Two Sample Proportion Test Calculator

Test Statistic Formula:

\[ Z = \frac{p_1 - p_2}{\sqrt{p(1 - p)(\frac{1}{n_1} + \frac{1}{n_2})}} \]

Sample 1 Proportion (p₁):

Sample 1 Size (n₁):

Sample 2 Proportion (p₂):

Sample 2 Size (n₂):

Test Statistic (Z-score):

Pooled Proportion:

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1. What is a Two Sample Proportion Test?

Definition: This statistical test compares two independent proportions to determine if they are significantly different from each other.

Purpose: Used in A/B testing, medical trials, social sciences, and any research comparing proportions between two groups.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ Z = \frac{p_1 - p_2}{\sqrt{p(1 - p)(\frac{1}{n_1} + \frac{1}{n_2})}} \]

Where:

\( Z \) — Test statistic (standard deviations from mean)
\( p_1, p_2 \) — Proportions for two samples (0 to 1)
\( p \) — Pooled proportion (weighted average)
\( n_1, n_2 \) — Sample sizes (positive integers)

Explanation: The difference between proportions is divided by the standard error of the difference to get a Z-score.

3. Interpreting the Results

Details:

|Z| > 1.96 suggests significant difference at 95% confidence
|Z| > 2.58 suggests significant difference at 99% confidence
Positive Z means p₁ > p₂, negative means p₁ < p₂

4. Using the Calculator

Tips: Enter proportions as decimals (e.g., 0.25 for 25%) and sample sizes as whole numbers. All values must be valid (0 ≤ p ≤ 1, n ≥ 1).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between one-tailed and two-tailed tests?
A: Two-tailed tests check for any difference, while one-tailed tests check if one proportion is specifically greater than the other.

Q2: When is this test appropriate?
A: When you have two independent samples with binary outcomes (success/failure) and sample sizes are large enough (n*p > 5 and n*(1-p) > 5).

Q3: What's the pooled proportion?
A: It's the combined success rate across both samples, used to calculate the standard error under the null hypothesis of equal proportions.

Q4: How do I get p-values from the Z-score?
A: Use a Z-table or normal distribution calculator. For example, Z=1.96 corresponds to p=0.05 in a two-tailed test.

Q5: What are alternatives for small samples?
A: Fisher's exact test is better for small sample sizes where normal approximation may not hold.