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Procedure: test of significance of mean

Short answer

State null and alternative hypotheses about the population mean. Choose α, then compute z = (x̄−μ₀)/(σ/√n) for known σ or large n, or t = (x̄−μ₀)/(s/√n) with n−1 df for small samples with unknown σ; compare to critical value and reject H0 if the statistic falls in the rejection region.

Describe the procedure of the test of significance of mean for sample.

A test of significance of a mean decides whether the sample mean x̄ differs from a claimed population mean μ₀ only by chance or because the true mean is different.

Procedure

  1. State the hypotheses. H0: μ = μ₀ (null). H1: μ ≠ μ₀ (two-tailed), or μ > μ₀ / μ < μ₀ (one-tailed).
  2. Choose the level of significance α (usually 5% or 1%).
  3. Select the test statistic.
    • σ known, or n ≥ 30 (large sample): z = (x̄ − μ₀)/(σ/√n); use s in place of σ if σ is unknown.
    • σ unknown, n < 30, population normal: t = (x̄ − μ₀)/(s/√n) with n − 1 degrees of freedom.
  4. Find the critical value from the normal table (z) or t-table (t, d.f. = n − 1) for α and the tail type. For example, at α = 0.05: z = ±1.96 (two-tail), 1.645 (one-tail).
  5. Compute the statistic by substituting x̄, μ₀, s (or σ) and n.
  6. Decide. If the calculated value falls in the rejection region (|z| or |t| greater than the critical value for two-tail), reject H0. Otherwise do not reject H0.
  7. Conclude in words about the original problem, for example "the sample mean differs significantly from the claimed mean".

Two-tailed layout

   reject H0     accept H0      reject H0
  <----|-----------------------|---->
   −critical        0        +critical
   (area α/2)               (area α/2)

✓ Answer checked by Claude Sonnet 5.5

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