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Significance Tests for a Difference of Proportions

Chapter 10 - Day 3 - Lesson 10.1

Learning Targets
  • Calculate the standardized test statistic and P-value for a test about a difference between two proportions.

  • Perform a significance test about a difference between two proportions.

Activity: Are Some Groups Underrepresented?
Answer Key:

In this Activity, we ask student to do a full four-step significance test for the difference of proportions….before we have taught them how to do a full four-step significance test for the difference of proportions. This experience is very similar to what students will go through on the AP Exam Free Response question #6 – the Investigative Task. The Investigative Task will ask students to think about a new concept that they didn’t specifically cover in the AP Stats course. But it is always something that can be done by using good statistical thinking and reasoning. This Activity is exactly that – it is asking students to take what they already know about significance tests and apply it to this new context. This whole idea is worth discussing with students.

The Big Mistake That All Students Will Make

In the formula for the standard deviation of the sampling distribution, students will most likely use the two different sample proportions. After all, this is exactly what they did yesterday when constructing a confidence interval for the difference of proportions. We let students make this mistake and we even had them come to the board and write up their incorrect solution. Then we revealed the truth.

In a two sample z-test for difference of proportions, we assume the null hypothesis is true (p1 = p2). So when we get to the formula for SD, it does not make sense to use different values (p1hat and p2hat) because we are assuming the two proportions are equal. Therefore we should be plugging in the same value for p1 and p2 in the formula for SD. So what value do we use? Well, the best guess is to take the two samples and combine them as if they came from the same population (after all we are assuming the two proportions are equal). Therefore we use the combined sample proportion in the formula for SD.

4-step White Board Critique

We used the Check Your Understanding to do a White Board Critique. Here is how it works:

  1. Assign students a 4-step problem to work on in pairs.

  2. Monitor the room to support student learning. As each pair finishes, send them to the white board to write up 1 of the steps STATE, PLAN, DO, or CONCLUDE. The first two pairs should handle the PLAN and the DO because they take the longest to write up.

  3. Once all 4 steps are on the board, call the class back together as a group. Ask them to critique the solution on the board as if it were a quiz or test question. Make any revisions with a red marker. This is your opportunity to make clear your expectations for a 4-step problem on an assessment.

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