AP Statistics · Unit 3 · Topic 3.12

Setting Up a Test for a Difference in Proportions Study Lab

Before you calculate anything, a significance test needs clear parameters, hypotheses that match the question, and conditions checked under the assumption that H₀ is true. Practice choosing Hₐ from the wording, then see why the pooled proportion appears in the test.

1. Set up the test

Every two-proportion test starts the same way: define the parameters, write hypotheses about p₁ − p₂, and check conditions using the pooled proportion. Pick a study, choose the alternative hypothesis the question calls for, and check the conditions.

H₀: p₁ − p₂ = 0 p̂c = (x₁ + x₂) ÷ (n₁ + n₂)

Preset studies

The question

Which alternative hypothesis fits?

State: parameters and hypotheses

Plan: check conditions with the pooled proportion

The test assumes H₀ is true, so both groups share one proportion. Its best estimate combines the two samples.

Successes x₁

Successes x₂

Sample size n₁

Sample size n₂

If H₀ were true

Null model N(0, SE)Observed p̂₁ − p̂₂

p̂₁

p̂₂

Pooled p̂c

Null SE

2. Key ideas

Parameters

Define both proportions

Name p₁ and p₂ in context, with the population or treatment and what counts as a success. Hypotheses are always about parameters, never about p̂₁ and p̂₂.

Null hypothesis

No difference

H₀: p₁ − p₂ = 0 (equivalently p₁ = p₂). The test asks how surprising the data would be if there were no difference.

Alternative

Direction comes from the question

Use > or < when the question names a direction (increase, lower, more), and ≠ when it asks about any difference. Decide before looking at the data.

Pooling

One shared proportion

If H₀ is true, both groups have the same p. The pooled proportion p̂c = (x₁ + x₂)/(n₁ + n₂) estimates it, and it is used in the conditions and the standard error.

Conditions

Random, 10%, large counts

Random samples or random assignment; each sample ≤ 10% of its population when sampling; and n₁p̂c, n₁(1 − p̂c), n₂p̂c, n₂(1 − p̂c) all at least 10.

Significance level

Set α in advance

α (often 0.05) is the cutoff for how small the p-value must be to reject H₀. Choose it before collecting data.

3. Quick reference

Writing hypotheses for a two-proportion z-test.

Question wording	Hₐ	Sides
"Is there a difference…?" "Do they differ…?"	p₁ − p₂ ≠ 0	Two-sided
"Is group 1 higher / more likely / improved…?"	p₁ − p₂ > 0	One-sided (right)
"Is group 1 lower / less likely / reduced…?"	p₁ − p₂ < 0	One-sided (left)
Pooled proportion	p̂c = (x₁ + x₂) ÷ (n₁ + n₂)	Used only in tests of H₀: p₁ = p₂
Null standard error	√( p̂c(1 − p̂c)(1/n₁ + 1/n₂) )	—
Large counts (test)	n₁p̂c, n₁(1 − p̂c), n₂p̂c, n₂(1 − p̂c) ≥ 10	Some teachers accept observed counts; pooled is the standard

4. On the FRQ

How to earn the points

Define both parameters in context, including the order: "p₁ = the true proportion of patients who get vaccinated after a reminder text; p₂ = … without a reminder."
Write hypotheses with parameter symbols: H₀: p₁ − p₂ = 0, Hₐ: p₁ − p₂ > 0.
Name the test: two-sample z-test for a difference in proportions.
Check large counts with the pooled proportion and show all four products.
Address random sampling or random assignment, and the 10% condition when sampling.

Common mistakes

Writing hypotheses with p̂ (statistics) instead of p (parameters).
Picking the direction of Hₐ after seeing which sample proportion is larger.
Using p̂₁ and p̂₂ separately in the test's standard error instead of the pooled p̂c.
Writing H₀: p₁ − p₂ > 0. The null always has the equals sign.
Forgetting to define which group is 1 and which is 2.

5. Practice questions

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

0 / 0 correct

6. Key terms

The vocabulary for topic 3.12. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 3 terminology guide.