AP Statistics Unit 7 Inference for Means

Topic 7.8: Setting Up a Test for the Difference of Two Population Means

Writing Null and Alternative Hypotheses, Choosing Direction, and Defining Parameters
Video Follow-Along Worksheet
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Learning Objectives: Essential Knowledge:
🖩 Practice on the TI-84 →

Key Vocabulary

Null Hypothesis A statement of no treatment effect, no difference, or no change in the population means.
Alternative Hypothesis A statement that there is a treatment effect or a difference in the population means.
Parameter A population value, such as a true mean or a true difference in means, that the test is about.
One-Sided Alternative An alternative hypothesis that uses > or < because the claim specifies a direction.
Two-Sided Alternative An alternative hypothesis that uses ≠ because the claim is only that the means are different.
Treatment Effect A change caused by the explanatory condition, such as being exposed to someone yawning.
Equality The type of statement required in the null hypothesis.
Strict Inequality The type of symbol required in the alternative hypothesis: >, <, or ≠.
Order of Subtraction The order in which the means are subtracted, which determines the direction of a one-sided inequality.
Subscripts Labels that identify which group each parameter refers to, such as Y for yawning exposure and N for no yawning exposure.
Statistic A sample value like x̄1 that should not appear in the hypotheses.

Video 1: Writing Hypotheses for Two Means

[0:00 – 6:05]
1. [0:08] We will learn how to state a hypothesis and an hypothesis in a test for a difference in means.
Context — Contagious Yawning Experiment [0:35 – 1:01] From 27 volunteers, 14 people heard a bedtime story while the storyteller yawned occasionally, and 13 people heard the same story without any yawning. Students recorded the number of times each person yawned and asked whether yawning is contagious.
2. [0:35 – 0:48] In the experiment, volunteers were exposed to yawning, and the remaining volunteers were not.
3. [1:01] The study asks whether yawning is contagious and whether people yawn , on average, when watching someone yawn.
4. [1:20 – 1:28] The null hypothesis represents the skeptical view that there is no effect, no difference between groups, and no change.
5. [1:34] In words, the mean number of yawns for people exposed to someone yawning is the as the mean number of yawns for people not exposed.
6. [1:48 – 2:03] In symbols, the null hypothesis can set the two population means equal or state that their difference is .
Null Hypothesis Forms: [1:20 – 2:03]
  1. No treatment effect, no difference, and no change all describe the null hypothesis.
  2. For the yawning study, H0: μY = μN or H0: μY − μN = 0.
  3. The null must include equality.
7. [2:22] The alternative hypothesis says there is a treatment effect or a between groups.
8. [2:29] For the yawning study, the mean number of yawns for the exposed group is hypothesized to be than the mean for the group with no yawning exposure.
9. [2:40] Because the claim gives a direction, the symbolic alternative must use an sign.
10. [2:55 – 3:10] A alternative is used when we want to show that one group has a larger or smaller mean than the other.
11. [3:15] A two-sided alternative is used when we want to show the groups have means without regard to which one is larger.
12. [3:25] The symbol used for a two-sided alternative is .
Alternative Hypothesis Rules: [2:55 – 4:25]
  1. Use a one-sided Ha when the claim is directional: μ1 − μ2 > 0 or μ1 − μ2 < 0.
  2. Use a two-sided Ha when the claim is only that the means differ: μ1 − μ2 ≠ 0.
  3. The direction of a one-sided inequality must match the context of the claim.
  4. Never use statistics like x̄1 or x̄2 in the hypotheses.
13. [3:28 – 3:40] In the yawning example, it would not make sense to use the opposite inequality or a two-sided alternative because the claim has a specific .
14. [3:48] Just as with confidence intervals, you must pay attention to the in which the means are subtracted.
15. [4:01] If the difference were rewritten as μN − μY, the one-sided inequality would have to .
16. [4:08 – 4:17] In summary, the null is a statement of , and the alternative always contains a inequality.
17. [4:25] Never refer to sample in the hypotheses; the statements must be about parameters.
18. [4:30] Remember to the parameters or subscripts that you use.
Practice — Chemistry vs. Physics Textbooks [4:35 – 5:22] A statistics major sampled words from a chemistry textbook and a physics textbook to decide whether the mean word lengths were different. The class then evaluated several multiple-choice hypothesis pairs.
Textbook n x̄ s
Chemistry2005.713.02
Physics2006.033.58
19. [4:50] In answer choice B, the null hypothesis was invalid because it did not set the difference equal to .
20. [5:08] For the textbook practice problem, the alternative hypothesis should be because the question asked whether the mean word lengths were different.
21. [5:22] Answer choice E is correct because it uses population , equality in the null, and inequality in the alternative.

Key Takeaways

[Summary]
  1. The null hypothesis for a difference in two population means states that the means are equal or that the difference is zero.
  2. The alternative hypothesis always uses a strict inequality: greater than, less than, or not equal to.
  3. Use a one-sided alternative for a directional claim and a two-sided alternative when the claim is simply that the means differ.
  4. If you reverse the order of subtraction in a one-sided alternative, you must reverse the inequality.
  5. Write hypotheses using parameters, define the subscripts, and never use sample statistics in the hypotheses.

Reflection Questions

[AI-Graded]
R1. Explain how to write the null and alternative hypotheses for the contagious-yawning study. Define μY and μN, describe the skeptic's view, write the hypotheses in words or symbols, and explain why the hypotheses must be about population parameters instead of sample statistics.
R2. Explain how to decide whether an alternative hypothesis should be one-sided or two-sided. Use the contagious-yawning study and the chemistry-vs.-physics textbook practice problem to compare directional and non-directional claims, and explain how reversing the order of subtraction changes a one-sided inequality.

Exit Ticket

A statistics major wants to test whether chemistry and physics textbooks differ in mean word length. A random sample of 200 words from a chemistry textbook had x̄C = 5.71 letters and sC = 3.02 letters. A random sample of 200 words from a physics textbook had x̄P = 6.03 letters and sP = 3.58 letters.

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