--- title: "Estimation vs testing; CI for hardness" subject: "Probability and Statistics" chapter: 6 key: "semester_5/probability_and_statistics:estimation-vs-hypothesis-ci-magnesium" source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/estimation-vs-hypothesis-ci-magnesium site: BCT Engineering --- # Estimation vs testing; CI for hardness Probability and Statistics · Chapter 6: Inference concerning mean - Asked in: 2074 Asoj, 5 marks - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/estimation-vs-hypothesis-ci-magnesium - Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-6 ## Question Discuss difference between estimation and hypothesis test of significance of population. The following hardness readings were obtained: 66.3, 63.5, 64.9, 61.9, 64.3, 64.7, 65.1, 64.5, 68.4, 63.2. Find 99% confidence interval for true hardness of magnesium alloy. ## Short answer Estimation gives a parameter value or interval from a sample; hypothesis testing decides if a claim is supported. For hardness: mean = 64.68, t(0.005,9)=3.250, 99% confidence interval is (62.86, 66.50). ## Answer #### Estimation versus testing of hypothesis **Estimation** finds a value, or an interval of values, for an unknown population parameter from a sample. **Testing of hypothesis** decides, using sample evidence, whether a stated claim about the parameter should be accepted or rejected. | Basis | Estimation | Hypothesis testing | |---|---|---| | Purpose | To find the value of the parameter | To check a claim about the parameter | | Starting point | No prior claim about the parameter | A definite claim (H0) is made first | | Result | Point value or confidence interval | Decision: reject or fail to reject H0 | | Question answered | "What is the value of θ?" | "Is θ equal to (or greater/less than) θ₀?" | | Error control | Confidence level (1 − α) | Level of significance α and power | | Example | 99% interval for mean hardness | Test whether mean hardness = 65 | Both use the sampling distribution of the statistic and are closely linked: a two-tailed test at level α rejects H0: μ = μ₀ exactly when μ₀ lies outside the (1 − α) confidence interval. #### 99% confidence interval for mean hardness Population SD is unknown and n is small (10), so the t-distribution is used. ```text Σx = 646.8, n = 10 x̄ = 64.680 s² = Σ(x−x̄)²/(n−1) = 28.1760/9 = 3.1307 s = 1.7694 SE = s/√n = 1.7694/√10 = 0.5595 t(0.005, 9) = 3.250 margin = 3.250 × 0.5595 = 1.818 limits = 64.680 ± 1.818 ``` **Answer: the 99% confidence interval for the true hardness is (62.86, 66.50)**, i.e. about 62.9 to 66.5. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which describes estimation as opposed to hypothesis testing? A. It finds the value or interval for a parameter with no prior claim B. It uses only alpha and power C. It starts with a claim H0 D. It decides reject or fail to reject Answer: A. Estimation gives a value or confidence interval; testing checks a definite claim. (Key point: Estimation vs testing) 2. Why is the t-distribution used for the hardness CI? A. n is large B. The data are binomial C. Sigma is unknown and n = 10 is small D. Sigma is known Answer: C. With unknown population SD and a small sample, the t-distribution with n-1 d.f. is used. (Key point: Distribution used) 3. Which critical value is used for the 99% CI? A. t(0.01, 10) = 2.764 B. z = 2.576 C. t(0.025, 9) = 2.262 D. t(0.005, 9) = 3.250 Answer: D. 99% two-sided leaves 0.005 in each tail, with df = 9, giving 3.250. (Key point: Critical value) 4. What is the 99% confidence interval? A. (63.1, 66.3) B. (60.0, 69.4) C. (62.86, 66.50) D. (64.68, 68.00) Answer: C. 64.680 +- 3.250 x 0.5595 = 64.680 +- 1.818. (Key point: Result) ## Related in Probability and Statistics - [Point vs interval estimation; estimation vs testing](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apoint-vs-interval-estimation-hypothesis-testing) — Probability and Statistics, chapter 6 - [Point/interval estimation; CI for accident proportion](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apoint-interval-estimation-accidents-ci) — Probability and Statistics, chapter 7 - [Hypothesis testing; difference of two proportions](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Ahypothesis-procedure-diff-two-proportions) — Probability and Statistics, chapter 7, asked 2× - [Estimation; good estimator; CI for mean](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Aestimation-good-estimator-ci-400-students) — Probability and Statistics, chapter 6 - [Test: mean weight of ten objects is 65 g](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Aweights-ten-objects-mean-65) — Probability and Statistics, chapter 6 - [Hypothesis testing steps; computer assemblies](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Asteps-hypothesis-testing-computers-readjust) — Probability and Statistics, chapter 7 ## Related in other subjects - [Hardness from ion analysis (Ca 65, Mg 35)](https://www.bctengineer.com/question.md?key=semester_5%2Fwater_supply_engineering%3Ahardness-c71) — Water Supply Engineering, chapter 4 - [Testing of DC machines: retardation vs Swinburne's test](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3Adc-testing-retardation-vs-swinburne) — Electrical Machines I, chapter 4 - [Tensile test engineering and true values](https://www.bctengineer.com/question.md?key=semester_4%2Fmanufacturing_and_production_processes%3Atensile-test-true-stress-strain-numerical) — Manufacturing and Production Processes, chapter 2 - [True length of suspended distance AB (66)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atape-suspended-ab-114-095) — Surveying I, chapter 2