Z Proportion Test Calculator Two

Two-Proportion Z-Test Formula:

\[ Z = \frac{p_1 - p_2}{\sqrt{p(1-p)(\frac{1}{n_1} + \frac{1}{n_2})}} \] \[ p = \frac{x_1 + x_2}{n_1 + n_2} \]

Z-Score:

Proportion 1 (p1):

Proportion 2 (p2):

Pooled Proportion (p):

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

The two-proportion z-test compares proportions between two independent groups. It determines whether the difference between sample proportions is statistically significant.

2. How Does the Calculator Work?

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

\[ Z = \frac{p_1 - p_2}{\sqrt{p(1-p)(\frac{1}{n_1} + \frac{1}{n_2})}} \] \[ p = \frac{x_1 + x_2}{n_1 + n_2} \]

Where:

\( p_1 \) — Proportion in group 1 (x1/n1)
\( p_2 \) — Proportion in group 2 (x2/n2)
\( p \) — Pooled proportion
\( n_1 \) — Sample size of group 1
\( n_2 \) — Sample size of group 2
\( x_1 \) — Successes in group 1
\( x_2 \) — Successes in group 2

Explanation: The test compares the difference between proportions to what would be expected by chance alone.

3. Interpretation of Results

Details: The Z-score indicates how many standard deviations the observed difference is from the expected difference of zero (no effect). Typically:

|Z| > 1.96 suggests significance at α=0.05 (two-tailed)
|Z| > 2.58 suggests significance at α=0.01 (two-tailed)

4. Using the Calculator

Tips: Enter the number of successes (x1, x2) and sample sizes (n1, n2) for both groups. All values must be non-negative integers with successes ≤ sample sizes.

5. Frequently Asked Questions (FAQ)

Q1: When should I use this test?
A: Use when comparing proportions between two independent groups with sufficiently large sample sizes (typically n*p > 5 and n*(1-p) > 5 for both groups).

Q2: What's the difference between this and chi-square?
A: The two-proportion z-test is mathematically equivalent to a chi-square test for 2×2 tables, but provides directionality (which proportion is larger).

Q3: What if my sample sizes are small?
A: For small samples, consider Fisher's exact test instead.

Q4: How do I calculate the p-value from the Z-score?
A: Use a standard normal distribution table or function. For example, a Z of 1.96 corresponds to a two-tailed p-value of 0.05.

Q5: What are common applications?
A: Comparing conversion rates, response rates, success rates, or any other proportions between two groups in clinical trials, marketing studies, etc.