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How do you state the hypotheses and check the conditions for a test comparing two means?

Topic 7.8 Setting Up a Test for the Difference of Two Population Means: state the hypotheses about the difference of two means, decide between a two-sample and a paired procedure, identify the significance level, and check the conditions.

A focused answer to AP Statistics Topic 7.8, on writing the hypotheses for a difference of two means, deciding between a two-sample and a paired t-test, choosing the significance level, and checking the conditions.

Generated by Claude Opus 4.89 min answerUpdated 2026-06-04

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this topic is asking
Decide the design first
Hypotheses for each design
Significance level and conditions
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What this topic is asking

The College Board (Topic 7.8) wants you to set up a test for the difference of two means: write the hypotheses, decide between a two-sample and a paired procedure based on the design, identify the significance level, and check the conditions.

Decide the design first

This decision drives everything that follows, the hypotheses, the standard error, and the degrees of freedom, so make it explicitly. The classic cue for pairing is that each subject contributes a before/after pair, or that subjects are deliberately matched (twins, left/right). The cue for independence is two separate groups of different units. Choosing the wrong design is one of the most consequential errors in the unit.

Hypotheses for each design

Both null hypotheses state "no difference," but they are about different parameters: two population means versus one mean of differences. Hypotheses are about parameters ( $\mu_1, \mu_2$ , or $\mu_d$ ), never the sample means. Define the parameters in words and fix the subtraction order so the alternative direction matches the question.

Significance level and conditions

Choose $\alpha$ in advance as the Type I error rate. Then check conditions appropriate to the design.

For a two-sample test: random (each sample random/randomised) and the two samples independent; normal/large sample for each group (population normal, $n \ge 30$ , or a graph with no strong skew or outliers); each sample at most $10\%$ of its population.

For a paired test: the differences come from a random sample (or randomised experiment); the differences are approximately normal (or the number of pairs is $\ge 30$ , or a graph of the differences shows no strong skew or outliers); the number of pairs is at most $10\%$ of the population of pairs. Note that for a paired test you check the differences, not the two original groups, a frequent slip.

Setting up a paired t-test

A trainer tests whether a stretching routine reduces sprint times. Each of $15$ athletes runs a timed sprint, does the routine, and runs again; the difference (before minus after) is recorded. A dotplot of the $15$ differences is roughly symmetric with no outliers. Set up the test at $\alpha = 0.05$ .

step 1 Identify the design

Each athlete is measured twice (before and after), so the data are paired. Analyze the $15$ within-athlete differences with a paired (one-sample) t-test.

step 2 Define the parameter and hypotheses

Let $\mu_d$ be the true mean difference (before minus after) in sprint time. "Reduces" means after is smaller, so before minus after is positive:

H_0: \mu_d = 0 \qquad H_a: \mu_d > 0.

step 3 Significance level

Use $\alpha = 0.05$ : the tolerated probability of concluding the routine reduces times when it actually has no effect (a Type I error).

step 4 Conditions

Random: assume the athletes are a random sample (or treat the assignment of measurement order as randomised). Normal/large: $n = 15$ pairs $< 30$ , but the dotplot of differences is roughly symmetric with no outliers, so the t-procedure is appropriate. $10\%$ : $15$ pairs is plausibly under $10\%$ of the relevant population.

step 5 Readiness

Conditions met; a paired t-test on the differences is appropriate, with $df = n - 1 = 14$ .

Try this

Q1. Two separate random groups are compared on a mean response. State the design and the null hypothesis. [1 point]

Cue. Independent samples, two-sample t-test; $H_0: \mu_1 = \mu_2$ .

Q2. For a paired test, which data do you check the normal/large condition on? [1 point]

Cue. The list of within-pair differences, not the two original groups.

Exam-style practice questions

Practice questions written in the style of College Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AP 2019 (style)1 marksSection I (multiple choice). To test whether two independent groups have different mean responses, the hypotheses are (A)

H_0: \bar{x}_1 = \bar{x}_2

(B)

H_0: \mu_1 = \mu_2

H_a: \mu_1 \ne \mu_2

(C)

H_0: \mu_1 \ne \mu_2

(D)

H_0: \mu_d = 0

(paired)

Show worked answer →

The correct answer is (B).

For two independent means, the null of "no difference" is $H_0: \mu_1 = \mu_2$ (equivalently $\mu_1 - \mu_2 = 0$ ), with $H_a: \mu_1 \ne \mu_2$ for "different."

(A) uses sample means, not parameters. (C) wrongly puts the inequality in the null. (D) is the paired hypothesis, used only when the data are matched, not for independent groups.

AP 2020 (style)3 marksSection II (free response). A researcher compares two diets. Design 1:

40

subjects randomly assigned to diet A and

40

to diet B, comparing mean weight loss. Design 2: each of

40

subjects tries both diets (in random order), comparing the mean difference in loss. (a) For each design, state the appropriate test and the hypotheses. (b) Explain why the procedures differ. (c) State a condition that design 2 must satisfy that design 1 does not require.

Show worked answer →

A 3-point design-and-setup question.

(a) (2 points) Design 1 (independent groups): two-sample t-test; $H_0: \mu_A = \mu_B$ versus $H_a: \mu_A \ne \mu_B$ . Design 2 (each subject does both, matched): paired t-test on the differences; $H_0: \mu_d = 0$ versus $H_a: \mu_d \ne 0$ , where $\mu_d$ is the mean within-subject difference.
(b) (1 point) Design 1 has two independent samples, so variances add (two-sample SE). Design 2 pairs each subject with themselves, removing between-subject variability, so it analyzes one list of differences with a one-sample procedure.
(c) (implicit in b) Design 2 requires the differences to be approximately normal (or $n \ge 30$ pairs) and the pairs to be a random sample; design 1 requires each of the two samples to satisfy the normal/large condition and be independent of the other.

Markers reward matching each design to the correct procedure and hypotheses and explaining the role of pairing.

Related dot points

Sources & how we know this

AP Statistics Course and Exam Description — College Board (2020)