Section 11.1

Comparing Two Population Proportions

Test whether two population proportions are equal using independent samples.

Independent vs Dependent Sampling

Independent Sampling

Individuals in one sample do not dictate who is in the second sample. Two separate groups with no connection.

Dependent (Matched-Pairs)

Individuals are related: husband/wife, twins, or same person before/after treatment.

This section: We focus on independent samples for comparing two proportions.

Pooled Estimate of p

When , we pool the data to get a common estimate:

This pooled proportion is used in the test statistic denominator.

Test Statistic

Requirements

Independent random samples

and

Each sample ≤ 5% of its population

Confidence Interval for

To estimate the difference between two proportions:

Note: For CIs, use individual values (not pooled) since we're not assuming equality.

Confidence Interval for p₁ - p₂

Estimate the difference between two population proportions

Sample 1

Successes (x₁)

Sample Size (n₁)

p̂₁ = 0.2250

Sample 2

Successes (x₂)

Sample Size (n₂)

p̂₂ = 0.1667

Confidence Level

Requirements Met

Calculations

Point Estimate

Standard Error

Margin of Error

FORMULA (Not Pooled!)

95% Confidence Interval for p₁ - p₂

(-0.0211, 0.1378)

Interval Visualization

No Significant Difference

We are 95% confident that p₁ - p₂ is between -0.0211 and 0.1378. Since the interval contains 0, we cannot conclude there is a significant difference between the proportions.

Two-Proportion Test Calculator

Two-Proportion Z-Test Calculator

Compare two population proportions from independent samples

Sample 1

Successes (x₁)

Sample Size (n₁)

p̂₁ = 0.2250

Sample 2

Successes (x₂)

Sample Size (n₂)

p̂₂ = 0.1667

Significance Level (α)

Alternative Hypothesis

Hypotheses

Null:

Alternative:

Requirements Met

Calculations

Pooled Proportion

Standard Error

Difference

Test Statistic

Standard Normal Distribution

Classical Approach

Critical Region: z < -1.960 or z > 1.960

z₀ = 1.4265 is NOT in rejection region

P-Value Approach

P-value = 0.1537

P-value ≥ α = 0.05

Fail to Reject H₀

At the α = 0.05 significance level, there is insufficient evidence to conclude that the two population proportions are different.

Sample Size Calculator

Sample Size Calculator for p₁ - p₂

Determine required sample size for a desired margin of error

Population 1

Prior Estimate (p̂₁)

Population 2

Prior Estimate (p̂₂)

Desired Margin of Error (E)

Confidence Level

Formula

z-value

Margin of Error

Variance Sum

Required Sample Size (each group)

n = 769

Total participants needed: 1,538

To estimate the difference with a margin of error of ±5.0% and 95% confidence, you need at least 769 participants in each group (1,538 total).

Quick Reference Table

Margin of Error	90% Conf	95% Conf	99% Conf
1%	13,531	19,208	33,179
3%	1,504	2,135	3,687
5%	542	769	1,328
10%	136	193	332

Common Pitfalls

Using Pooled p̄ for Confidence Intervals

Use pooled p̄ only for hypothesis tests. For CIs, use individual p̂ values.

Confusing Independent vs Dependent

Matched pairs (before/after) require a different method (Section 11.2).

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