--- title: "Procedure: test of significance of mean" subject: "Probability and Statistics" chapter: 6 key: "semester_5/probability_and_statistics:test-of-significance-mean-procedure" source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/test-of-significance-mean-procedure site: BCT Engineering --- # Procedure: test of significance of mean Probability and Statistics · Chapter 6: Inference concerning mean - Asked in: 2068 Chaitra, 4 marks - Video: [Large sample test- Test of significance for single mean](https://www.youtube.com/watch?v=YqrJUdmHsK0) — Mythri Maths Tutorial - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/test-of-significance-mean-procedure - Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-6 ## Question Describe the procedure of the test of significance of mean for sample. ## 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. ## Answer 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 ```text reject H0 accept H0 reject H0 <----|-----------------------|----> −critical 0 +critical (area α/2) (area α/2) ``` ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does H0 state in a test of significance of a mean? A. mu ≠ mu0 B. mu = mu0, the claimed population mean C. x̄ = mu0 exactly in the sample D. mu > mu0 Answer: B. H0 is a statement about the population mean: mu = mu0. The sample mean is only evidence against it. (Key point: Null hypothesis) 2. When is the t-statistic with n − 1 d.f. used instead of z? A. n ≥ 30 in all cases B. sigma known and n large C. sigma unknown, n < 30, normal population D. two-tailed tests only Answer: C. For small samples with unknown sigma from a normal population, t = (x̄ − mu0)/(s/sqrt(n)) with n − 1 d.f. is used. (Key point: Choice of statistic) 3. At alpha = 0.05, what are the z critical values for two-tailed and one-tailed tests? A. ±1.645 and 1.96 B. ±2.576 and 1.96 C. ±1.96 and 2.326 D. ±1.96 and 1.645 Answer: D. Two-tailed puts 0.025 in each tail (±1.96); one-tailed puts 0.05 in one tail (1.645). (Key point: Critical values) 4. In a two-tailed test, when is H0 rejected? A. When |statistic| is greater than the critical value B. When |statistic| is less than the critical value C. When the statistic is zero D. When alpha is 5% Answer: A. A statistic falling in the rejection region, |stat| > critical value, leads to rejecting H0. 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