AP Statistics Unit 6 Inference for Proportions

Topic 6.11: Carrying Out a Test for the Difference of Two Population Proportions

Calculating the Test Statistic, p-Value, and Conclusion
Video Follow-Along Worksheet
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Learning Objectives: Essential Knowledge:

Key Vocabulary

Standardized Test Statistic The value z that compares the observed difference in sample proportions to what is expected under the null hypothesis.
Pooled Proportion The combined success proportion, written as p̂c, used in the standard error for a significance test.
p-Value The probability, assuming the null is true, of getting a result as extreme or more extreme than the observed result.
Right-Tail p-Value Used when the alternative is greater than, such as p1 > p2.
Two-Sided p-Value Used when the alternative is not equal to, so extreme results in both directions count.
Significance Level The cutoff, written as α, used to decide whether the evidence is strong enough to reject the null hypothesis.
Reject H0 A decision made when the p-value is less than or equal to α, supporting the alternative hypothesis.
Fail to Reject H0 A decision made when the p-value is greater than α, meaning the data do not provide convincing evidence for the alternative.

Video 1: Calculating the Test Statistic and p-Value

[0:00 – 10:28]
1. [0:08] This topic is about carrying out a significance test for a difference in .
2. [0:18] One goal is to calculate an appropriate test .
3. [0:25] Another goal is to calculate a .
Context — Effective Eye Drops? [0:30] In a clinical trial, 82 of 130 patients randomly assigned azithromycin drops were cured within a week, compared with 74 of 149 assigned placebo drops.
4. [1:57] In the eye-drop study, the difference in group proportions is 82/130 - 74/149 = .
5. [2:44] The standardized test statistic equals the statistic minus the parameter divided by the of the statistic.
Test Statistic Formula: [2:54 – 3:25] For a two-sample z-test for a difference in proportions,
z = [ (p̂1 - p̂2) - 0 ] / √[ p̂c(1 - p̂c)(1/n1 + 1/n2) ]
where p̂c = (X1 + X2) / (n1 + n2).
6. [3:55] For the eye-drop data, the pooled proportion is 156/279 = .
7. [4:22] Plugging the values into the formula gives a standardized test statistic of .
8. [6:13] Using Table A or technology, the p-value for the eye-drop study is .
Practice Context — A Bright Idea [6:25] Of 400 randomly selected Soltown residents, 314 purchased sunglasses in the last 12 months. Of 550 randomly selected Brightville residents, 452 purchased sunglasses.
9. [7:05] Because the sunglasses question asks whether there is a difference, the alternative is .
10. [7:48] For the practice data, the pooled proportion is .
11. [8:29] The practice problem gives a z statistic of .
12. [9:05] The two-sided p-value for the practice problem is .

Video 2: Interpreting the p-Value and Stating a Conclusion

[0:00 – 10:03]
Context — Eye Drops Revisited [1:29 – 2:09] From earlier work: H0: p1 - p2 = 0, Ha: p1 - p2 > 0, with p̂1 - p̂2 = 0.134, z = 2.25, and p-value = 0.0122.
13. [2:35] The p-value measures how likely it is to get evidence as strong or stronger by chance alone when the hypothesis is true.
p-Value Interpretation Template: [2:50] Assuming the null hypothesis is true, there is a p-value probability of getting a difference in sample proportions as extreme or more extreme than the observed result by chance alone in the random assignment or random samples.
14. [3:49] For a greater-than alternative, the interpretation says the observed difference in proportions or by chance alone.
15. [4:23] If the p-value is less than or equal to the significance level, we the null hypothesis.
16. [4:56] If the p-value is greater than the significance level, we the null hypothesis.
17. [5:22] When no significance level is given, the video uses α = .
Context — Sunglasses Revisited [6:42] For the Soltown and Brightville comparison, the p-value is 0.1556 and the significance level is 0.10.
18. [7:47] Because the alternative is two-sided, the interpretation says the sample difference of -0.037 or one more in either direction.
19. [8:16] Since 0.1556 is greater than 0.10, we the null hypothesis.

Video 3: A Complete Significance Test from Start to Finish

[0:00 – 8:33]
Context — 2012 AP Exam Survey [0:24 – 0:54] In December 2008, 676 of 1009 randomly selected U.S. adults said television commercials are an effective way to promote a new product. In December 2007, 622 of 1020 randomly selected U.S. adults said yes.
20. [1:30] Because the question asks whether the proportion changed, the alternative should be .
21. [2:47] The named procedure is a two-sample for a difference in proportions.
22. [3:30] The pooled proportion for the survey data is .
23. [5:01] The standardized test statistic for the survey example is .
24. [5:30] The two-sided p-value for the survey example is .
25. [6:36] Because 0.0048 is less than 0.05, we the null hypothesis.
Complete Test Checklist: [7:36] A full significance test should include hypotheses, parameter definitions, significance level, named procedure, condition checks, test statistic, p-value, and a conclusion in context.

Key Takeaways

[Summary]
  1. For a significance test about two population proportions, compute the pooled proportion first and use it in the standard error.
  2. Use the standard normal distribution to turn the z test statistic into a p-value.
  3. The alternative hypothesis determines whether the p-value comes from the left tail, right tail, or both tails.
  4. Interpret the p-value by assuming the null hypothesis is true and describing chance variation.
  5. Compare the p-value to α to reject or fail to reject H0, then state the conclusion in context.

Reflection Questions

[AI-Graded]
R1. In the pink-eye eye-drop study, explain how to calculate the pooled proportion, the standardized test statistic, and the p-value. Include the numerical results and explain why the p-value is found in the right tail. Then interpret the p-value in context.
R2. For the Soltown vs Brightville sunglasses study, interpret the p-value of 0.1556 and state the conclusion at the α = 0.10 significance level. Explain why the conclusion is fail to reject H0 rather than saying the two population proportions are proven equal.

Exit Ticket

A school compares two study apps. In random samples, 84 of 120 students using App A met a benchmark, and 66 of 120 students using App B met the benchmark. Assume each sample is less than 10% of its population. The school wants to test whether App A has a higher true benchmark-success proportion than App B.

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