AP Statistics Unit 6 Inference for Proportions

Topic 6.10: Setting Up a Test for the Difference of Two Population Proportions

Writing Hypotheses, Choosing the Procedure, and Checking Conditions
Video Follow-Along Worksheet
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Learning Objectives: Essential Knowledge:
🖩 Practice on the TI-84 →

Key Vocabulary

Null Hypothesis A claim of no difference or no effect, usually written as p1 - p2 = 0.
Alternative Hypothesis The claim supported by evidence, written with >, <, or ≠.
One-Sided Test A test with a directional alternative (p1 > p2 or p1 < p2).
Two-Sided Test A test with p1 ≠ p2.
Two-Sample z-Test The significance test used for comparing two population proportions.
Pooled Proportion The combined success proportion, written as p̂c, used for expected-count checks.
Expected Counts The values n1p̂c, n1(1 - p̂c), n2p̂c, and n2(1 - p̂c).
10% Condition For random samples without replacement, each sample size is at most 10% of its population.

Video 1: Stating Null and Alternative Hypotheses

[0:00 – 9:19]
1. [0:05] This lesson is about setting up a test for a difference in .
2. [0:13] The first goal is to state the hypothesis for a difference in proportions.
3. [0:20] The second goal is to state the hypothesis.
4. [1:25] Significance tests are used to test a about a population parameter.
Context — Effective Eye Drops? [1:42] In a clinical trial, 82 of 130 patients assigned azithromycin drops were cured within a week, compared with 74 of 149 assigned placebo drops.
5. [2:36] The null hypothesis is a claim of no difference, so a common form is p1 - p2 = .
6. [3:25] In this example, p1 is the true cure proportion for drops.
7. [3:34] And p2 is the true cure proportion for drops.
8. [4:13] Because researchers suspected azithromycin works better, the alternative is p1 p2.
Hypothesis Structure Summary: [4:53] Null hypotheses use equality; alternatives use strict inequalities. Direction comes from the research question stated before data collection.
9. [5:00] The null is a statement of , while the alternative has a strict .
10. [5:24 – 5:32] Alternatives with > or < are -sided; alternatives with ≠ are -sided.
11. [5:54] Never write hypotheses using sample statistics like p̂; hypotheses must use population .
12. [6:16] Always any parameter symbols you use.
Practice Context — A Bright Idea [6:33] A marketing agent compares sunglasses-purchase proportions in Soltown and Brightville and asks whether there is a difference.
13. [7:34] For this practice, the null can be written as pS - pB = .
14. [8:00] Because the question asks about "a difference," the alternative uses equal.
15. [8:18] Here, pS is the proportion of all residents who purchased sunglasses in the last 12 months.

Video 2: Identifying the Test and Checking Conditions

[0:00 – 9:37]
Context — Effective Eye Drops Revisited [0:28] The same azithromycin vs placebo trial is used to identify the correct significance test and verify conditions.
16. [1:52] The appropriate procedure is a two-sample test for a difference in proportions.
17. [2:59] Before checking large counts, compute the combined (or ) proportion of successes, p̂c.
Pooled Proportion Formula: [3:32] p̂c = (X1 + X2) / (n1 + n2), using successes and totals from both groups.
18. [3:32] For the eye-drop data, p̂c = 156/279 = .
19. [4:17] With random samples and no replacement, each sample size should be at most % of its population.
20. [4:17] For shape, expected successes and expected in both groups should each be at least 10.
21. [5:06] In experiments, independence is checked by confirming subjects were randomly to groups.
22. [8:35] The named procedure for this lesson is a two-sample z-test for a difference in population .

Key Takeaways

[Summary]
  1. For a difference in two population proportions, the null uses equality: p1 - p2 = 0.
  2. The alternative uses a strict inequality (>, <, or ≠) based on the original question of interest.
  3. The correct significance test is a two-sample z-test for a difference in population proportions.
  4. Check independence and large counts, using the pooled proportion for expected-success and expected-failure counts.

Reflection Questions

[AI-Graded]
R1. In the pink-eye eye-drop study, define p1 and p2, then state appropriate null and alternative hypotheses for testing whether azithromycin is more effective than placebo. Explain why the alternative is one-sided.
R2. For the Soltown vs Brightville sunglasses question, state null and alternative hypotheses in symbols and context. Explain why the alternative should be two-sided and why hypotheses should use parameters, not sample statistics.

Exit Ticket

A district compares two reading apps. In random samples, 78 of 120 students using App A met a benchmark, and 66 of 120 students using App B met a benchmark. Assume each sample is less than 10% of its population. The district wants to test whether App A has a higher true success proportion than App B.

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