Z Test Calculator Two Sample

Two-sample Z-test Formula:

\[ Z = \frac{x_1 - x_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}} \]

Sample 1 Mean (x₁):

Sample 1 Standard Deviation (σ₁):

Sample 1 Size (n₁):

Sample 2 Mean (x₂):

Sample 2 Standard Deviation (σ₂):

Sample 2 Size (n₂):

Z-Score:

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

The two-sample z-test is a statistical method used to determine whether two population means are different when the variances are known and the sample sizes are large (typically n > 30). It compares the means of two independent groups to assess if they are significantly different from each other.

2. How Does the Calculator Work?

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

\[ Z = \frac{x_1 - x_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}} \]

Where:

\( x_1 \) — Mean of sample 1
\( x_2 \) — Mean of sample 2
\( \sigma_1 \) — Standard deviation of sample 1
\( \sigma_2 \) — Standard deviation of sample 2
\( n_1 \) — Size of sample 1
\( n_2 \) — Size of sample 2

Explanation: The z-score represents how many standard deviations the difference between means is away from zero (no difference). A higher absolute z-score indicates stronger evidence against the null hypothesis.

3. When to Use This Test

Details: Use this test when comparing means from two independent samples with known population standard deviations and large sample sizes. Common applications include clinical trials, quality control, and social science research.

4. Using the Calculator

Tips: Enter all required parameters (means, standard deviations, and sample sizes) for both groups. Standard deviations must be positive and sample sizes must be at least 1.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between z-test and t-test?
A: Z-tests are used when population standard deviations are known and sample sizes are large, while t-tests are used when standard deviations are unknown and estimated from the sample.

Q2: How do I interpret the z-score?
A: Compare your z-score to critical values from the standard normal distribution. Typically, |z| > 1.96 indicates significance at α = 0.05 level.

Q3: What if my sample sizes are small?
A: For small samples (n < 30), consider using a two-sample t-test instead, which accounts for additional uncertainty in small samples.

Q4: Can I use this for paired data?
A: No, for paired or matched samples, use a paired z-test or paired t-test instead.

Q5: What assumptions does this test make?
A: Assumes independent samples, normally distributed populations (or large sample sizes), and known population standard deviations.