1. What is the Two-Proportion Z-Test?

The two-proportion Z-test compares proportions between two independent groups to determine if they are significantly different. It's commonly used in A/B testing, medical studies, and social sciences.

2. How Does the Calculator Work?

The calculator uses the two-proportion Z-test formula:

Where:

• \( \hat{p}_1, \hat{p}_2 \) — Sample proportions
• \( x_1, x_2 \) — Success counts
• \( n_1, n_2 \) — Sample sizes
• \( \hat{p} \) — Pooled proportion
• normcdf — Standard normal cumulative distribution function

3. Interpreting the Results

Z-Score: Measures how many standard deviations the observed difference is from the null hypothesis (no difference).

P-Value: Probability of observing a difference this extreme if the null hypothesis is true. Typically, p < 0.05 is considered statistically significant.

4. Using the Calculator

Tips: Enter success counts and sample sizes for both groups. All values must be valid (successes ≤ sample size, sample sizes > 0).

5. Frequently Asked Questions (FAQ)

Q1: When should I use this test?
A: When comparing proportions between two independent groups with sufficiently large sample sizes (n > 30).

Q2: What's the difference between one-tailed and two-tailed?
A: Two-tailed tests for any difference (default), one-tailed tests for directional difference (requires justification).

Q3: What are the assumptions?
A: Independent samples, normal approximation valid (n×p > 5 and n×(1-p) > 5 for both groups).

Q4: What if my sample size is small?
A: Consider Fisher's exact test for small samples (n < 30).

Q5: How do I report the results?
A: Report both the Z-score and p-value, e.g., "Z = 2.45, p = 0.014".