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Two-Sample Z-Test

Created:August 22, 2024

This Two-Sample Z-Test Calculator helps you compare means between two independent groups when both population standard deviations are known. For example, you could compare the average output between two production lines, given known variability in each line's process. The calculator performs comprehensive statistical analysis including descriptive statistics and hypothesis testing. It also generates publication-ready APA format reports. To learn about the data format required and test this calculator, click here to populate the sample data.

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Note: Column names will be converted to snake_case (e.g., "Product ID" → "product_id") for processing.

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Two-Sample Z-Test

Definition

Two-Sample Z-Test is a statistical test used to determine whether the means of two populations are significantly different from each other when both population standard deviations are known. It's particularly useful for large samples and when working with known population parameters.

Formula

Test Statistic:

z=(xˉ1−xˉ2)−(μ1−μ2)σ12n1+σ22n2z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}

Where:

  • xˉ1,xˉ2\bar x_1, \bar x_2 = sample means
  • μ1,μ2\mu_1, \mu_2 = population means
  • σ1,σ2\sigma_1, \sigma_2 = known population standard deviations
  • n1,n2n_1, n_2 = sample sizes

Confidence Interval for Mean Difference:

Two-sided: (xˉ1−xˉ2)±zα/2σ12n1+σ22n2\text{Two-sided: }(\bar{x}_1 - \bar{x}_2) \pm z_{\alpha/2} \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}One-sided: (xˉ1−xˉ2)±zασ12n1+σ22n2\text{One-sided: }(\bar{x}_1 - \bar{x}_2) \pm z_{\alpha} \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}
Where zα/2z_{\alpha/2} and zαz_{\alpha} are the critical values for the two-tailed and one-tailed tests, respectively.

Key Assumptions

Known Population Standard Deviations: Both σ₁ and σ₂ must be known
Independent Samples: The two samples must be independent
Random Sampling: Both samples should be randomly selected
Normality: Both populations should be normal or have large samples (n>30n > 30)

Practical Example

Comparing the efficiency of two production lines with known process variations:

Step 1: State the Data
  • Line 1: n1n_1 = 50, xˉ1\bar x_1 = 95.2 units/hour, σ1\sigma_1 = 4.0
  • Line 2: n2n_2 = 45, xˉ2\bar x_2 = 93.8 units/hour, σ2\sigma_2 = 3.8
Step 2: State Hypotheses
  • H0:μ1−μ2=0H_0: \mu_1 - \mu_2 = 0 (no difference)
  • Ha:μ1−μ2≠0H_a: \mu_1 - \mu_2 \neq 0 (there is a difference)
  • α=0.05\alpha = 0.05
Step 3: Calculate Test Statistic

Z-statistic:

z=(95.2−93.8)−04.0250+3.8245=1.73z = \frac{(95.2 - 93.8) - 0}{\sqrt{\frac{4.0^2}{50} + \frac{3.8^2}{45}}} = 1.73
Step 4: Calculate P-value

For two-tailed test:

p-value=2(1−Φ(∣z∣))=2(1−Φ(1.73))=0.084p\text{-value} = 2(1 - \Phi(|z|)) = 2(1 - \Phi(1.73)) = 0.084
Step 5: Calculate Confidence Interval
CI: (95.2−93.8)±1.964.0250+3.8245=[0.7,2.3]\text{CI: }(95.2 - 93.8) \pm 1.96 \sqrt{\frac{4.0^2}{50} + \frac{3.8^2}{45}} = [0.7, 2.3]
Step 6: Draw Conclusion

Critical value at 5% significance level:

zα/2=1.96z_{\alpha/2} = 1.96

Since ∣z∣<zα/2|z| < z_{\alpha/2} and p-value>0.05p\text{-value} > 0.05, we fail to reject H0H_0. There is no significant difference between the two production lines.

Effect Size

Cohen's d for two-sample z-test:

d=∣xˉ1−xˉ2∣σ12+σ222d = \frac{|\bar{x}_1 - \bar{x}_2|}{\sqrt{\frac{\sigma_1^2 + \sigma_2^2}{2}}}

Interpretation guidelines:

  • Small effect: ∣d∣≈0.2|d| \approx 0.2
  • Medium effect: ∣d∣≈0.5|d| \approx 0.5
  • Large effect: ∣d∣≈0.8|d| \approx 0.8

Power Analysis

Required sample size per group for equal sample sizes:

Two-sided: n=2(z1−α/2+z1−β)2(σ12+σ22)(μ1−μ2)2\text{Two-sided: } n = \frac{2(z_{1-\alpha/2} + z_{1-\beta})^2(\sigma_1^2 + \sigma_2^2)}{(\mu_1-\mu_2)^2}One-sided: n=2(z1−α+z1−β)2(σ12+σ22)(μ1−μ2)2\text{One-sided: } n = \frac{2(z_{1-\alpha} + z_{1-\beta})^2(\sigma_1^2 + \sigma_2^2)}{(\mu_1-\mu_2)^2}

Where:

  • α\alpha = significance level
  • β\beta = probability of Type II error
  • μ1−μ2\mu_1 - \mu_2 = minimum detectable difference

Decision Rules

Reject H0H_0 if:

  • Two-sided test: ∣z∣>zα/2|z| > z_{\alpha/2}
  • Left-tailed test: z<−zαz < -z_{\alpha}
  • Right-tailed test: z>zαz > z_{\alpha}
  • Or if p-value<αp\text{-value} < \alpha

Reporting Results

Standard format:

"A two-sample z-test was conducted to compare [variable] between [group 1] (M₁ = [mean₁], σ₁ = [std₁], n₁ = [n₁]) and [group 2] (M₂ = [mean₂], σ₂ = [std₂], n₂ = [n₂]). Results indicated [significant/no significant] difference between the groups, z = [z-value], p = [p-value], d = [Cohen's d]."

Verification

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