AP Statistics Unit 6 Inference for Proportions

Topic 6.4: Setting Up a Test for a Population Proportion

Null and Alternative Hypotheses, One-Sample z Tests, and Condition Checks
Video Follow-Along Worksheet
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Learning Objectives:
🖩 Practice on the TI-84 →

Key Vocabulary

Null Hypothesis The initial or “no difference / no change” claim; for proportions, H0: p = p0
Alternative Hypothesis The claim we hope to support with evidence; for proportions, Ha: p > p0, p < p0, or p ≠ p0
One-Sided Alternative Uses > or < and looks for evidence in only one direction
Two-Sided Alternative Uses ≠ and looks for evidence on either side of the null value
Procedure For one categorical variable, use a one-sample z test for a population proportion
Large Counts Condition For tests, assume H0 is true and verify that np0 ≥ 10 and n(1 - p0) ≥ 10

Video 1: Stating Null & Alternative Hypotheses

[0:00-8:33]
1. [0:20] In this video, you learn how to state a hypothesis and an hypothesis for a population proportion.
Lemonade Example: [1:00] Two student researchers randomly selected 30 students. Each tasted lemonade from a green cup and a white cup, even though both cups contained the same lemonade. Of the 30 students, 18 said the green cup tasted more natural.
Hypothesis Structure: [2:00] H0: p = p0
Ha: p > p0, p < p0, or p ≠ p0
Always define p in context as a population proportion.
2. [1:36] The null hypothesis is often a claim of “no ” or “no .”
3. [2:00] For the lemonade study, the null hypothesis is H0: = .
4. [2:28] Until we have convincing evidence otherwise, we the null hypothesis is correct.
5. [3:09] Because researchers wanted to know if more than 50% would choose the green cup, the alternative is Ha: p 0.5.
6. [3:46] The null hypothesis always contains an , while the alternative hypothesis contains a strict .
7. [4:24] If the alternative uses > or <, the test is . If it uses ≠, the test is .
8. [4:54] The choice of alternative should be based on the and stated data collection begins.
9. [5:12] Never include a statistic such as in the hypotheses. Hypotheses should be about the , not the sample.
Practice Problem: [5:49] A newspaper reports that 40% of adults in the United States would say football is their favorite sport. A mayor wonders if the proportion in the town differs from the national proportion.
10. [6:26] For the mayor’s study, the null is H0: p = and the alternative is Ha: p 0.40.
11. [7:10] Define the parameter as the proportion of all adults in the town who football is their favorite sport.
12. [7:52] Summary: a null hypothesis for a population proportion is always in the form p equals some between 0 and 1.

Video 2: Identifying the Procedure & Checking Conditions

[0:00-7:26]
13. [0:21] In this video, you learn how to identify the correct test and how to verify the .
Recall the Lemonade Study: [1:20] The hypotheses are H0: p = 0.5 and Ha: p > 0.5, where p is the proportion of all students at the school who would choose the green cup.
14. [2:04] When testing a claim about the proportion of successes in a single population, use a for a population proportion.
Conditions for a One-Sample z Test for p: [3:08] 1. Random sample or randomized experiment
2. When sampling without replacement, n ≤ 10% of the population size
3. Assuming H0 is true, verify np0 ≥ 10 and n(1 - p0) ≥ 10
15. [3:08] To check independence, the data should come from a sample or a randomized experiment.
16. [3:08] When sampling without replacement, we check that n is less than or equal to % of the population size.
17. [3:31] For the large counts condition in a significance test, we use the null value , not p̂.
18. [3:31] Specifically, we verify that np0 and n(1 - p0) are both at least .
19. [4:30] In the lemonade study, the first condition is met because the students were selected .
20. [4:30] For the 10% condition, with n = 30 we need to assume the school has more than students.
21. [4:58] Using H0: p = 0.5, the expected number of successes is 30(0.5) = and the expected number of failures is also .
22. [4:58] When checking the third condition, we use the counts under the null, not the observed sample counts.
Practice Conditions Check: [5:28] For the mayor’s study, n = 100 and the null hypothesis value is p0 = 0.40.
23. [6:03] In the mayor’s study, np0 = and n(1 - p0) = , so the large counts condition is met.
24. [6:39] Key takeaway: for a single categorical variable, the appropriate significance test is a one-sample test for a population proportion.

Reflection Questions

[AI-Graded]
R1. A snack company claims that 50% of all teens would prefer its new granola bar to the old version. A researcher wants to investigate whether the true proportion is higher than 50%. Define the parameter, state the null and alternative hypotheses, and explain why the alternative is one-sided instead of two-sided.
R2. A town health department wants to test whether the proportion of all adults in the town who would say walking is their main exercise differs from 0.35. A random sample of 120 adults will be selected. Identify the correct inference procedure and verify whether the conditions for the test are met. Be sure to address the random condition, the 10% condition, the large counts condition, and why the large counts check uses 0.35 rather than the sample proportion.

Exit Ticket

A state report says that 65% of all high school students get at least 8 hours of sleep on school nights. A principal wonders whether the proportion at her school is lower than 65%. She randomly selects 90 students from the school, and 50 say they usually get at least 8 hours of sleep.

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