AP Statistics, AP exams

Unit 4: Inference for Quantitative Data: Means

Unit 4 extends inference to quantitative data. It covers t procedures for one mean, for matched pairs, and for the difference between two means, including the conditions each requires and the reasons the t distribution replaces the normal model. Recognizing whether a study calls for a paired or a two-sample procedure is a frequently tested judgment.

Position: 4 of 5 units
Topics covered: 6 topics in this guide

What is covered

Topics inside this unit.

These topics describe the scope students most often review for this part of AP Statistics. Treat them as a study checklist rather than an exhaustive list of what a single exam form can sample.

Sampling distribution of a sample mean: Center and variability of the sampling distribution, and when it is approximately normal.
Confidence intervals for a mean: Using t procedures, choosing degrees of freedom, and interpreting the interval in context.
Significance tests for a mean: Setting up hypotheses, computing the test statistic, and stating a contextual conclusion.
Matched pairs procedures: Recognizing a paired design and analyzing the differences rather than the two groups separately.
Comparing two means: Two-sample t intervals and tests, and the conditions required for each.
Choosing an appropriate procedure: Deciding among one-sample, paired, and two-sample approaches based on how the data were collected.

Study tips

How to practice Inference: Means.

Section-specific guidance that follows from how this material is usually tested.

Practice identifying the design before choosing a procedure; paired-versus-two-sample errors are the most common loss in this unit.
State and check conditions in writing every time, since the rubric awards those points separately from the calculation.
Interpret the p-value as a probability under the null hypothesis, in one carefully worded sentence you can reproduce.

Where this fits

Unit 4: Inference for Quantitative Data: Means is 4 of 5 units in the AP Statistics guide. Use the full guide to balance it against everything else you have to review.

Back to the AP Statistics guide Official exam information

Sample Inference: Means questions

Practice questions, not yet expert-reviewed

Original practice questions written to the published framework for this section. No expert has reviewed them, and they are not taken from any official exam. Try each one before you open the answer.

Question 1Foundational
Why do inference procedures for a population mean normally use a t distribution rather than a standard normal distribution?
1. ABecause the population standard deviation is unknown and is estimated from the sample
2. BBecause the sample size is always small
3. CBecause the population is never normally distributed
4. DBecause the t distribution has less spread than the normal distribution
Show answer
Answer: A. Because the population standard deviation is unknown and is estimated from the sample
Replacing the unknown population standard deviation with the sample standard deviation adds extra variability, and the t distribution accounts for it with heavier tails than the normal. That spread is larger, not smaller, and it shrinks toward the normal as the degrees of freedom grow.
Key insight: Estimating the standard deviation from data costs precision, and the t distribution prices that in.
Question 2Foundational
A one-sample t interval for a mean is computed from a random sample of 25 observations. How many degrees of freedom does it use?
1. A24
2. B25
3. C23
4. D26
Show answer
Answer: A. 24
For a one-sample t procedure the degrees of freedom are n - 1 = 25 - 1 = 24. Using n itself is the classic off-by-one error, and it makes the interval slightly too narrow because it borrows precision the data do not supply.
Key insight: One-sample t procedures use n - 1 degrees of freedom, one fewer than the sample size.
Question 3Intermediate
Each of 30 runners has a race time recorded before a training program and again after it. Which inference procedure fits this design?
1. AA paired t procedure on the differences
2. BA two-sample t procedure comparing two independent groups
3. CA one-sample z procedure for a proportion
4. DA chi-square test of independence
Show answer
Answer: A. A paired t procedure on the differences
The two measurements come from the same runner, so the observations are not independent and each runner supplies one difference. Analyzing 30 differences with a one-sample t procedure is the paired approach; treating the before and after lists as two independent samples ignores the pairing and inflates the estimated variability.
Key insight: Paired data give one difference per subject; two-sample procedures require independent groups.
Question 4Intermediate
Is the following statement true or false: the p-value of a significance test is the probability that the null hypothesis is true?
1. ATrue
2. BFalse
Show answer
Answer: B. False
A p-value is computed assuming the null hypothesis is true; it is the probability of getting a statistic at least as extreme as the observed one under that assumption. It says nothing about the probability that the hypothesis itself is true, which classical inference never assigns.
Key insight: The p-value is conditional on the null hypothesis, not a probability about the null hypothesis.
Question 5Advanced
A random sample of 16 batteries has a mean life of 21.0 hours and a standard deviation of 2.0 hours. What is the value of the t statistic for testing the null hypothesis that the population mean is 20.0 hours?
1. A2.00
2. B0.50
3. C8.00
4. D0.25
Show answer
Answer: A. 2.00
The statistic is (x-bar - μ0)/(s/√n) = (21.0 - 20.0)/(2.0/4) = 1/0.5 = 2.00. Dividing by s alone gives 0.50, and dividing by s/n rather than s/√n gives 8.00.
Key insight: The denominator of a t statistic is the standard error s/√n, never the sample standard deviation alone.

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Unit 4: Inference for Quantitative Data: Means