Confidence Intervals for Differences in Population Proportions

Introduction

Key Concepts

Understanding Population Proportions

In statistics, a population proportion refers to the fraction of individuals in a population that possess a particular characteristic. It is denoted as $p$ for one population and $p_1$, $p_2$ for two distinct populations. For example, $p_1$ could represent the proportion of students who prefer online classes, while $p_2$ represents those who prefer in-person classes.

Difference Between Population Proportions

The difference between two population proportions is expressed as $p_1 - p_2$. Estimating this difference is crucial when comparing the prevalence of a characteristic between two distinct populations. For instance, assessing if the proportion of users who prefer a new product differs between two age groups involves calculating $p_1 - p_2$.

Confidence Interval Basics

A confidence interval provides a range of plausible values for a population parameter, based on sample data. The confidence level, typically expressed as 95%, signifies the probability that the interval contains the true population parameter. For differences in population proportions, the confidence interval offers a range within which $p_1 - p_2$ likely falls.

Constructing Confidence Intervals for Differences in Proportions

Constructing a confidence interval for the difference between two population proportions involves several steps:

1. Sample Proportions: Calculate the sample proportions, $\hat{p}_1$ and $\hat{p}_2$, from the respective samples.
2. Difference in Sample Proportions: Compute the difference $\hat{p}_1 - \hat{p}_2$.
3. Standard Error: Determine the standard error (SE) of the difference: $$SE = \sqrt{ \frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2} }$$ where $n_1$ and $n_2$ are the sample sizes.
4. Z-Score: Identify the z-score corresponding to the desired confidence level, such as $1.96$ for 95% confidence.
5. Margin of Error: Calculate the margin of error (ME): $$ME = z \times SE$$
6. Confidence Interval: Construct the confidence interval: $$ (\hat{p}_1 - \hat{p}_2) \pm ME $$

Assumptions for Valid Confidence Intervals

Several assumptions ensure the validity of the confidence interval for differences in proportions:

• Random Sampling: Samples must be randomly selected from their respective populations.
• Independence: The two samples must be independent of each other.
• Normality: The sampling distribution of the difference in proportions should be approximately normal. This is typically satisfied if both $n_1\hat{p}_1$, $n_1(1 - \hat{p}_1)$, $n_2\hat{p}_2$, and $n_2(1 - \hat{p}_2)$ are at least 10.

Example Calculation

Suppose a survey is conducted to compare the proportion of Collegeboard AP students who prefer studying in the library versus at home. From a sample of 200 students, 120 prefer the library ($\hat{p}_1 = 0.60$). From another sample of 150 students, 90 prefer studying at home ($\hat{p}_2 = 0.60$). To construct a 95% confidence interval for $p_1 - p_2$:

1. Difference in sample proportions: $0.60 - 0.60 = 0$.
2. Standard Error: $$SE = \sqrt{ \frac{0.60 \times 0.40}{200} + \frac{0.60 \times 0.40}{150} } = \sqrt{ \frac{0.24}{200} + \frac{0.24}{150} } = \sqrt{0.0012 + 0.0016} = \sqrt{0.0028} \approx 0.0529$$
3. Z-score for 95% confidence: $1.96$.
4. Margin of Error: $$ME = 1.96 \times 0.0529 \approx 0.1037$$
5. Confidence Interval: $$0 \pm 0.1037 = (-0.1037, 0.1037)$$

Interpretation: We are 95% confident that the true difference in population proportions $p_1 - p_2$ lies between -0.1037 and 0.1037. This interval includes zero, suggesting no significant difference between the two preferences.

Interpreting Confidence Intervals

When interpreting confidence intervals for differences in proportions:

• Includes Zero: If the interval contains zero, there is no evidence of a significant difference between the two population proportions.
• Excluding Zero: If zero is not within the interval, a significant difference exists.
• Direction of Difference: The sign of the interval endpoints indicates the direction of the difference.

Common Mistakes to Avoid

• Miscalculating Standard Error: Ensure both sample sizes and proportions are correctly incorporated into the SE formula.
• Ignoring Assumptions: Always verify that the assumptions for constructing the confidence interval are met.
• Incorrect Z-Score: Use the appropriate z-score corresponding to the desired confidence level.
• Sample Size: Small sample sizes can lead to inaccurate confidence intervals due to violated normality assumptions.

Applications in AP Statistics

Understanding confidence intervals for differences in population proportions enables AP Statistics students to perform comparative analyses in various contexts, such as:

• Comparing voting preferences between demographic groups.
• Assessing the effectiveness of different teaching methods.
• Evaluating the prevalence of certain behaviors across populations.

Comparison Table

Summary and Key Takeaways