2 Sample Z Test Calculator

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 2 Sample Z Test?

The 2 Sample Z Test is a statistical method used to determine whether the means of two populations are significantly different from each other when the population standard deviations are known. It's commonly used for large sample sizes (typically n > 30).

2. How Does the Calculator Work?

The calculator uses the 2 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 population 1
\( \sigma_2 \) — Standard deviation of population 2
\( n_1 \) — Size of sample 1
\( n_2 \) — Size of sample 2

Explanation: The numerator measures the difference between sample means, while the denominator calculates the standard error of this difference.

3. Interpretation of Results

Details: The z-score indicates how many standard deviations the observed difference is from the expected difference (usually zero under the null hypothesis). 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 means, standard deviations, and sample sizes for both groups. All values must be valid (sample sizes > 0, standard deviations ≥ 0).

5. Frequently Asked Questions (FAQ)

Q1: When should I use a Z test vs a T test?
A: Use Z test when population standard deviations are known or sample sizes are large (n > 30). Use T test for small samples with unknown population standard deviations.

Q2: What does a negative z-score mean?
A: A negative z-score indicates that the first sample mean is lower than the second sample mean.

Q3: How is this different from a 1-sample Z test?
A: The 1-sample version compares a sample mean to a known population mean, while the 2-sample version compares two sample means.

Q4: What assumptions does this test make?
A: Assumes independent samples, normal distribution (or large samples), and known population standard deviations.

Q5: Can I use this for proportions?
A: For comparing proportions, you would use a slightly different formula specifically designed for proportion tests.