🧮 Z-Test

Z-Test#

A hypothesis test using the normal distribution when variance is known or n is large.

Important

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What it is#

A z-test is a hypothesis test for whether a sample mean (or proportion) differs from a known population value, built on the standard normal distribution. It applies when the sample is large (\(n > 30\), so the CLT holds) and the population variance is known — or well approximated by a large sample.

The statistic#

For a one-sample mean,

\[z = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}},\]

the gap between the sample mean \(\bar{X}\) and the hypothesised \(\mu\), measured in standard errors. There are three common forms: one-sample (mean vs population), two-sample (two independent means), and the proportion z-test.

The procedure#

State \(H_0\) (no difference) and \(H_1\); pick \(\alpha\); compute \(z\); and compare to the critical value (\(\pm 1.96\) at \(\alpha = 0.05\), two-tailed) or read off a p-value — reject when \(|z|\) exceeds it.

Examples#

With \(\mu = 100, \sigma = 15, n = 50\) and a sample mean of 105, \(z = 5 / 2.12 \approx 2.36 > 1.96\), so reject \(H_0\). For a proportion, 320 of 500 (0.64) against a hypothesised 0.60 gives \(z = 0.04 / 0.022 \approx 1.82 < 1.96\) — fail to reject.

Z-test vs t-test#

The choice turns on what you know about the variance. Use a z-test when \(\sigma\) is known and the sample is large, working from the normal distribution; use a t-test when \(\sigma\) is unknown and estimated from the sample \(s\), working from the heavier-tailed Student’s t — the gap between them vanishes as \(n\) grows.

Theme: Statistical Inference & Power · All terminology

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Mind map — connected ideas

Z-Score · Two-Proportion Z-Test · Critical Value · Standard Error (SE) · Hypothesis Testing · T-Test

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More in Statistical Inference & Power

A Priori Power Analysis · Chi-square (χ²) Test · Clopper–Pearson Interval · Compromise Power Analysis · Confidence Intervals (CIs) · Effect Size (δ) · Hypothesis Testing · Kolmogorov–Smirnov (KS) Test · Minimum Detectable Lift (MDL) · P-Value (probability value) · Post Hoc Power Analysis · Power (1 – β) · Power Analysis · Sample size