Testing a Claim about a Difference Between Two Proportions (Lesson 10.2)
Chapter 10 - Day 2
Chapter 10
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8
Day 9
Day 10
Day 11
Day 12
Day 13
Day 14
All Chapters
Learning Targets
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State hypotheses and check conditions for performing a significance test about a difference between two proportions.
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Calculate the standardized test statistic and P-value for a significance test about a difference between two proportions.
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Use the four-step process to perform a significance test about a difference between two proportions.
Experience First
In the state of Michigan, there is no bigger rivalry than Michigan versus Michigan State (GO BLUE!). You may want to edit the Word document to include colleges from your region.
In Lesson 10.1, we decided to allow students to use an applet for the calculations for the confidence interval in order to avoid some messy formulas and focus more on interpreting the results. We will do the same here with the significance test in this lesson.
Students might need some support in coming up with the hypotheses for this significance test. We suggest starting with the alternative hypothesis and looking at the wording of the question:
Do the data provide convincing evidence that a higher proportion of recent graduates from Michigan get jobs than recent graduates from MSU?
Students should be able to turn this into the alternative hypothesis: p1 > p2. Help them to see that this is exactly the same as writing p1 - p2 > 0. Then from last chapter, we know that the null hypothesis always says the parameter is equal to some value, so we get a null hypothesis of p1 - p2 = 0 (or p1 = p2).
Students will use the Two sample z test for proportions applet (part of the Traditional Inference applets at www.statsmedic.com/applets. Notice that the data will need to be input into a two-way table.
Formalize Later
When debriefing the activity, here are a few items to highlight:
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When defining the parameter, it is important to include the word "difference" and to clearly indicate the order of subtraction.
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It is perfectly acceptable to subtract in the other order (MSU - Michigan), as long as you are consistent with this choice throughout. The correct z-test statistic would be z = -1.923 and the P-value would be the same.
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The Random and Large Counts condition need to be checked for two samples (twice the work!).
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Don't forget about the interpretation of the P-value (after all it is the holy grail). Students don't need to write this down, but it will help them to better understand the conclusion:"Assuming the null hypothesis is true (the proportion of grads that get jobs is the same at Michigan and MSU), there is a 0.027 probability of getting a difference of sample proportions of 0.04 or greater purely by chance."
Because we are not requiring students to know or use the formulas here, it is not necessary to teach them the idea of the combined sample proportion. For those interested, this proportion is calculated by pooling the two samples into one. This is done because we assume the null hypothesis is true, that there is no difference between the two population proportions. This combined sample proportion is then used in the calculation of the standard error (and can also be used to check the Large Counts condition).