Confidence Intervals for Differences in Population Proportions

A Confidence Interval (CI) for the difference in proportions is a symmetric range centered at the observed difference between two sample proportions ($\hat{p}_1 - \hat{p}_2$).

The primary goal is to capture the true population difference ($p_1 - p_2$) with a specified level of confidence, such as 95% or 99%.

The interval is constructed by taking the point estimate and adding or subtracting a Margin of Error (ME), which represents the maximum expected distance between the estimate and the true parameter.

The method relies on the Sampling Distribution of the Difference in Proportions, which describes how the difference between two sample proportions behaves across many repeated samples.

According to the Central Limit Theorem, if sample sizes are sufficiently large, this sampling distribution will be approximately normal with a mean equal to the true difference ($p_1 - p_2$).

The spread of this distribution is measured by the Standard Error (SE), which estimates the standard deviation of the sampling distribution using sample data: $SE = \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$.

Step 1: Identify Parameters: Define $p_1$ and $p_2$ as the true proportions of the two populations being compared.

Step 2: Verify Conditions: Ensure the data meets the requirements for Randomness, Independence (10% rule), and Normality (Large Counts).

Step 3: Calculate the Point Estimate: Find the difference between the two sample proportions: $\text{Estimate} = \hat{p}_1 - \hat{p}_2$.

Step 4: Determine the Critical Value ($z^*$): Select the z-score corresponding to the desired confidence level (e.g., $z^* = 1.96$ for 95% confidence).

Step 5: Construct the Interval: Apply the formula: $(\hat{p}_1 - \hat{p}_2) \pm z^* \cdot \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$.

Check for Zero: If the interval contains zero (e.g., $-0.05$ to $0.10$), you cannot conclude there is a statistically significant difference between the populations at that confidence level.

Directionality: Always specify the direction of the subtraction (e.g., Group A minus Group B) to ensure the interpretation of positive or negative values is clear.

Condition Verification: When checking the Large Counts condition, you must verify $n\hat{p} \ge 10$ and $n(1-\hat{p}) \ge 10$ for both samples individually; failing to check all four values is a common way to lose marks.

Width Relationships: Remember that increasing the confidence level makes the interval wider, while increasing the sample sizes ($n_1, n_2$) makes the interval narrower.