When do I use a paired t test?

Use it when observations are naturally matched, such as before-and-after measurements on the same unit. Form one difference per pair, then perform one-sample t inference on those differences.

Why use t instead of z for a population mean?

When the population standard deviation is unknown and estimated by s, t inference accounts for the extra uncertainty. Its tails depend on degrees of freedom and approach the normal distribution as degrees of freedom grow.

Which calculator can I use on AP Statistics in 2027?

A graphing calculator with statistical capabilities is expected on both sections. Bluebook includes Desmos graphing. Nongraphing handheld calculators are not allowed in 2027.

AP Statistics

AP Statistics Unit 4 review: Inference for Quantitative Data: Means

Unit 4 uses sample means to estimate and test population means, paired differences and differences between independent groups. It is 10–20% of multiple choice. Expect to identify the design, check conditions and connect a t statistic or interval to a claim in context.

Try the practice questions Topic list

By Marteen Tang, founder of Ashore · Updated Nov 9, 2026 · Sources at the bottom of the page

Unit 4 on the May 2027 exam

AP Statistics Unit 4 exam facts
	2027 exam
Weight	10–20% of the multiple-choice section
Exam date	Tuesday, May 11, 2027, afternoon session
Multiple choice	42 questions in 90 minutes, worth 50% of the exam score
Free response	4 questions in 90 minutes, worth 50%. All answers are entered in Bluebook: two multi-focus questions, one inference question and a final multi-focus question.
Calculator	A graphing calculator with statistical capabilities is expected on both sections. Bluebook includes Desmos graphing. Nongraphing handheld calculators are not allowed in 2027.
Class time	About 18 class periods in College Board's pacing guide
2026 results	17% of students scored a 5 on the whole exam, and about 62% scored 3 or higher

Sources: College Board's course and exam description, exam page, 2027 exam dates and score distributions.

What Unit 4 covers

College Board's course description splits the unit into 10 topics. The second column is what you should be able to do for each one.

AP Statistics Unit 4 topics
Topic	What you should be able to do
4.1 Sampling Distributions for Sample Means	Calculate the center and spread of sample-mean distributions and assess normality conditions.
4.2 Constructing a Confidence Interval for a Population Mean or Population Mean Difference	Construct a t interval for a population mean or the mean of paired differences.
4.3 Justifying a Claim Based on a Confidence Interval for a Population Mean or Population Mean Difference	Interpret the interval and evaluate a claim about the population mean.
4.4 Setting Up a Test for a Population Mean or Population Mean Difference	Define the parameter and hypotheses and verify conditions for one-sample or paired t inference.
4.5 Carrying Out a Test for a Population Mean or Population Mean Difference	Calculate a t statistic and p-value and justify a decision in context.
4.6 Sampling Distributions for the Difference Between Two Sample Means	Describe the distribution of the difference between independent sample means.
4.7 Constructing a Confidence Interval for the Difference Between Two Population Means	Construct a two-sample t interval using separate sample variances.
4.8 Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Means	Interpret a confidence interval for the difference between two population means.
4.9 Setting Up a Test for the Difference Between Two Population Means	State hypotheses and verify conditions for independent two-sample t inference.
4.10 Carrying Out a Test for the Difference Between Two Population Means	Calculate a two-sample t statistic and p-value and connect them to the population claim.

Choose paired or independent inference before calculating

Check random sampling or random assignment, independence and a suitable sampling distribution. For samples without replacement, verify the 10% condition.

Choose paired or independent inference before calculating
Design	Statistic and standard error	Condition to examine
One quantitative sample	x̄; SE = s/√n	Approximately normal population, n ≥ 30, or a smaller sample free of strong skewness and outliers. Extreme population skew can require more than 30.
Paired measurements	Mean difference d̄; SE = sd/√n pairs	Use at least 30 differences, or check that a smaller difference sample has no strong skewness or outliers.
Independent groups	x̄₁ − x̄₂; SE = √(s₁²/n₁ + s₂²/n₂)	Independent groups; both n ≥ 30 or both populations approximately normal. If either n < 30, check both samples for strong skewness and outliers.
One-sample or paired test	t = (estimate − null value)/SE	Use df = n − 1, with n equal to the number of differences for paired data.
Confidence interval	Estimate ± t*SE	Use the appropriate t critical value and degrees of freedom.
Two-sample degrees of freedom	Use the procedure’s Welch calculation or an approved conservative approach	Do not assume pooled variances without justification.
Interpretation	Interval and test concern population means	They do not describe a range containing most individual observations.

