Normal-distribution tests
Objective. Test means or model parameters using tail probabilities.
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Learn: the key idea
A normal-distribution test checks whether a sample mean, or an estimated model parameter, is consistent with a claimed population value by converting to a standardised z-value and comparing it against a critical value from tail probabilities. Under H0, the sample mean X̄ from a normal population with known variance σ² is distributed N(μ, σ²/n), so z = (x̄ − μ) ÷ (σ/√n). If z falls beyond the critical value for the chosen significance level, the evidence rejects H0 in favour of H1. For example, testing whether a machine's mean fill of 500 ml has drifted uses this standardised comparison.
Worked example
A population is N(μ, 16). A sample of 25 values has mean 42. Test H0: μ = 40 against H1: μ > 40 using a normal-distribution test, given critical z = 1.645.
- 1Standard error = σ/√n = 4/√25 = 4/5 = 0.8
- 2z = (x̄ − μ) ÷ SE = (42 − 40) ÷ 0.8
- 3z = 2 ÷ 0.8 = 2.5
- 4Since 2.5 > 1.645, reject H0: mean has increased
Answer: z = 2.5, reject H0 (significant evidence μ > 40)
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Start practisingCommon mistake. Students compare the raw sample mean to μ without standardising by σ/√n, so sampling variability is ignored. Always convert to a z-value using the standard error before comparing with critical values.
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