AP Statistics · Unit 4 · Topic 4.4

Setting Up a Test for a Mean Study Lab

Before any calculation, a significance test needs a clear setup: define the parameter in context, write H₀ and Hₐ, name the procedure, and check the conditions. Work through each scenario, check your choices, and see what the alternative hypothesis means on the t distribution.

1. Set up the test

Pick a scenario, make your four choices, then check them.

H₀: μ = μ₀ Hₐ: μ < μ₀, μ > μ₀, or μ ≠ μ₀ Paired: H₀: μ_d = 0

Scenarios

1. Parameter

2. Procedure

3. Null hypothesis H₀

4. Alternative hypothesis Hₐ

Your setup, written out

Conditions for this scenario

What would count as evidence for Hₐ?

Significance level α

Procedure

n, df

Direction of Hₐ

t values beyond

2. Key ideas

Parameter

Hypotheses are about μ, never x̄

Hypotheses describe the population, so they use the parameter μ (or μ_d). Define it in context: "μ = the true mean delivery time of all pizzas from this chain."

Null

H₀ has the equals sign

The null hypothesis is the claim of "no effect" or "no change": μ = μ₀, or for paired data μ_d = 0. We assume it is true while we check how surprising the data are.

Alternative

Hₐ is what you suspect

Read the question for direction words: "more than," "less than," "different from." Choose Hₐ before looking at the data. Use ≠ when either direction matters.

Paired or not?

Same subjects twice → paired

If each subject gives two measurements (before/after, left/right, two conditions), or subjects are matched in pairs, test μ_d using the differences. Define the order of subtraction.

Conditions

Random, 10%, Normal/Large

Random sample or random assignment; n ≤ 10% of the population when sampling without replacement; population normal, n ≥ 30, or a graph of the sample (or differences) with no strong skew or outliers.

Significance level

α is set in advance

α is the cutoff for "convincing." Common choices are 0.05 and 0.01. A smaller α demands stronger evidence before rejecting H₀.

3. Quick reference

Wording in the question and the hypotheses it leads to.

The question says…	Parameter	H₀	Hₐ	Procedure
"…the mean is greater than 30 minutes"	μ	μ = 30	μ > 30	One-sample t test
"…the mean is less than the advertised 20 hours"	μ	μ = 20	μ < 20	One-sample t test
"…the mean differs from 500 mL"	μ	μ = 500	μ ≠ 500	One-sample t test
"…scores improve after the course" (same students)	μ_d, d = after − before	μ_d = 0	μ_d > 0	Paired t test
"…two brands wear differently" (both on each car)	μ_d, d = A − B	μ_d = 0	μ_d ≠ 0	Paired t test
Two separate random samples (e.g., juniors vs. seniors)	μ₁ − μ₂	μ₁ − μ₂ = 0	depends	Two-sample t test (topics 4.9–4.10)

4. On the FRQ

How to earn the points

Write both hypotheses with the parameter symbol and define it in words: "where μ = the true mean delivery time for all pizzas from this chain."
For paired data, define the difference: "μ_d = the true mean difference in score (music − silence) for volunteers like these."
Name the test: "one-sample t test for a mean" or "paired t test for a mean difference."
Check each condition with a reason tied to the problem, such as "25 < 10% of all deliveries" or "the dotplot of differences shows no outliers."
State α if it is given; otherwise use α = 0.05 and say so.

Common mistakes

Writing hypotheses about x̄ ("H₀: x̄ = 30"). We already know x̄; the question is about μ.
Putting the suspected claim in H₀, or using ≥ or ≤ in H₀ (on the AP exam, H₀ uses =).
Choosing the direction of Hₐ after seeing the sample result.
Using a two-sample test when the same subjects were measured twice.
Saying "the population is normal" without evidence. For small samples, look at a graph of the data.

5. Practice questions

AP-style multiple choice. Pick an answer to see the explanation.

0 / 0 correct

6. Key terms

The vocabulary for topic 4.4. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 4 terminology guide.