Hypothesis Testing and Confidence Intervals for Two Proportions

9.1 Two Proportions

Introduction to Comparing Two Proportions

In statistical analysis, comparing two proportions from different populations is a common task, especially when evaluating the effectiveness of treatments or differences between groups. This section focuses on constructing confidence intervals and performing hypothesis tests for the difference between two population proportions.

• Objective: To determine if there is a significant difference between two population proportions.

• Applications: Used in medical studies, market research, and quality control.

Objectives

• Hypothesis Test: Test a claim about two proportions.

• Confidence Interval: Estimate the difference between two proportions.

Both objectives use similar statistical processes.

Requirements for Valid Comparison

• Samples must be from simple random samples.

• Samples must be independent.

• Number of successes and failures in each sample should be greater than 5.

Definitions

• Pooled Sample Proportion (π): Combines the two sample proportions into one proportion for hypothesis testing.

$\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$ $\hat{q} = 1 - \hat{p}$

Notation

• $p_1$, $p_2$: Population proportions for groups 1 and 2.

• $n_1$, $n_2$: Sample sizes for groups 1 and 2.

• $x_1$, $x_2$: Number of successes in each sample.

• $\hat{p}_1 = \frac{x_1}{n_1}$, $\hat{p}_2 = \frac{x_2}{n_2}$: Sample proportions.

Test Statistic for Two Proportions

To test the difference between two proportions, we use the following test statistic:

$z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p}\hat{q}(\frac{1}{n_1} + \frac{1}{n_2})}}$

P-value and Critical Value

• P-value: Calculated using statistical software or calculators; represents the probability of observing the test statistic under the null hypothesis.

• Critical Value: The threshold value for significance, determined by the chosen significance level (e.g., 0.05).

Confidence Interval for the Difference in Proportions

The confidence interval for the difference $p_1 - p_2$ is given by:

$(\hat{p}_1 - \hat{p}_2) \pm E$

Where $E$ is the margin of error:

$E = z_{\alpha/2} \sqrt{\frac{\hat{p}_1 \hat{q}_1}{n_1} + \frac{\hat{p}_2 \hat{q}_2}{n_2}}$

Round interval limits to three significant digits.

Hypothesis Test Procedure

When testing a claim about two proportions, the null hypothesis is always:

$H_0: p_1 = p_2$

The alternative hypothesis depends on the claim (e.g., $p_1 > p_2$, $p_1 < p_2$, or $p_1 \neq p_2$).

• Use calculators or software to compute the test statistic and p-value.

• Compare the p-value to the significance level to decide whether to reject the null hypothesis.

Example: Comparing Proportions of License Plate Violations

Suppose we want to compare the proportion of commercial trucks and passenger cars in Connecticut that have only rear license plates.

Vehicle Type	Vehicles with Only Rear Plates	Total Vehicles
Passenger Cars	239	2049
Commercial Trucks	45	334

• Hypothesis: Passenger car owners violate license plate laws at a higher rate than commercial truck owners.

• Significance Level: 0.05

Let $p_1$ be the proportion for passenger cars, $p_2$ for commercial trucks.

• Test the claim with a hypothesis test and a confidence interval.

• Calculate sample proportions: $\hat{p}_1 = \frac{239}{2049}$, $\hat{p}_2 = \frac{45}{334}$

• Set up hypotheses: $H_0: p_1 = p_2$, $H_a: p_1 > p_2$

• Compute test statistic and p-value using 2-PropZTest.

• If p-value > 0.05, fail to reject the null hypothesis.

Confidence Interval Interpretation

• If the confidence interval contains zero, there is no significant difference between the two proportions.

• Use 2-PropZInt to compute the confidence interval for the difference.

Requirements Not Met

• If samples are not random, results may not be valid.

• If $n_1p_1 \leq 5$ or $n_2p_2 \leq 5$, consider using exact methods such as Fisher's exact test or bootstrapping methods.

Summary Table: Key Steps in Two-Proportion Hypothesis Testing

Step	Description
1. State Hypotheses	Formulate null and alternative hypotheses.
2. Check Requirements	Random samples, independence, sufficient successes/failures.
3. Calculate Test Statistic	Use pooled proportion and formula for z.
4. Find P-value	Use calculator/software.
5. Make Decision	Compare p-value to significance level.
6. Construct Confidence Interval	Estimate difference between proportions.

Additional info:

• Bootstrapping is a resampling method used when sample sizes are small or requirements for normal approximation are not met.

• Fisher's exact test is recommended for small sample sizes or when expected counts are low.