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Unit 3 · Inference for Proportions Flashcards Cheat Sheet Essentials Visual Review MC Practice FRQ Practice

AP Statistics Unit 3 Essentials

The must-know terms and core concepts for Unit 3: Inference for Proportions. Every vocabulary word and idea you need to master.

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Key Concept 1
Inference rests on the sampling distribution of p̂
Every method in Unit 3 comes from one fact: for a random sample, the sample proportion p̂ varies from sample to sample with mean p and standard deviation √(p(1 − p)/n), and is approximately normal when the Large Counts condition holds. Because you know how p̂ behaves, you can reverse the logic and reason from a single sample back to the population — but only after checking the Random, 10%, and Large Counts conditions.
Sampling Distribution Conditions p̂
Key Concept 2
Intervals estimate a parameter; tests weigh evidence against a claim
A confidence interval (estimate ± margin of error) gives a plausible range for the true proportion, and its confidence level describes the long-run capture rate of the method. A significance test starts from a null hypothesis, computes a test statistic and a p-value — the probability of data at least as extreme as observed if H₀ is true — and rejects H₀ when that p-value is small. Interpreting both correctly, and always in context, is the most-tested skill on the exam.
Confidence Intervals Significance Tests p-values
Key Concept 3
Extend to two proportions and to two-way tables — and mind the errors
The same framework extends to comparing two proportions (with a pooled proportion for the test) and to the chi-square test for homogeneity or independence, which compares observed and expected counts across a two-way table. Alongside the mechanics you must reason about Type I and Type II errors and power: rejecting a true H₀, failing to reject a false one, and the probability of correctly detecting a real effect.
Two Proportions Chi-Square Errors & Power
Estimator / point estimate
A statistic used to estimate a parameter (p̂ for p), and the single value it produces.
Foundations
Sampling distribution of p̂
The distribution of the sample proportion: mean p, SD √(p(1 − p)/n), approximately normal under Large Counts.
Foundations
Random / 10% / Large Counts
The three conditions for proportion inference: random data, sample under 10% of the population, and np̂ ≥ 10 and n(1 − p̂) ≥ 10.
Foundations
Standard error
The estimated standard deviation of a statistic, such as √(p̂(1 − p̂)/n) for a proportion.
Foundations
Confidence interval
An interval of the form estimate ± margin of error that gives a plausible range for a parameter.
Confidence Intervals
Margin of error
The critical value times the standard error; it shrinks with larger n and grows with higher confidence.
Confidence Intervals
Critical value (z*)
The multiplier determined by the confidence level (1.96 for 95%).
Confidence Intervals
Confidence level
The long-run percentage of such intervals that would capture the true parameter.
Confidence Intervals
Null hypothesis (H₀)
A specific claim about a parameter, such as p = p₀, assumed true when computing a p-value.
Significance Tests
Alternative hypothesis (Hₐ)
The claim you seek evidence for: p < p₀, p > p₀, or p ≠ p₀.
Significance Tests
Test statistic
A standardized measure of how far the data fall from H₀, such as z = (p̂ − p₀)/√(p₀(1 − p₀)/n).
Significance Tests
p-value
The probability, assuming H₀ is true, of a result at least as extreme as the one observed.
Significance Tests
Significance level (α)
The pre-set threshold for rejecting H₀; also the probability of a Type I error.
Significance Tests
Statistically significant
A result whose p-value is at or below α, leading to rejection of H₀.
Significance Tests
Type I error
Rejecting H₀ when it is actually true; its probability is α.
Errors & Power
Type II error
Failing to reject H₀ when Hₐ is actually true; its probability is β.
Errors & Power
Power
The probability of correctly rejecting a false H₀, equal to 1 − β.
Errors & Power
Difference of two proportions
p̂₁ − p̂₂, with SD √(p₁(1 − p₁)/n₁ + p₂(1 − p₂)/n₂).
Two Proportions
Pooled proportion
(x₁ + x₂)/(n₁ + n₂), the combined estimate used in a two-proportion significance test.
Two Proportions
Chi-square test
A test comparing observed counts to expected counts in a two-way table.
Chi-Square
Expected count
(row total × column total)/grand total for a cell, under the assumption of no association.
Chi-Square
Test for homogeneity
A chi-square test comparing the distribution of a variable across several groups or populations.
Chi-Square
Test for independence
A chi-square test of whether two variables are associated within a single population.
Chi-Square
Degrees of freedom
For a chi-square two-way table, (rows − 1)(columns − 1), which sets the reference distribution.
Chi-Square