The Z-Test

A z-test checks whether an observed result differs from a claimed value by more than chance would explain — using the standard normal distribution as the yardstick.

When a z-test applies

Two conditions: the population standard deviation σ is known (or the sample is large, n ≥ 30), and the sampling distribution is roughly normal. When σ is unknown and the sample is small, the t-test with its degrees of freedom takes over — that’s the exam-question distinction.

Worked example

A factory claims bags weigh μ₀ = 500g with σ = 10. Your sample of n = 25 bags averages 495g. Then z = (495 − 500) / (10/√25) = −2.5. The negative z table gives P(Z < −2.5) = 0.00621 — a two-tailed p-value of ~0.012. At the usual 0.05 threshold, you reject the null hypothesis: the bags are lighter than claimed.

The three flavors

Left-tailed (is it less?), right-tailed (is it more?), two-tailed (is it different?) — the same z, different regions of the table. Two-tailed doubles the tail probability, which is why ±1.96 marks the classic 5% cutoff.