There are two important values that we need to calculate in order to carry out a test for the difference in two population proportions. We need to calculate a z-score and a p-value. Both need to be included in our response to receive full credit.
The z-score for our test is our critical value. We can calculate it by using the formula for any z-score:
That formula looks pretty 🤢, so luckily if we run our test on a calculator, the z-score is part of our calculator output.
As with any p-value, we are using the normal curve and our z-score to calculate the probability of obtaining such a value (or one less than/greater than) that value. Also, using a calculator here is much easier than doing it by hand.
Once we have our p-value and z-score, we can then make a judgment to either reject or fail to reject our null hypothesis.
If we are using our p-value, we are inspecting if it is lower than our given significance level (0.05 if not specified). If our p-value is really low, this means that it was an extremely rare occurrence (WOW value!) and we have reason to believe our null hypothesis (center of our sampling distribution) wasn't accurate.
Therefore, if the p is low, reject the Ho.
If you are using the z-score, you are going to use the empirical rule of a normal distribution to check if our z-score represents an extremely rare value on our sampling distribution. Any z-score higher than 2 or 3 gives us reason to reject the Ho.
Let's continue our example of MJ vs. Lebron to see if we can verify our conclusion from Unit 6.9. Recall that Michael Jordan made 836/1623 shots and Lebron James made 622/1493 shots. We are seeing if our data gives significant statistical evidence that Michael Jordan is actually better than Lebron.
To avoid the lengthy and cumbersome formulas above, we are going to use a graphing calculator and the appropriate "2 Prop Z Test" to test our hypothesis:
The two big values that we need to write down from our output is our z-score and p-value.
To conclude our test, we can look at how low our p-value is, or how high our z-score is.
Our p-value is the best way to conclude. In this problem, we can see that our p-value is essentially 0. Therefore, we can say:
"Since our p-value is less than 0.05 (~0<0.05), we reject our Ho. We have convincing evidence that the true population proportion of shots made by MJ is higher than the true population proportion of shots made by Lebron."
Notice that this supports our conclusion from the confidence interval.
If not given the data to calculate a p-value or if only given the z-score, we can still make a judgment on the null hypothesis. In the example above, we end up with a z-score of 5.5, which is pretty extreme. Therefore, we can conclude that we can reject our null hypothesis in favor of our alternate hypothesis.
🎥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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