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Michael Brenndoerfer

The Z-Test › Two-Sample Z-Test

Michael BrenndoerferUpdated January 12, 20262 min read

Machine Learning from ScratchAll chapters

Part 2 of 9Hypothesis Testing

Chapter 14The Z-Test › Two-Sample Z-Test

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When comparing means from two independent populations with known variances, we use the two-sample z-test.

The FormulaLink Copied

z = \frac{(\bar{x}_1 - \bar{x}_2) - D_0}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}

Where:

$\bar{x}_1, \bar{x}_2$ : Sample means from groups 1 and 2
$D_0$ : Hypothesized difference (usually 0 for testing equality)
$\sigma_1, \sigma_2$ : Known population standard deviations
$n_1, n_2$ : Sample sizes

Why Do Variances Add?Link Copied

When you subtract two independent random variables, their variances add:

\text{Var}(\bar{X}_1 - \bar{X}_2) = \text{Var}(\bar{X}_1) + \text{Var}(\bar{X}_2) = \frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}

This might seem counterintuitive (why add when we're subtracting?), but it makes sense when you think about it: the difference between two uncertain quantities is more uncertain than either quantity alone. Uncertainty compounds.

Example: Comparing Two Production LinesLink Copied

A factory has two production lines making the same component. Historical quality data shows Line A has $\sigma_A = 2.5$ units and Line B has $\sigma_B = 3.0$ units. A sample of 40 items from Line A has mean 50.2, and a sample of 50 items from Line B has mean 48.7. Do the lines produce different mean outputs?

In[9]:

Code

import numpy as np
from scipy import stats

# Data
x_bar_A, sigma_A, n_A = 50.2, 2.5, 40
x_bar_B, sigma_B, n_B = 48.7, 3.0, 50

# Standard error of the difference
se_diff = np.sqrt((sigma_A**2 / n_A) + (sigma_B**2 / n_B))

# Observed difference
observed_diff = x_bar_A - x_bar_B

# Z-statistic (testing H0: μA = μB, i.e., D0 = 0)
z = observed_diff / se_diff

# Two-sided p-value
p_value = 2 * stats.norm.sf(abs(z))

# 95% CI for the difference
z_crit = stats.norm.ppf(0.975)
ci_lower = observed_diff - z_crit * se_diff
ci_upper = observed_diff + z_crit * se_diff

print("Two-Sample Z-Test: Production Lines")
print("=" * 50)
print(f"Line A: x̄ = {x_bar_A}, σ = {sigma_A}, n = {n_A}")
print(f"Line B: x̄ = {x_bar_B}, σ = {sigma_B}, n = {n_B}")
print()
print(f"Observed difference (A - B): {observed_diff:.2f}")
print(f"Standard error of difference: {se_diff:.4f}")
print(f"Z-statistic: {z:.3f}")
print(f"P-value (two-sided): {p_value:.4f}")
print()
print(f"95% CI for difference: [{ci_lower:.3f}, {ci_upper:.3f}]")
print()
print(f"Decision: {'Reject H₀' if p_value < 0.05 else 'Fail to reject H₀'}")

Out[9]:

Console

Two-Sample Z-Test: Production Lines
==================================================
Line A: x̄ = 50.2, σ = 2.5, n = 40
Line B: x̄ = 48.7, σ = 3.0, n = 50

Observed difference (A - B): 1.50
Standard error of difference: 0.5799
Z-statistic: 2.587
P-value (two-sided): 0.0097

95% CI for difference: [0.363, 2.637]

Decision: Reject H₀

The p-value of about 0.008 provides strong evidence that the two lines have different mean outputs. The 95% CI for the difference [0.39, 2.61] doesn't include 0, consistent with our rejection of the null hypothesis.