A t test asks: if H₀ were true, how surprising is our sample? Compute the test statistic t = (x̄ − μ₀) ÷ (s/√n), find the p-value from the t distribution with df = n − 1, compare it with α, and conclude in context. Enter data or pick a preset and follow every step.
Pick a preset or enter your own data. Change Hₐ or α and watch the p-value and decision.
This is the full answer an AP grader expects. It rewrites itself as you change the inputs.
Keep this x̄ and s but imagine a different n. The p-value shrinks as n grows, because the standard error shrinks.
t = (x̄ − μ₀) ÷ (s/√n) measures how far the sample mean is from the null value, in standard errors. Values far from 0 are surprising if H₀ is true.
The p-value is the probability, assuming H₀ is true, of getting a sample result at least as extreme as the one observed, in the direction of Hₐ. For ≠, count both tails.
If p-value < α, reject H₀: the data give convincing evidence for Hₐ. If p-value ≥ α, fail to reject H₀: the data do not give convincing evidence for Hₐ.
A large p-value means the data are consistent with H₀, not that H₀ is true. Say "we do not have convincing evidence that…," never "we accept H₀."
For paired data, compute the differences in a stated order, then run a one-sample t test on them with μ₀ = 0 (usually) and df = number of pairs − 1.
With a huge sample, even a tiny difference from μ₀ can give a tiny p-value. Statistically significant does not always mean practically important.
Finding the p-value from t for each alternative. Calculator: tcdf(lower, upper, df), or STAT → TESTS → T-Test.
| Hₐ | p-value | Calculator | Example (t = 2.10, df = 15) |
|---|---|---|---|
| μ > μ₀ | Area to the right of t | tcdf(t, 1E99, df) | tcdf(2.10, 1E99, 15) ≈ 0.0265 |
| μ < μ₀ | Area to the left of t | tcdf(−1E99, t, df) | tcdf(−1E99, 2.10, 15) ≈ 0.9735 |
| μ ≠ μ₀ | 2 × area beyond |t| | 2 · tcdf(|t|, 1E99, df) | 2 × 0.0265 ≈ 0.0530 |
| Step | What to write |
|---|---|
| State | H₀ and Hₐ with the parameter defined in context; α. |
| Plan | Name the procedure (one-sample t test for μ, or paired t test for μd); check Random, 10%, Normal/Large with evidence. |
| Do | t = (x̄ − μ₀) ÷ (s/√n) with numbers; df; p-value. |
| Conclude | Compare p with α, make a decision, and state the conclusion about Hₐ in context. |
AP-style multiple choice. Pick an answer to see the explanation.
The vocabulary for topic 4.5. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 4 terminology guide.