--- title: "Decision criteria in hypothesis test with diagram" subject: "Probability and Statistics" chapter: 6 key: "semester_5/probability_and_statistics:decision-criteria-hypothesis-diagram" source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/decision-criteria-hypothesis-diagram site: BCT Engineering --- # Decision criteria in hypothesis test with diagram Probability and Statistics · Chapter 6: Inference concerning mean - Asked in: 2074 Chaitra, 4 marks - Video: [One Tailed and Two Tailed Tests, Critical Values, & Significance Level - Inferential Statistics](https://www.youtube.com/watch?v=XHPIEp-3yC0) — The Organic Chemistry Tutor - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/decision-criteria-hypothesis-diagram - Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-6 ## Question Write the Decision criteria in test of Hypothesis with diagram. ## Short answer If the test statistic falls in the rejection region (beyond the critical value), reject H₀; otherwise do not reject. For two-tailed tests, reject if |Z| > Zα/2; for one-tailed, Z > Zα or Z < −Zα. ## Answer In hypothesis testing, the decision is made by comparing the calculated value of the test statistic with the **critical value(s)**, which separate the **acceptance region** from the **rejection (critical) region**. The size of the rejection region is the level of significance α. #### Decision rule - If the calculated statistic falls in the **rejection region**, reject H₀ (the result is significant). - If it falls in the **acceptance region**, do not reject H₀ (accept H₀; the result is not significant). - Equivalent p-value rule: reject H₀ if p-value ≤ α. #### Two-tailed test (H₁: μ ≠ μ₀) ```text Acceptance region (area 1 − α) .-"""""""""""""""-. .' '. / \ .' '. _.-' '-._ ___.--'' ''--.___ |////////| |////////| |////////| |////////| +--------+---------------------------------+--------+-- −Zα/2 0 Zα/2 Reject H0 (area α/2) Reject H0 (area α/2) ``` Reject H₀ if Z < −Z_α/2 or Z > Z_α/2 (±1.96 at α = 0.05). #### One-tailed tests ```text Right-tailed (H1: μ > μ0) Left-tailed (H1: μ < μ0) .-"""-. .-"""-. .' '. .' '. .' '. .' '. _.-' '-._ _.-' '-._ ___.--' |////| |////| '--.___ ---------------------+----+-- --+----+--------------------- Zα (area α) −Zα (area α) ``` - Right-tailed: reject H₀ if Z > Z_α (1.645 at 5%). - Left-tailed: reject H₀ if Z < −Z_α (−1.645 at 5%). #### Steps in applying the criterion 1. State H₀ and H₁. 2. Fix α. 3. Compute the test statistic. 4. Find the critical value from the table. 5. Compare and decide. 6. State the conclusion. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does the size of the rejection region equal? A. The power of the test B. The level of significance alpha C. The confidence level 1 - alpha D. The sample size Answer: B. The rejection (critical) region has total area alpha, the level of significance. (Key point: Rejection region size) 2. In the decision rule, what do you do if the calculated statistic falls in the rejection region? A. Increase alpha B. Accept H1 only if p-value > alpha C. Do not reject H0 D. Reject H0; the result is significant Answer: D. A statistic in the critical region leads to rejection of H0 and the result is called significant. (Key point: Decision rule) 3. For a two-tailed test at alpha = 0.05, how is the rejection region placed? A. 2.5% in each tail, beyond +-1.96 B. All 5% in the right tail C. 5% in each tail D. All 5% in the left tail Answer: A. Alpha is split as alpha/2 in each tail, giving Z < -1.96 or Z > 1.96. (Key point: Two-tailed critical values) 4. By the p-value rule, when is H0 rejected? A. p-value = 1 - alpha B. p-value > 0.5 C. p-value <= alpha D. p-value > alpha Answer: C. A small p-value (at most alpha) means the statistic lies in the rejection region. 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