers

The topics below follow the published Statistics course framework for Unit 6. This unit is worth 12–15% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Confidence intervalsSignificance testsErrors & powerTwo-sample

Lessons in this unit

Confidence Intervals for a Proportion15 min · 3 objectivesConstruct a one-sample z-interval for a population proportion · Interpret a confidence interval and a confidence level correctly · Explain how confidence level and sample size affect the margin of error
Significance Tests for a Proportion15 min · 3 objectivesState null and alternative hypotheses for a proportion · Compute the one-sample z test statistic and p-value · Make and interpret a decision in context
Errors, Significance & Power14 min · 3 objectivesDistinguish Type I and Type II errors and their consequences · Relate the significance level α to the probability of a Type I error · Identify the factors that increase the power of a test
Comparing Two Proportions14 min · 3 objectivesConstruct a two-sample z-interval for a difference of proportions · Perform a two-proportion z test using the combined (pooled) proportion · Interpret a difference of proportions in context

Formulas in Unit 6

One-sample z-interval for a proportion

p-hat ± z\* · sqrt( p-hat(1−p-hat) / n )

z\* is the critical value (1.96 for 95%, 2.576 for 99%, 1.645 for 90%). The margin of error is everything after the ± sign.

One-sample z test for a proportion

z = (p-hat − p₀) / sqrt( p₀(1−p₀) / n )

Use the null value p₀ (not p-hat) in the standard error, since the p-value is computed assuming H₀ is true. Check Random, 10%, and Large Counts (np₀ ≥ 10, n(1−p₀) ≥ 10).

Errors and power at a glance

P(Type I) = α · P(Type II) = β · power = 1 − β

Type I = reject a TRUE null (false positive). Type II = fail to reject a FALSE null (false negative). Increasing n raises power without raising α.

Two-proportion inference

CI: (p-hat₁ − p-hat₂) ± z\*·sqrt( p-hat₁(1−p-hat₁)/n₁ + p-hat₂(1−p-hat₂)/n₂ ) · TEST SE: sqrt( p-hat_c(1−p-hat_c)(1/n₁ + 1/n₂) )

p-hat_c = (x₁ + x₂)/(n₁ + n₂) is the pooled (combined) proportion, used only in the test. The interval uses unpooled separate proportions.

Every term in Unit 6

All 20 terms we publish for Inference for Proportions, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Confidence interval structure: Statistic ± (critical value)(standard error). Every interval in the course has this form.
Interpreting a confidence level: "If we repeated this sampling many times, about 95% of the intervals produced would capture the true parameter." A statement about the method, not one interval.
Interpreting a confidence interval: "We are 95% confident that the interval from a to b captures the true population proportion of …" — in context, about the parameter, not the statistic.
Conditions for a one-proportion z-interval: Random sample, 10% condition, and Large Counts using np̂ ≥ 10 and n(1 − p̂) ≥ 10.
Conditions for a one-proportion z-test: Same, but Large Counts uses the null value p₀ rather than p̂, because the test assumes the null is true.
Margin of error: The critical value times the standard error. Reducing it requires a larger sample or a lower confidence level.
Effect of sample size on the interval: Quadrupling n halves the margin of error, because standard error falls as √n.
Null and alternative hypotheses: H₀ states no effect or no difference using the parameter symbol; Hₐ states what you are testing for. Both are about parameters, never statistics.
p-value definition: The probability of getting a result at least as extreme as the one observed, ASSUMING the null hypothesis is true.
Interpreting a p-value in context: "Assuming the true proportion is p₀, there is a p% chance of observing a sample proportion this far from p₀ or farther."
Conclusion wording: Reject H₀ when p < α: "we have convincing evidence that…". Otherwise fail to reject: "we do not have convincing evidence" — never "we accept H₀".
Type I and Type II errors: Type I rejects a true null (a false positive, probability α); Type II fails to reject a false null. Lowering α raises the chance of a Type II error.
Power: The probability of correctly rejecting a false null, 1 − P(Type II). Increased by a larger sample, a larger effect size, or a larger α.
Two-proportion z-test pooling: Under the null the proportions are equal, so combine both samples into a pooled p̂ for the standard error. The interval does not pool.
The four-step template: State the parameter and hypotheses, Plan by naming the procedure and checking conditions, Do the mechanics, Conclude in context. Each step carries points.
A p-value is not the probability the null is true: It is the probability of data this extreme GIVEN the null. The reversal is the most penalized misinterpretation in the course.
Why we never accept the null: Failing to reject means the evidence was insufficient, which is not the same as evidence of no effect.
Using a confidence interval to test: If the null value falls outside a 95% interval, a two-sided test at α = 0.05 would reject it. The two procedures agree.
Choosing sample size for a margin of error: Set the margin-of-error expression less than the target and solve for n, using p̂ = 0.5 when no estimate is available, since it maximizes the required size.
One-sided vs two-sided alternatives: Decide from the question before seeing the data. Choosing a one-sided alternative after looking at the sample is not legitimate.

What examiners penalize here

Always describe errors *in context*. On free response, spell out what rejecting or failing to reject means for the real situation, and state a consequence. "Type I error: concluding the water is unsafe when it is actually safe, causing a needless costly shutdown" earns full credit.
Match the standard error to the procedure: a two-proportion **test** pools into p-hat_c because H₀ assumes equality; a two-proportion **confidence interval** keeps the proportions separate because it makes no equality assumption. Using the wrong one is a common lost point.

Practice Statistics

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

38 practice questionsEvery one with a written explanation FRQ trainerWritten answers scored against the rubric Statistics study guideAll units, weights and score data in one place All 9 Statistics unitsThe full course

Questions about this unit

How much of the AP Statistics exam is Unit 6?

Unit 6, Inference for Proportions, is worth 12–15% of the Statistics multiple-choice section according to the published course framework. Across all 9 units that makes it a substantial share — heavier than an even split would give it.

What topics are covered in Statistics Unit 6?

Inference for Proportions covers Confidence intervals, Significance tests, Errors & power and Two-sample. We publish 20 terms with definitions for this unit, all of them on this page.

How should I study Statistics Unit 6?

Read the 4 lessons below first — about 60 minutes — then drill the 20 terms in cram mode until you can produce each definition from memory rather than just recognize it. Recognition is what makes a unit feel finished when it is not. Finish with practice questions and read the explanation for every one you get right by elimination as well as the ones you miss.

All 9 units of AP Statistics

Unit names, topics and exam weights follow the published College Board course framework for AP Statistics. AP® is a trademark registered by the College Board, which does not endorse this site.