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Unit 6

6.10 Setting Up a Test for the Difference of Two Population Proportions

3 min readjune 5, 2020

harrison-burnside

Harrison Burnside


Another way to check a statistical claim is to perform a significance test for the difference in two population proportions. As with any significance test, we have to write hypotheses, check our conditions and then calculate and conclude.

Hypotheses and Parameters

The first thing we need to do when setting up a significance test for the difference in two population proportions is to write out our hypotheses. Our null hypotheses will always have our two population proportions being equal, while our alternate has them either greater than, less than or not equal to each other.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-23%20at%2011.57-eQyBDBBrfpMo.png?alt=media&token=7cf28edc-aee1-40fe-8a41-38d90f4e795a
It is also important in this stage of setting up the test to identify what p1 and p2 represent. We have to define our parameters so the reader knows what we are truly comparing.

Conditions

We also must check our conditions for inference. The same three conditions apply as did for confidence intervals with one little small change in the normal check.

🌟Random

Probably the most important condition is that we need to be sure that both of our samples come from random samples. If we don't take a random sample from our population, then our findings suffer from sampling bias and we are stuck and we can't generalize our findings to our population. 😞

🌟Independence

To check that our sample is independent, we need to make sure that both of our populations are at least 10 times that of our samples. Also, if we are dealing with a randomized experiment, the random assignment of treatments classifies our samples as independently selected.

🌟Normal

When dealing with proportions, we always check our normal condition by using the Large Counts Condition, which states that our expected successes and failures is at least 10. With a 2 proportion z test, we have to combine our proportions to create a combined p-hat. This is what we use to find our expected failures and successes.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-23%20at%2012.01-6Pfyi1xuQs0Q.png?alt=media&token=ae1ef28e-a0b6-453f-aabc-ee22951a8f2e
Then we have to verify that each of our expected failures and successes are at least 10.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-23%20at%2012.03-q6oVEihOeF6o.png?alt=media&token=668e9c68-3186-4da9-bd21-8526a9bcc08c
This is because we are using a pooled sample.

Example

Let's return to our MJ vs. Lebron problem from earlier. Recall that MJ made 836/1623 shots and Lebron made 622/1493 shots. Instead of testing this claim with a confidence interval, let's test it using a 2 Prop Z Test to verify our results.

Hypotheses and Parameters

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-23%20at%2012.08-UcnU917akZoW.png?alt=media&token=e884006f-1807-48e5-8b0d-c27511b7baf2
Another great idea when writing our hypotheses is to use meaningful subscripts such as MJ and L that clarify which proportion matches which population.

Conditions

  • Random: Even though the problem never stated that they were random (and we discussed the problems with this in Unit 6.9) we are going to assume it is random.
  • Independent: It is reasonable to believe (and obviously true) that MJ took at least 16, 230 shots in his career and Lebron took at least 14,930 shots in his career, so the samples are independent.
  • Normal: This is the one that will be a bit different. First, we have to calculate our pooled p-hat. Using the formula above, we get 0.468
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-23%20at%2012.16-qUoLRkLqkcPO.png?alt=media&token=3bc1e071-d7ab-44f6-a30a-b087794cfff1
Next, we have to check our large counts condition using this pooled p-hat.
1623(0.468)>10, 1623(0.532)>10, 1493(0.468)>10, 1493(0.532)>10.
Now that we have checked conditions, we are ready to calculate and test our claim.

Resources:

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