Z-Test and Hypothesis Testing

Maths: Statistics for machine learning

3 min read

Published Oct 22 2025, updated Oct 23 2025

Machine LearningMathsNumPyPandasPythonStatistics

A Z-test is a statistical hypothesis test used to determine whether there is a significant difference between a sample mean and a population mean, or between the means of two samples, when the population standard deviation (σ) is known and/or the sample size is large (n ≥ 30).

In simple terms:

“A Z-test checks if a sample mean is far enough away from what we expect — in standard deviation units — to suggest a real difference rather than random variation.”

When to Use a Z-Test

Use a Z-test when:

The population standard deviation (σ) is known
The sample size is large (n ≥ 30) (Central Limit Theorem applies)
The data are approximately normally distributed

If σ is unknown and n is small, use a T-test instead.

Types of Z-Tests

Test Type	Purpose	Example
One-sample Z-test	Compare a sample mean to a population mean	“Is the average height of students different from 170 cm?”
Two-sample Z-test	Compare two independent sample means	“Do males and females have different average test scores?”
Z-test for proportions	Compare sample proportion(s) to a population or another group	“Did more than 60% of users click the ad?”

Hypothesis Setup

Symbol	Meaning	Example
H₀ (Null Hypothesis)	No difference or effect	μ = μ_₀ (e.g. mean = 50)
H₁ (Alternative Hypothesis)	There is a difference	μ ≠ μ_₀ (two-tailed) or μ > μ_₀ / μ < μ_₀ (one-tailed)

You start by assuming H₀ is true, then use your data to test if it should be rejected.

Z-Test Formula

For a one-sample Z-test:

Where:

X = sample mean
μ₀ = population mean under H₀
σ = population standard deviation
n = sample size

Decision Rule

Choose a significance level (α) (commonly 0.05).
Find the critical Z-value:
- Two-tailed test: ±1.96 (for α = 0.05)
- One-tailed test: ±1.645 (for α = 0.05)
Compute your Z-statistic using the formula.
Compare:
- If |Z| > Z-critical, reject H₀ (significant difference).
- Otherwise, fail to reject H₀.

Example

A sample of 50 students has an average test score of 78.
The population mean is 75, with a known σ = 10.
At α = 0.05, is the difference significant?

Compare with critical Z (±1.96 for 95% confidence):

|2.12| > 1.96 → Reject H₀

Interpretation:
There is a statistically significant difference between the sample mean and population mean.

P-Value Approach

You can also interpret the Z-test using the p-value:

Find p-value from Z (e.g., p = 0.034).
Compare with α = 0.05.
If p ≤ α → Reject H₀, else Fail to reject H₀.

Same conclusion as using Z-critical values.

Visual Understanding

The Z-distribution (standard normal) is centred at 0.
The tails represent rare or extreme outcomes.
If your Z-statistic lies in the tails (beyond ±Z-critical), your result is unlikely under H₀.

A two-tailed test shades both ends of the normal curve (extreme low and high).
A one-tailed test shades only one end.