A t-test is a significance test for one or more population means that uses the t-distribution because the population standard deviation is unknown and must be estimated with the sample standard deviation; it's the standard tool in AP Stats Unit 7 (Inference for Quantitative Data: Means).
A t-test is a hypothesis test you run when your question is about a mean (or a difference between means) and you don't know the population standard deviation, which is basically always in real life. Since you have to estimate the standard deviation from your sample, there's extra uncertainty in your test statistic. The t-distribution accounts for that extra uncertainty by having fatter tails than the normal curve, and those tails slim down as your degrees of freedom grow.
In AP Stats you'll meet three flavors. A one-sample t-test checks whether a population mean equals some claimed value. A two-sample t-test compares the means of two independent groups. A paired t-test is secretly a one-sample test in disguise, because you take the differences within each pair first and then test whether the mean difference is zero. All three follow the same four-step rhythm: state hypotheses, check conditions (random, independent via the 10% condition, and normal/large sample), compute the t-statistic and p-value, and conclude in context.
t-tests live in Unit 7 of the AP Statistics CED, Inference for Quantitative Data: Means. This is where everything from earlier units pays off. Sampling distributions from Unit 5 explain why the t-statistic works, the logic of significance testing from Unit 6 (which you learned with proportions) transfers directly, and the random sampling ideas from Unit 3 justify your conditions. The big skill the CED cares about isn't crunching the number. It's choosing the right test, verifying conditions, and writing a conclusion that links the p-value to a decision about the null hypothesis in the context of the problem. If you can do a full one-sample or paired t-test cleanly, you've mastered the inference template the whole second half of the course is built on.
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t-Distribution (Unit 7)
The t-test gets its name from the distribution it uses. Because you estimate the standard deviation from the sample, your statistic follows a t-distribution instead of a normal one. Think of the t-distribution as a normal curve that admits it's a little less sure of itself.
Degrees of Freedom (Unit 7)
Every t-test needs degrees of freedom (n − 1 for one-sample and paired tests) to pick the right t-curve. More degrees of freedom means a curve closer to normal, which is why big samples make the t and z approaches nearly identical.
P-value (Unit 6)
The t-test produces a p-value, and you interpret it exactly the way you learned with proportions in Unit 6. It's the probability of getting a sample result at least as extreme as yours, assuming the null hypothesis is true. The test changes; the logic doesn't.
10% Condition (Units 5-7)
When you sample without replacement, the 10% condition (sample is less than 10% of the population) is how you justify treating observations as independent, which the t-test's standard error formula assumes.
Confidence Interval (Units 6-7)
A t-test and a t-interval are two sides of the same coin. If a 95% confidence interval for a mean doesn't contain the null value, a two-sided test at α = 0.05 would reject it. The exam loves asking you to connect the two.
Inference for means is a staple of the AP Statistics exam. On the multiple-choice section, expect stems that ask you to identify the correct test (one-sample, two-sample, or paired), check whether conditions are met, compute or interpret a t-statistic, or count degrees of freedom. On the free-response section, a full significance test for a mean is one of the most common question types, and it's graded on the complete four-step process. That means you must name the test, define your parameter, state hypotheses with symbols, verify random/independence/normality conditions with evidence, report the t-statistic, df, and p-value, and write a conclusion that compares the p-value to α and answers the question in context. Skipping the conditions check or giving a conclusion without context is the classic way to lose points on an otherwise correct answer.
Both test hypotheses, but the choice comes down to what you're testing and what you know. Use a z-test for proportions (the standard deviation of a proportion comes straight from the hypothesized p, so it's known). Use a t-test for means, because you almost never know the population standard deviation σ and have to estimate it with the sample standard deviation s. On the AP exam, means → t, proportions → z. If a problem about a mean hands you σ (rare and artificial), a z-test would technically apply, but in practice every means problem you'll see is a t-test.
A t-test is the significance test for means when the population standard deviation is unknown, which is essentially every means problem on the AP exam.
The three types are one-sample (test a mean against a claimed value), two-sample (compare two independent group means), and paired (test the mean of within-pair differences).
Before running any t-test, verify three conditions: random sampling or assignment, independence (often via the 10% condition), and normality (population roughly normal, n ≥ 30, or a graph of the data showing no strong skew or outliers).
Degrees of freedom (n − 1 for one-sample and paired tests) determine which t-curve you use, and bigger df makes the t-distribution look more like the normal curve.
Paired data is the trap to watch for; if the two sets of measurements come from the same subjects or matched pairs, take differences and run a one-sample t-test on them instead of a two-sample test.
On FRQs, the conclusion must compare the p-value to α, state a decision about the null hypothesis, and answer the question in the context of the problem to earn full credit.
It's a significance test for a population mean (or difference in means) that uses the t-distribution because the population standard deviation is unknown and estimated from the sample. It's the core test of Unit 7, Inference for Quantitative Data: Means.
Use a t-test when your hypothesis is about a mean, and a z-test when it's about a proportion. The shortcut for the AP exam is means → t, proportions → z, because with means you have to estimate the standard deviation, which is exactly the situation the t-distribution was built for.
No. The sample size of 30 is just one way to satisfy the normality condition via the Central Limit Theorem. With smaller samples you can still run a t-test if the population is stated to be roughly normal, or if a graph of your sample data shows no strong skew or outliers, and you should say so in your conditions check.
Ask whether each value in one group is naturally matched to a specific value in the other. Same subjects measured twice (before/after), twins, or matched pairs means paired, so you take differences and run a one-sample t-test on them. Two separate, independent groups means a two-sample t-test.
Readers score the full four-step process. You need correctly stated hypotheses with a defined parameter, a named test with conditions checked and justified, the correct t-statistic, degrees of freedom, and p-value, and a conclusion that compares the p-value to α and answers in context. Each piece earns credit, so never skip the conditions or the context.