Z-Test

Definition

Core Statement

A Z-Test is a statistical test used to determine whether a sample mean differs significantly from a population mean when the Population Standard Deviation ( $σ$ ) is Known and/or the sample size is Large ( $n \geq 30$ ).

Z = \frac{\bar{x} - μ_{0}}{σ / \sqrt{n}}

Z-Test vs T-Test

Feature	Z-Test	T-Test
Standard Deviation	Known ( $σ$ )	Unknown (use sample $s$ )
Sample Size	Large ( $n \geq 30$ )	Any (esp. Small $n < 30$ )
Distribution	Standard Normal ( $Z$ )	Student's $t$ (fatter tails)

Practical Reality

In real life, we almost never know the true population $σ$ . Therefore, T-tests are almost always preferred, even for large samples (where t approaches Z anyway). Z-tests are mostly taught for theoretical introduction.

One-Sample Z-Test Example

IQ Test

National Average IQ = 100, $σ = 15$ . (We know this from census).
A school samples 50 students with Mean IQ = 105.

Calculation:

S E = 15 / \sqrt{50} \approx 2.12

Z = \frac{105 - 100}{2.12} = \frac{5}{2.12} \approx 2.36

Decision:

Critical Z (95% confidence) = 1.96.
$2.36 > 1.96$ . Reject Null.
This school is significantly smarter than average.

Two-Proportion Z-Test

This is the most common real-world use case for Z-tests (A/B Testing).
Used for comparing two conversion rates ( $p_{1}$ vs $p_{2}$ ).

Z = \frac{{\hat{p}}_{1} - {\hat{p}}_{2}}{\sqrt{\hat{p} (1 - \hat{p}) (\frac{1}{n_{1}} + \frac{1}{n_{2}})}}

Python Implementation

from statsmodels.stats.weightstats import ztest

# 1. One Sample Mean
# (Requires raw data, treats sample std as population std if n is large)
stat, p = ztest(data, value=100)

# 2. Proportions (A/B Test) - The more common use
from statsmodels.stats.proportion import proportions_ztest
count = [30, 45]     # Successes
nobs = [1000, 1000]  # Total observations
z_stat, p_val = proportions_ztest(count, nobs)

One-Sample t-test - The alternative for unknown $σ$ .
Central Limit Theorem (CLT) - Justifies the Normal approx.
A/B Testing - Uses Two-Proportion Z-test.

Z-Test

Definition

Z-Test vs T-Test

One-Sample Z-Test Example

Two-Proportion Z-Test

Python Implementation

Related Concepts