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.
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.
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.
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.
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. 😞
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.
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.
Then we have to verify that each of our expected failures and successes are at least 10.
This is because we are using a pooled sample.
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.
Another great idea when writing our hypotheses is to use meaningful subscripts such as MJ and L that clarify which proportion matches which population.
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
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.
🎥Watch: AP Stats - Inference: Hypothesis tests for Proportions
✍️ Free Response Questions (FRQs)
👆 Unit 1: Exploring One-Variable Data
1.4Representing a Categorical Variable with Graphs
1.5Representing a Quantitative Variable with Graphs
1.6Describing the Distribution of a Quantitative Variable
1.7Summary Statistics for a Quantitative Variable
1.8Graphical Representations of Summary Statistics
1.9Comparing Distributions of a Quantitative Variable
✌️ Unit 2: Exploring Two-Variable Data
2.0 Unit 2 Overview: Exploring Two-Variable Data
2.1Introducing Statistics: Are Variables Related?
2.2Representing Two Categorical Variables
2.3Statistics for Two Categorical Variables
2.4Representing the Relationship Between Two Quantitative Variables
2.8Least Squares Regression
🔎 Unit 3: Collecting Data
3.5Introduction to Experimental Design
🎲 Unit 4: Probability, Random Variables, and Probability Distributions
4.1Introducing Statistics: Random and Non-Random Patterns?
4.7Introduction to Random Variables and Probability Distributions
4.8Mean and Standard Deviation of Random Variables
4.9Combining Random Variables
4.11Parameters for a Binomial Distribution
📊 Unit 5: Sampling Distributions
5.0Unit 5 Overview: Sampling Distributions
5.1Introducing Statistics: Why Is My Sample Not Like Yours?
5.4Biased and Unbiased Point Estimates
5.6Sampling Distributions for Differences in Sample Proportions
⚖️ Unit 6: Inference for Categorical Data: Proportions
6.0Unit 6 Overview: Inference for Categorical Data: Proportions
6.1Introducing Statistics: Why Be Normal?
6.2Constructing a Confidence Interval for a Population Proportion
6.3Justifying a Claim Based on a Confidence Interval for a Population Proportion
6.4Setting Up a Test for a Population Proportion
6.6Concluding a Test for a Population Proportion
6.7Potential Errors When Performing Tests
6.8Confidence Intervals for the Difference of Two Proportions
6.9Justifying a Claim Based on a Confidence Interval for a Difference of Population Proportions
6.10Setting Up a Test for the Difference of Two Population Proportions
😼 Unit 7: Inference for Qualitative Data: Means
7.1Introducing Statistics: Should I Worry About Error?
7.2Constructing a Confidence Interval for a Population Mean
7.3Justifying a Claim About a Population Mean Based on a Confidence Interval
7.4Setting Up a Test for a Population Mean
7.5Carrying Out a Test for a Population Mean
7.6Confidence Intervals for the Difference of Two Means
7.7Justifying a Claim About the Difference of Two Means Based on a Confidence Interval
7.8Setting Up a Test for the Difference of Two Population Means
7.9Carrying Out a Test for the Difference of Two Population Means
✳️ Unit 8: Inference for Categorical Data: Chi-Square
📈 Unit 9: Inference for Quantitative Data: Slopes
🧐 Multiple Choice Questions (MCQs)
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