One Sample T-test in AP Statistics

In AP Statistics, a one-sample t-test is a significance test that uses a sample mean (x̄) and sample standard deviation (s) to test a claim about a single population mean (μ), using the statistic t = (x̄ − μ₀)/(s/√n) with n − 1 degrees of freedom.

What is One Sample T-test?

A one-sample t-test checks whether a sample gives convincing evidence against a claimed value of one population mean. You start with a null hypothesis like H₀: μ = μ₀ and an alternative like Hₐ: μ > μ₀, μ < μ₀, or μ ≠ μ₀. Then you measure how far your sample mean landed from that claimed value.

The test statistic is t = (x̄ − μ₀) / (s/√n). Think of it as "how many standard errors away from the claim is my sample mean?" You use a t distribution instead of a normal distribution because you don't know the population standard deviation σ. You're estimating it with s, and that extra uncertainty gives the t distribution its fatter tails. It has n − 1 degrees of freedom. Before you trust any of this, you check three conditions. The data must come from a random sample or experiment. The 10% condition (n ≤ 10% of the population when sampling without replacement) protects independence. The normal/large sample condition requires n ≥ 30, or a sample graph with no strong skewness or outliers.

Why One Sample T-test matters in AP® Statistics

This term lives in Topic 4.4, Setting Up a Test for a Population Mean or Population Mean Difference. That topic is about the setup half of inference. You state hypotheses about μ, name the right procedure, and verify conditions before any calculation happens. The one-sample t-test is the default tool whenever the question involves one quantitative variable and one mean, so naming it correctly is often the first point an FRQ awards. It's also the engine behind tests on a population mean difference. Once you turn paired data into a single list of differences, you run the exact same test. For the full walkthrough of setting up tests for means, head to the Topic 4.4 study guide.

How One Sample T-test connects across the course

Matched Pairs and Dependent Samples (Unit 4)

A paired t-test is just a one-sample t-test wearing a disguise. When the data are dependent, like before/after scores on the same people, you subtract within each pair to get one list of differences. Then you test μ_diff with the same formula, using n − 1 df where n is the number of pairs.

Randomness, Independence, and the 10% Condition (Unit 4)

These conditions are what make the standard error s/√n honest. Randomness lets you generalize beyond the sample. The 10% condition makes sure the observations are close enough to independent that the formula for the spread of x̄ still works.

Skewness and the Normal/Large Sample Condition (Units 1, 4)

The shape skills from describing distributions come back here. With a small sample (n < 30), you graph the data. Strong skewness or outliers mean the t procedure may not be trustworthy, because the sampling distribution of x̄ might not be close to normal.

One-Tailed vs. Two-Tailed Tests (Unit 4)

The wording of the claim decides the alternative hypothesis. "Greater than" or "less than" gives a one-tailed test, and "different from" gives a two-tailed test. That choice changes how you find the p-value from the same t statistic.

Is One Sample T-test on the AP® Statistics exam?

Multiple-choice stems often describe a scenario and ask you to pick the right procedure, the correct hypotheses, the degrees of freedom, or which condition is in doubt. A common one is "the sample is small and the dotplot is strongly skewed." Released FRQs don't usually say "one-sample t-test" in the prompt. The test shows up anyway whenever a question gives you one quantitative sample and a claimed mean, and the scoring expects you to name the procedure yourself. A full-credit response has four parts. First, define μ in context and state H₀ and Hₐ. Second, name the test ("one-sample t-test for a mean") and check random, 10%, and normal/large sample conditions. Third, give t, df, and the p-value. Fourth, compare the p-value to α and conclude about the population mean in context. Watch for paired designs, where the expected procedure is a one-sample t-test on the differences.

One Sample T-test vs Two-sample t-test

A two-sample t-test compares the means of two independent groups, like a random sample of juniors vs. a random sample of seniors. A one-sample t-test handles one mean. That includes the mean of differences from paired or dependent data. The test is if each value in group A is naturally linked to a specific value in group B (same person, twins, before/after). In that case, subtract and run a one-sample t-test on the differences. Treating paired data as two independent samples is a classic point-losing mistake.

Key things to remember about One Sample T-test

• A one-sample t-test tests a claim about a single population mean μ using the sample mean x̄ and sample standard deviation s.

• The test statistic is t = (x̄ − μ₀)/(s/√n), and it follows a t distribution with n − 1 degrees of freedom.

• You use t instead of z because σ is unknown and estimated by s, which adds uncertainty and fattens the tails.

• Before running the test, check the random condition, the 10% condition for independence, and the normal/large sample condition (n ≥ 30 or no strong skew or outliers in the data).

• Paired or matched-pairs data are analyzed with a one-sample t-test on the differences, not a two-sample t-test.

• Your conclusion must compare the p-value to α and state what it means for the population mean in the context of the problem.

Frequently asked questions about One Sample T-test

What is a one-sample t-test in AP Stats?

It's a significance test for a claim about one population mean. You compute t = (x̄ − μ₀)/(s/√n), find a p-value from a t distribution with n − 1 degrees of freedom, and decide whether there's convincing evidence against H₀: μ = μ₀.

Do you need n ≥ 30 to use a one-sample t-test?

No. If n is under 30, you can still use it as long as a graph of the sample data shows no strong skewness or outliers. The n ≥ 30 rule is just one way to satisfy the normal/large sample condition.

How is a one-sample t-test different from a two-sample t-test?

A one-sample t-test handles one mean, including the mean of differences from paired data. A two-sample t-test compares means from two independent groups. If the observations come in natural pairs, like before and after on the same subjects, take the differences and use the one-sample version.

Why use a t-test instead of a z-test for a mean?

In real problems you almost never know the population standard deviation σ, so you estimate it with s. That extra variability means the standardized statistic follows a t distribution, which has heavier tails than the normal and gets closer to normal as n grows.

What are the conditions for a one-sample t-test?

There are three. The data must come from a random sample or randomized experiment. Under the 10% condition, n must be no more than 10% of the population when sampling without replacement. Under the normal/large sample condition, you need n ≥ 30 or a sample graph with no strong skew or outliers.