--- title: "Regression: chemical yield vs temperature" subject: "Probability and Statistics" chapter: 5 key: "semester_5/probability_and_statistics:yield-temperature-regression-200c" source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/yield-temperature-regression-200c site: BCT Engineering --- # Regression: chemical yield vs temperature Probability and Statistics · Chapter 5: Correlation and regression - Asked in: 2070 Chaitra, 4 marks - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/yield-temperature-regression-200c - Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-5 ## Question Observation on the yield of a chemical reaction taken at various temperatures was recorded as follows: | X (°C) | 150 | 150 | 200 | 250 | 250 | 300 | 150 | |---|---|---|---|---|---|---|---| | Y% | 75.4 | 81.2 | 85.5 | 89 | 90.5 | 96.7 | 75.4 | Fit a simple linear regression and estimate value of yield at 200°C. ## Short answer ŷ = 58.581 + 0.1266X. At X=200°C, estimated yield ≈ 83.9%. ## Answer Let X = temperature (°C), Y = yield (%), n = 7. Fit Ŷ = a + bX. Sums (n = 7): ```text x y xy x² y² 150 75.4 11310 22500 5685.16 150 81.2 12180 22500 6593.44 200 85.5 17100 40000 7310.25 250 89 22250 62500 7921 250 90.5 22625 62500 8190.25 300 96.7 29010 90000 9350.89 150 75.4 11310 22500 5685.16 --------------------------------------------- 1450 593.7 125785 322500 50736.15 ``` Σx = 1450, Σy = 593.7, Σxy = 125785, Σx² = 322500, Σy² = 50736.15 Means: x̄ = 207.1429, ȳ = 84.8143 Sxx = Σx² − (Σx)²/n = 22142.8571 Sxy = Σxy − (Σx)(Σy)/n = 2804.2857 Syy = Σy² − (Σy)²/n = 381.9086 Slope b = Sxy/Sxx = 2804.2857/22142.8571 = 0.1266 Intercept a = ȳ − bx̄ = 84.8143 − (0.1266)(207.1429) = 58.5806 **Fitted line: Ŷ = 58.581 + 0.1266 X** (r = 0.964) Estimate at X = 200°C: Ŷ = 58.5806 + 0.12665 × 200 = 58.5806 + 25.3290 = **83.91%** **Answer: Ŷ = 58.58 + 0.1266X; the estimated yield at 200°C is about 83.9%.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which is the model fitted? A. Temperature on yield B. Yield Y on temperature X: Y = a + bX C. Y = bX with no intercept D. Yield squared on X Answer: B. Yield depends on temperature so Ŷ = a + bX. (Key point: Model) 2. How is the slope obtained? A. b = Sxy/Sxx = 2804.29/22142.86 ≈ 0.1266 B. b = Sxx/Sxy C. b = Sxy/Syy D. b = ȳ/x̄ Answer: A. b = Sxy/Sxx ≈ 0.1266 % per °C. (Key point: Slope) 3. What is the estimated yield at 200°C? A. About 83.9% B. About 89.6% C. About 58.6% D. About 84.8% Answer: A. Ŷ = 58.58 + 0.1266(200) ≈ 83.9%. (Key point: Estimate) ## Related in Probability and Statistics - [Regression of rice yield on tillers](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Arice-yield-tillers-regression) — Probability and Statistics, chapter 5 - [Multiple regression: yield on rainfall, fertilizer](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Amultiple-regression-yield-rainfall-fertilizer) — Probability and Statistics, chapter 5 - [Correlation vs regression; marks correlation](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Acorrelation-vs-regression-statistics-maths) — Probability and Statistics, chapter 5 - [Regression: operator experience and performance](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Aexperience-performance-regression-r2) — Probability and Statistics, chapter 5 - [Regression of wear on oil viscosity](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Afretting-wear-regression-viscosity) — Probability and Statistics, chapter 5, asked 2× - [Correlation vs regression; stopping distance line](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Acorrelation-regression-plot-stopping-distance) — Probability and Statistics, chapter 5 ## Related in other subjects - [Linear speed-density: free speed 90, jam 150](https://www.bctengineer.com/question.md?key=semester_7%2Ftransportation_engineering_ii%3Alinear-speed-density-free-90-jam) — Transportation Engineering II, chapter 1, asked 2× - [Subnet 120.89.96.0/19 for 100-800 hosts](https://www.bctengineer.com/question.md?key=semester_7%2Fcomputer_networks_and_security%3Asubnetting-120-89-96-four-depts) — Computer Networks and Security, chapter 4, asked 2× - [Subnet 120.89.96.0/19 for 500-2000 hosts](https://www.bctengineer.com/question.md?key=semester_7%2Fcomputer_networks_and_security%3Asubnetting-120-89-96-500-2000) — Computer Networks and Security, chapter 4 - [Annual savings of 90% vs 85% efficient motor](https://www.bctengineer.com/question.md?key=semester_6%2Findustrial_power_distribution_and_illumination%3Amotor-efficiency-cost-savings) — Industrial Power Distribution and Illumination, chapter 10