Z-test Calculator for 2 Population Proportions

Z-test Formula for Two Proportions:

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

Proportion 1 (p₁):

(0-1)

Sample Size 1 (n₁):

(≥1)

Proportion 2 (p₂):

(0-1)

Sample Size 2 (n₂):

(≥1)

Z-score:

Pooled Proportion:

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1. What is the Z-test for Two Proportions?

The z-test for two proportions is a statistical method used to determine whether two population proportions are significantly different from each other. It compares the observed difference between sample proportions to what would be expected if the null hypothesis (that the population proportions are equal) were true.

2. How Does the Calculator Work?

The calculator uses the following formula:

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

Where:

\( p_1 \) — Sample proportion 1
\( p_2 \) — Sample proportion 2
\( n_1 \) — Sample size 1
\( n_2 \) — Sample size 2
\( p \) — Pooled proportion = (x₁ + x₂)/(n₁ + n₂)

Explanation: The test statistic z measures how many standard deviations the observed difference in proportions is from the expected difference (which is 0 under the null hypothesis).

3. Interpretation of Results

Details:

|z| > 1.96 suggests statistically significant difference at α=0.05 level
|z| > 2.58 suggests statistically significant difference at α=0.01 level
The sign indicates direction of difference (positive when p₁ > p₂)

4. Using the Calculator

Tips:

Enter proportions as decimals between 0 and 1
Sample sizes must be positive integers
For best results, sample sizes should be large (n > 30) and np > 5 for each group

5. Frequently Asked Questions (FAQ)

Q1: When should I use this test?
A: Use when comparing proportions from two independent groups with large sample sizes (normal approximation is valid).

Q2: What's the difference between z-test and chi-square test?
A: For 2×2 tables, they're equivalent (z² = χ²). The z-test provides directionality (which proportion is larger).

Q3: What if my sample sizes are small?
A: Consider Fisher's exact test for small samples where np < 5 for any group.

Q4: How do I interpret a negative z-score?
A: A negative z-score indicates that the first proportion (p₁) is smaller than the second (p₂).

Q5: What assumptions does this test make?
A: Assumes independent samples, normal approximation to binomial distribution (valid when np > 5 and n(1-p) > 5 for both groups).