--- title: "Hypothesis definition; two means large sample" subject: "Probability and Statistics" chapter: 6 key: "semester_5/probability_and_statistics:hypothesis-definition-difference-two-means-large" source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/hypothesis-definition-difference-two-means-large site: BCT Engineering --- # Hypothesis definition; two means large sample Probability and Statistics · Chapter 6: Inference concerning mean - Asked in: 2074 Chaitra, 6 marks - Video: [An Example for Hypothesis testing - Large Sample- two means](https://www.youtube.com/watch?v=dtRqm8kirSA) — Ashwini Madhikunta - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/hypothesis-definition-difference-two-means-large - Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-6 ## Question Define hypothesis. Describe the procedure of testing of hypothesis of significant difference between two population means for large samples. ## Short answer A hypothesis is a statement about a population parameter (H₀ assumes no difference). For large samples, Z = (x̄₁−x̄₂)/√(s₁²/n₁+s₂²/n₂) is used; reject H₀ if |Z| exceeds critical Z (e.g., 1.96). ## Answer #### Hypothesis A statistical hypothesis is an assumption or statement about a population parameter (or distribution) which is tested on the basis of sample evidence. There are two kinds: - **Null hypothesis H₀:** the statement of no difference or no effect (e.g. μ₁ = μ₂); it is the hypothesis tested. - **Alternative hypothesis H₁:** the statement accepted if H₀ is rejected (e.g. μ₁ ≠ μ₂, μ₁ > μ₂ or μ₁ < μ₂). #### Testing the difference between two population means (large samples) Independent random samples of sizes n₁, n₂ (both ≥ 30) give means x̄₁, x̄₂ and standard deviations s₁, s₂. 1. **State the hypotheses:** H₀: μ₁ = μ₂; H₁: μ₁ ≠ μ₂ (two-tailed) or μ₁ > μ₂ / μ₁ < μ₂ (one-tailed). 2. **Choose the level of significance** α (0.05 or 0.01). 3. **Select the test statistic:** by the central limit theorem the statistic is standard normal: Z = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂) (σ₁², σ₂² replace s₁², s₂² when known.) 4. **Find the critical region:** - Two-tailed: |Z| > 1.96 (α = 0.05) or |Z| > 2.576 (α = 0.01) - Right-tailed: Z > 1.645 (0.05) or 2.326 (0.01); left-tailed: Z < −1.645 or −2.326. 5. **Compute Z** from the sample data. 6. **Decision:** reject H₀ if the calculated Z falls in the critical region; otherwise do not reject. 7. **Conclusion:** interpret in the context of the problem. Equivalent: the (1 − α) confidence interval (x̄₁ − x̄₂) ± Z_α/2·SE can be used; H₀ is rejected if 0 lies outside the interval. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which statement best describes the null hypothesis H0 in a test of two means? A. The statement of no difference (mu1 = mu2) that is actually tested B. The statement accepted when H1 is rejected without evidence C. The statement that the means differ D. The statement that the sample sizes are equal Answer: A. H0 states no difference or no effect and is the hypothesis that is tested; H1 is accepted if H0 is rejected. (Key point: Null hypothesis) 2. For large independent samples, which statistic is used to test mu1 = mu2? A. Z = (x1bar - x2bar)/sqrt(s1^2/n1 + s2^2/n2) B. Z = (x1bar + x2bar)/sqrt(s1^2/n1 + s2^2/n2) C. F = s1^2/s2^2 D. t = (x1bar - x2bar)/(s1 + s2) Answer: A. By the central limit theorem the difference of means divided by its standard error is approximately standard normal. (Key point: Test statistic) 3. At alpha = 0.05 in a two-tailed large-sample test, when is H0 rejected? A. Z < -1.96 only B. |Z| < 1.96 C. Z > 1.645 D. |Z| > 1.96 Answer: D. Two-tailed at 5% uses critical values +-1.96; H0 is rejected when the calculated |Z| exceeds it. (Key point: Critical region) 4. How is a (1 - alpha) confidence interval for mu1 - mu2 related to the test? A. H0 is accepted if the interval is wide B. H0 is rejected if 0 lies outside the interval C. H0 is rejected if 0 lies inside the interval D. The interval is unrelated to the test Answer: B. If the interval for the difference excludes 0, the means differ significantly and H0 is rejected. 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