Common mistakes

Each of these loses points on questions you otherwise understand.

Stating hypotheses in terms of x̄. A hypothesis concerns a population mean μ or difference in means. CED printed p. 117.
Listing a condition without evidence from the study. Identify randomization and explain why the sampling distribution is suitable. CED p. 117.
Answering a convincing-evidence question with only two sample averages. Carry out the appropriate significance test and connect its p-value to α. CED p. 117.
Treating repeated measurements on the same units as independent groups. Compute one difference per pair and use its standard deviation. CED topics 4.2 and 4.4.

6 practice questions with answers

Written by Ashore for this page, in the style of the multiple-choice section. Try each one before you open the answer.

Question 1 · 4.2
A random sample of 25 observations has mean 42 and SD 10. Conditions for t inference hold. What is the standard error of the mean?
- A0.4
- B5
- C2
- D10
Show the answer
C. SE = s/√n = 10/√25 = 2. The sample SD describes individual observations; the standard error describes variability in sample means.
Question 2 · 4.2
For that sample, use x̄ = 42, SE = 2 and t* = 2.064 for a 95% interval. Which interval is correct?
- A(37.872, 46.128)
- B(39.936, 44.064)
- C(40.000, 44.000)
- D(21.360, 62.640)
Show the answer
A. The margin is 2.064(2) = 4.128. Add and subtract it from 42 to obtain (37.872, 46.128). The t critical value must multiply the standard error, not the sample SD.
Question 3 · 4.5
A test of H₀: μ = 50 uses n = 16, x̄ = 54 and s = 8. Conditions hold. What is the t statistic?
- A0.50
- B4.00
- C8.00
- D2.00
Show the answer
D. SE = 8/√16 = 2. Thus t = (54 − 50)/2 = 2.00, with 15 degrees of freedom. The numerator is the observed difference from the null mean.
Question 4 · 4.4
Twelve machines are measured before and after adjustment. The mean after-minus-before difference is −3 and its SD is 6. What is the SE for the mean difference?
- A6/√24
- B6/√12
- C3/√12
- D√(36 + 36)
Show the answer
B. There are 12 independent pairs and one difference per pair. Use the SD of those differences: SE = 6/√12 ≈ 1.732. Counting 24 separate measurements would ignore pairing.
Question 5 · 4.7
Two independent samples have n₁ = 36, s₁ = 12, n₂ = 64 and s₂ = 16. What is the SE of x̄₁ − x̄₂?
- A0
- B4
- C√8
- D√28
Show the answer
C. SE = √(12²/36 + 16²/64) = √(4 + 4) = √8 ≈ 2.828. Variances add for a difference of independent estimates; standard deviations do not subtract.
Question 6 · 4.5
A valid test of H₀: μ = 100 against Hₐ: μ > 100 gives p = 0.032. At α = 0.05, which conclusion is justified?
- AReject H₀; there is evidence that the population mean exceeds 100
- BReject H₀; there is a 96.8% probability the alternative is true
- CFail to reject H₀; the population mean has been shown to equal 100
- DFail to reject H₀; fewer than 5% of observations exceed 100
Show the answer
A. Because 0.032 < 0.05, reject H₀ and conclude there is evidence for μ > 100. The p-value is calculated assuming the null model; it is not the probability that either hypothesis is true.

How to study Unit 4

Before opening a calculator menu, determine whether observations are paired. If they are, write the subtraction order and create the difference list. If they are independent, preserve the two groups and use separate sample variances.

Practice the complete inference sequence: parameter, hypotheses or confidence level, method, conditions, calculation and conclusion. Translate the software’s t, degrees of freedom and p-value into a sentence about the actual measured variable and population.

Ashore's Stats course has a lesson and practice questions for every topic in this unit. The free starting check takes about 15 minutes and shows which ones are costing you points.

Questions

Still have a question? support@tryashore.com

It is 10–20% of the multiple-choice section, which is half the exam score. This is not a fixed share of the whole exam: free-response questions can combine units. Suggested pacing is 18 class periods.

By Marteen Tang · Founder of Ashore

Marteen Tang founded Ashore and works on its SAT and AP lessons, practice and plans. Updated Nov 9, 2026.

Sources

Find the Stats topics costing you points

The starting check takes about 15 minutes. Free, no card.

Start Stats free AP Statistics overview

Starting check

SAT Math · 1 of 5

Calculator allowed

A jacket's price is increased by 20% and then the new price is decreased by 20%. The final price is what percent of the original price?

A $96\%$
B $100\%$
C $104\%$
D $80\%$