Before you run a two-sample t-test, you have to set it up correctly: decide whether the data are paired or come from two independent groups, write hypotheses about population means, and check the conditions. Each step below gives instant feedback.
Before any calculation, a test needs three things: the right procedure, hypotheses about population means, and conditions that make the procedure trustworthy. Practice each one here.
Paired data come from one group measured twice, or from matched pairs. Two independent samples come from two separate groups with no link between individuals.
Pick a research question, then choose the alternative hypothesis that matches its wording.
Group 1 and group 2 are defined in the hypotheses on the right. The order of subtraction sets the direction.
Describe how the data were collected and what each sample looks like. Example dotplots show what each shape means.
If each value in one group is naturally linked to one value in the other (same person, twins, the same car), the data are paired: analyze the differences with a one-sample t procedure. Otherwise use a two-sample t procedure.
Write hypotheses with μ₁ and μ₂, never x̄₁ and x̄₂. Define each one in context: "μ₁ = the true mean height of plants given the new fertilizer."
The null hypothesis is almost always H₀: μ₁ − μ₂ = 0 (equivalently μ₁ = μ₂). It is the claim we look for evidence against.
"Greater," "taller," "more" → >. "Less," "lower," "reduce" → <. "Different," "changed," "any difference" → ≠. Decide this before looking at the data.
Random samples or random assignment; each sample at most 10% of its population when sampling; each n ≥ 30, or that sample's data show no strong skew or outliers.
Random sampling lets you generalize to the populations. Random assignment lets you conclude cause and effect. A convenience sample allows neither.
The conditions for a two-sample t-test, and how to show each one.
| Condition | Why it matters | How to check it | Example wording |
|---|---|---|---|
| Random | Avoids bias; makes the sampling distribution model correct | Two independent random samples, or a randomized experiment | "The 50 plants were randomly assigned to the two fertilizers." |
| 10% | Keeps observations within each sample close to independent | n₁ ≤ 0.10 N₁ and n₂ ≤ 0.10 N₂; not needed for random assignment | "30 is less than 10% of all juniors at the school." |
| Normal/Large sample | Makes the sampling distribution of x̄₁ − x̄₂ approximately normal, so t is valid | n₁ ≥ 30 and n₂ ≥ 30, or each small sample's graph shows no strong skew or outliers | "n₂ = 18 < 30, but the dotplot of sample 2 shows no strong skew or outliers." |
| Independent groups | Two-sample procedures assume no pairing | The groups are separate, with no matching between individuals | "Different students were in each group." |
AP-style multiple choice. Pick an answer to see the explanation.
The vocabulary for topic 4.9. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 4 terminology guide.