A two-sample t-test asks whether the difference between two sample means is too big to blame on chance alone. Enter the data and hypotheses to see the test statistic, the p-value as a shaded area, and the decision. A complete State, Plan, Do, Conclude answer is written out for you as you change things.
Enter the summary statistics, the hypothesized difference, the direction of the alternative, and α. The test statistic, p-value, and a complete four-step answer update as you type.
t measures how far the observed difference x̄₁ − x̄₂ is from the hypothesized difference, in standard errors. Far from 0 (in the direction of Hₐ) means the data are surprising if H₀ is true.
The p-value is the probability, assuming H₀ is true, of getting a difference in sample means at least as extreme as the one observed, in the direction of Hₐ. Two-sided tests count both tails.
If p ≤ α, reject H₀: there is convincing evidence for Hₐ. If p > α, fail to reject H₀: there is not convincing evidence for Hₐ. Never "accept" H₀.
Technology uses the Welch df. With a table, use the smaller of n₁ − 1 and n₂ − 1. The conservative df gives a slightly larger p-value, so it is less likely to reject.
Rejecting a true H₀ is a Type I error (probability α). Failing to reject a false H₀ is a Type II error. Larger samples raise power, the chance of correctly rejecting a false H₀.
With random assignment, a significant difference can be attributed to the treatments. With random samples, the conclusion applies to the populations sampled. Without either, be cautious.
How the alternative hypothesis decides the p-value, using t = 2.10 with df = 40 as an example.
AP-style multiple choice. Pick an answer to see the explanation.
The vocabulary for topic 4.10. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 4 terminology guide.