2 Sample Z Test Calculator

Two-Sample Z-Test Formula:

\[ Z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}} \]

Sample 1 Mean (x̄₁):

Sample 1 Standard Deviation (σ₁):

Sample 1 Size (n₁):

Sample 2 Mean (x̄₂):

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 significantly different when the standard deviations are known and sample sizes are large (typically n > 30). It's commonly used in hypothesis testing to compare two independent samples.

2. How Does the Calculator Work?

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

\[ Z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}} \]

Where:

\( \bar{x}_1 \) — Mean of sample 1
\( \bar{x}_2 \) — 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 numerator measures the difference between sample means, while the denominator calculates the standard error of this difference.

3. Interpretation of Results

Details: The calculated Z-score can be compared to critical values from the standard normal distribution. 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 standard deviations must be positive and sample sizes must be at least 1.

5. Frequently Asked Questions (FAQ)

Q1: When should I use a z-test vs 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 are the assumptions of the z-test?
A: The test assumes independent samples, normally distributed populations (or large samples), and known population standard deviations.

Q3: Can I use this for paired samples?
A: No, this calculator is for independent samples. For paired samples, use a paired t-test.

Q4: How do I interpret a negative Z-score?
A: A negative Z-score indicates that the first sample mean is lower than the second sample mean.

Q5: What if my sample sizes are unequal?
A: The z-test can handle unequal sample sizes, as the formula accounts for different n values.