Comparing Two Worlds: Confidence Intervals for a Difference Between Proportions
Welcome to one of the most useful tools in statistics! So far, you have learned how to estimate a single population proportion (like the percentage of all students who prefer pizza over tacos). But in the real world, we often want to compare two different groups. For example, do seniors prefer pizza more than freshmen do? Is a new medication more effective for women than for men?
In this chapter, we will learn how to build a "safety net"—a confidence interval—to estimate the true difference between two population proportions (\(p_1 - p_2\)).
1. The Foundation: The Sampling Distribution
Before we build an interval, we need to understand what happens if we take many, many pairs of samples. This is called the sampling distribution for the difference between sample proportions.
If we take a sample from Group 1 (\(\hat{p}_1\)) and a sample from Group 2 (\(\hat{p}_2\)), the difference (\(\hat{p}_1 - \hat{p}_2\)) is our point estimator. Here is what you need to know about its distribution:
- The Mean: The mean of all possible differences \(\hat{p}_1 - \hat{p}_2\) is exactly equal to the true difference between the population proportions, \(p_1 - p_2\). This makes it an unbiased estimator.
- The Standard Deviation: As long as the samples are independent, the standard deviation of the difference is:
\(\sigma_{\hat{p}_1 - \hat{p}_2} = \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}\) - The Shape: The distribution will be approximately Normal if the sample sizes are large enough (we check this using the Large Counts condition).
Quick Review: We use parameters (\(p\)) for populations and statistics (\(\hat{p}\)) for samples. Since we usually don't know the population parameters, we use the Standard Error (SE) in our actual calculations:
\(SE_{\hat{p}_1 - \hat{p}_2} = \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}\)
2. The Three Commandments: Checking Conditions
In AP Statistics, you must verify conditions before doing any inference. If you don't check these, your interval might be "statistically illegal!"
1. Randomization: The data must come from two independent random samples or from a randomized experiment. This allows us to generalize our results to the populations.
2. 10% Condition: If you are sampling without replacement, your sample sizes must be less than 10% of their respective populations (\(n_1 < 0.10N_1\) and \(n_2 < 0.10N_2\)). This ensures the observations are independent enough to use our formulas. Note: We don't check this for randomized experiments!
3. Normal (Large Counts): We need to ensure the sampling distribution is approximately Normal. Check that there are at least 10 "successes" and 10 "failures" in each group:
Group 1: \(n_1\hat{p}_1 \ge 10\) and \(n_1(1-\hat{p}_1) \ge 10\)
Group 2: \(n_2\hat{p}_2 \ge 10\) and \(n_2(1-\hat{p}_2) \ge 10\)
Key Takeaway: Always list these conditions and show your math (e.g., don't just say "it's Normal," show that \(50 \times 0.4 = 20\), which is \(\ge 10\)).
3. Constructing the Confidence Interval
The general formula for any confidence interval is:
Statistic \(\pm\) (Critical Value) \(\times\) (Standard Error)
For the difference between two proportions, the formula looks like this:
\((\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}\)
Step-by-Step Process:
- Identify the populations and parameters: Define what \(p_1\) and \(p_2\) represent in context.
- Name the procedure: Use the full name: "Two-proportion z-interval for \(p_1 - p_2\)."
- Verify conditions: Random, 10%, and Large Counts (as discussed above).
- Calculate:
- Find the point estimate: \(\hat{p}_1 - \hat{p}_2\).
- Find the critical value \(z^*\) using your calculator or Table A (e.g., for 95% confidence, \(z^* = 1.96\)).
- Plug the numbers into the formula to get your interval.
- Interpret: "We are \(C\%\) confident that the interval from [lower bound] to [upper bound] captures the true difference in the proportion of [Context Group 1] and [Context Group 2]."
Common Mistake to Avoid: "Calculator speak." While your graphing calculator can do this automatically (2-PropZInt), you must show the formula or the named procedure and the resulting interval to get full credit on the AP Exam.
4. Justifying a Claim: The Power of Zero
One of the most common questions on the AP Exam asks: "Based on your interval, is there evidence of a significant difference between the two groups?"
To answer this, look at the number 0. Zero represents "no difference."
- If 0 is NOT in the interval: (e.g., the interval is \((0.02, 0.15)\) or \((-0.20, -0.05)\)).
Conclusion: Since the entire interval is above (or below) zero, we have evidence of a convincing difference between the two proportions. - If 0 IS in the interval: (e.g., the interval is \((-0.05, 0.10)\)).
Conclusion: Since zero is a plausible value for the difference, we do not have evidence of a convincing difference between the two proportions.
Did you know? If the entire interval is positive (e.g., \(0.04\) to \(0.12\)), it suggests that \(p_1\) is likely larger than \(p_2\). If it's all negative, \(p_2\) is likely the larger one!
5. Margin of Error and Sample Size
The Margin of Error (ME) is the "plus or minus" part of your interval: \(z^* \times SE\). To make your interval more precise (smaller ME), you can:
- Increase the sample size (\(n\)): Larger samples give us more information and less variability.
- Decrease the confidence level: A 90% interval is narrower (more precise) than a 99% interval, but you are less sure that it actually "caught" the true difference.
Analogy: Think of a confidence interval like catching a fish. A wider net (higher confidence) makes you more likely to catch the fish, but a smaller net (lower confidence) gives you a more precise location of where the fish might be.
Summary Checklist
• Point Estimate: \(\hat{p}_1 - \hat{p}_2\)
• Standard Error: Combined variability of both samples.
• Conditions: Random, 10%, and Large Counts (10 successes/failures per group).
• Claim Justification: If the interval contains 0, there is no evidence of a difference. If it doesn't contain 0, there is evidence of a difference.
Don't worry if the formulas look intimidating! With practice, you'll realize they follow the same logic as the single-proportion intervals you've already mastered. Just take it one step at a time!