The t-test, explained with examples

By Ashraf Bhura, Co-Founder, STEM-Tutor · Updated October 2026

A t-test checks whether a mean, or a difference between means, is significantly different from a claimed value. It is the workhorse test for quantitative data in AP Statistics and in real research.

Why t instead of z?

To test a claim about a population mean μ, we compare the sample mean x̄ with the claimed value. If we knew the population standard deviation σ, we could use a z statistic. In practice σ is almost never known, so we estimate it with the sample standard deviation s. That extra uncertainty is handled by the t distribution, which looks like the normal curve but has heavier tails, especially for small samples. As the sample size grows, the t distribution gets closer to the normal distribution.

The t statistic

t = (x̄ − μ₀) / (s / √n), with df = n − 1 degrees of freedom.

The denominator s/√n is the standard error of the mean. The t statistic tells you how many standard errors the sample mean is from the claimed value. Large values of |t| give evidence against H₀.

Conditions

1. Random: a random sample or a randomised experiment.
2. 10% condition: when sampling without replacement, n is at most 10% of the population.
3. Normal/large sample: the population is approximately normal, or n ≥ 30. For small samples, check a dotplot or boxplot of the data for strong skewness or outliers.

The four steps

Every AP significance test follows the same structure: State the hypotheses and significance level with the parameter defined in context; Plan by naming the test and checking conditions; Do the calculation of t, df and the P-value; and Conclude by comparing the P-value with α and answering in context.

Worked examples

Example 1: a one-sample t-test

A company claims its batteries last 40 hours on average. A random sample of 16 batteries has mean 38.2 hours and standard deviation 3.0 hours. Is there evidence at α = 0.05 that the mean is less than 40 hours?

• State: H₀: μ = 40; H_a: μ < 40, where μ is the true mean life of all the company's batteries.
• Plan: one-sample t-test. Random sample; 16 is less than 10% of all batteries; n is small, so we need the sample data to show no strong skewness or outliers.
• Do: t = (38.2 − 40)/(3.0/√16) = −1.8/0.75 = −2.4, df = 15. P-value = P(t₁₅ ≤ −2.4) ≈ 0.015.
• Conclude: 0.015 < 0.05, so reject H₀. There is convincing evidence that the true mean battery life is less than 40 hours.

Example 2: a paired t-test

Eight students took a test before and after a revision course. The mean of the differences (after − before) was 4.5 points, with standard deviation 5.2. Test whether the course improves scores, at α = 0.05.

• The same students were measured twice, so the data are paired. Work with the differences as one sample.
• H₀: μ_diff = 0; H_a: μ_diff > 0.
• t = 4.5/(5.2/√8) ≈ 4.5/1.838 ≈ 2.45, df = 7. P-value ≈ 0.022.
• 0.022 < 0.05, so reject H₀: there is convincing evidence that the course increases scores on average for students like these. Without random selection of students, be careful generalising to everyone.

Example 3: a two-sample t-test

Random samples of commuters: by train, n = 25, mean 42 minutes, s = 8; by car, n = 30, mean 48 minutes, s = 10. Is there a difference in mean commute times? Use α = 0.05.

• H₀: μ_train − μ_car = 0; H_a: μ_train − μ_car ≠ 0.
• Standard error = √(8²/25 + 10²/30) = √(2.56 + 3.333) ≈ 2.428. t = (42 − 48)/2.428 ≈ −2.47.
• Using technology, df ≈ 53 and the two-sided P-value ≈ 0.017.
• 0.017 < 0.05, so reject H₀: there is convincing evidence that the mean commute times differ, with train commuters taking less time on average.

Paired or two-sample?

This choice is a favourite AP question. Use a paired test when each observation in one group is naturally matched with one in the other: the same people before and after, twins, or left and right hands of the same person. Use a two-sample test when the groups are independent, such as two separate random samples. Using a two-sample test on paired data throws away the matching and usually makes it harder to detect a real effect.

When the result is not significant

If the P-value is larger than α, write: “Because the P-value is greater than α, we fail to reject H₀. There is not convincing evidence that …”. This does not prove the claim is true. A small sample may simply be unable to detect a real but modest difference, which is a Type II error risk.

Degrees of freedom

For one-sample and paired tests, df = n − 1. For a two-sample test, the exact df comes from a complicated formula that calculators handle automatically. A conservative approach, if you are working by hand, is to use the smaller of n₁ − 1 and n₂ − 1. Fewer degrees of freedom give a slightly larger P-value, so the conservative choice is safe.

Using a calculator

On a TI-84, T-Test handles one-sample and paired data (enter the differences as one list), and 2-SampTTest handles independent samples; choose “Pooled: No” for AP Statistics. On the exam, always write the test name, t, df and P-value, not just the calculator's output.

Common mistakes

• Using z when σ is unknown; use t with s.
• Treating paired data as two independent samples.
• Writing hypotheses about x̄ instead of the population mean μ.
• Skipping the normality check for small samples.
• Concluding “accept H₀” or stating a conclusion without context.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Is the t-test on the AP Statistics exam?

Yes. Inference for means, including one-sample, paired and two-sample t procedures, is Unit 7 and appears on almost every free-response section.

Is the t-test in A level Maths?

No. A level Maths hypothesis testing uses the binomial and normal distributions. The t-test is in some Further Maths options and is used widely at university.

Key takeaways

• Use t when σ is unknown: t = (x̄ − μ₀)/(s/√n).
• One-sample and paired tests use df = n − 1.
• Paired data: analyse the differences as one sample.
• Check random, 10% and normal/large-sample conditions.
• State, Plan, Do, Conclude, always in context.

Related guides

• P-values
• Confidence intervals
• Central limit theorem