AP Statistics Unit 3: Inference for Categorical Data: Proportions Notes

Introduction

Proportion inference connects estimator behavior, design and large-count conditions, confidence intervals, hypothesis tests, decision errors, two-group comparisons, and retained chi-square procedures.

Every inference begins with a population parameter and a data-collection design. Confidence intervals estimate plausible parameter values; tests measure how surprising a statistic would be under a null model. Both require conditions and contextual interpretation.

One- and two-proportion procedures use different standard errors for estimation and testing. Chi-square procedures extend categorical inference to larger two-way tables, but the revised course retains homogeneity and independence—not goodness-of-fit.

Estimator and sampling model

uncertainty

Confidence interval

plausible parameter values

Null model and p-value

evidence decision

Two proportions and chi-square

comparative categorical inference

Current exam guidance: Aligned to the effective Fall 2026 AP Statistics framework, including its current calculator and inference expectations.

3.1

Estimators

What makes a statistic a useful estimator of a population parameter?

Short answer

An estimator is judged across repeated samples: its sampling-distribution center reveals bias and its spread reveals variability, while one observed value is only a point estimate.

Know firstParameter and statistic · Sampling distributions

Judge the rule across all possible samples

The population 1,1,1,0,0,0 has proportion p=0.50. Every unordered size-two SRS produces a sample proportion of 0, 0.5, or 1. Averaging the 15 possible estimates returns 0.50, so $\hat p$ is unbiased in this finite-population example even though most individual estimates miss the parameter.

Bias concerns the long-run center, not whether one estimate lies above or below the target. Variability concerns the spread of estimates. A large sample often reduces variability, but it cannot repair a biased selection mechanism aimed at the wrong population. Low bias and low variability are separate goals.

All size-two samples summarized by estimate

Sample composition	Number of samples	p-hat
two zeros	3	0
one zero, one one	9	0.5
two ones	3	1
weighted mean	15	0.5

Evaluate an estimator

Name the target parameter.
Define the statistic rule.
Construct or simulate its sampling distribution.
Compare its mean with the parameter.
Compare spread and bias without treating either as the whole story.

Keep these three ideas

Point estimate is one realized statistic.
Bias is a center property.
Variability is a spread property.

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3.1 EstimatorsRead

3.2 Sampling Distributions for Sample Proportions

3.3 Constructing a Confidence Interval for a Population Proportion

3.4 Justifying a Claim Based on a Confidence Interval for a Population Proportion

3.5 Setting Up a Test for a Population Proportion

3.6 p-Values

3.7 Carrying Out a Test for a Population Proportion

3.8 Potential Errors When Performing Tests

3.9 Sampling Distributions for the Difference Between Sample Proportions

3.10 Constructing a Confidence Interval for the Difference Between Two Population Proportions

3.11 Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Proportions

3.12 Setting Up a Test for the Difference Between Two Population Proportions

3.13 Carrying Out a Test for the Difference Between Two Population Proportions

3.14 Setting Up a Chi-Square Test for Homogeneity or Independence

3.15 Carrying Out a Chi-Square Test for Homogeneity or Independence

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