--- title: "Multiple regression: yield on rainfall, fertilizer" subject: "Probability and Statistics" chapter: 5 key: "semester_5/probability_and_statistics:multiple-regression-yield-rainfall-fertilizer" source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/multiple-regression-yield-rainfall-fertilizer site: BCT Engineering --- # Multiple regression: yield on rainfall, fertilizer Probability and Statistics · Chapter 5: Correlation and regression - Asked in: 2076 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/multiple-regression-yield-rainfall-fertilizer - Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-5 ## Question Fit the regression line of yield of crop ('000 tones) on amount of rainfall (mm) and amount of fertilizers used (kg). Also estimate the yield of crop for the year in which rainfall is 13 mm and fertilizer used is 9 kg. | Yield | 4 | 6 | 7 | 9 | 13 | 15 | |---|---|---|---|---|---|---| | Rainfall | 3 | 4 | 6 | 8 | 12 | 15 | | Fertilizer | 4 | 10 | 14 | 20 | 24 | 30 | ## Short answer Least squares plane: yield = 1.706 + 0.738·rainfall + 0.082·fertilizer. For 13 mm rainfall and 9 kg fertilizer, estimated yield is 12.03 thousand tonnes. ## Answer Let y = yield, x₁ = rainfall, x₂ = fertilizer. Fit the plane **y = a + b₁x₁ + b₂x₂** by least squares; n = 6. #### Normal equations Σy = na + b₁Σx₁ + b₂Σx₂ Σx₁y = aΣx₁ + b₁Σx₁² + b₂Σx₁x₂ Σx₂y = aΣx₂ + b₁Σx₁x₂ + b₂Σx₂² #### Calculation of sums ```text y x1 x2 x1y x2y x1² x2² x1x2 4 3 4 12 16 9 16 12 6 4 10 24 60 16 100 40 7 6 14 42 98 36 196 84 9 8 20 72 180 64 400 160 13 12 24 156 312 144 576 288 15 15 30 225 450 225 900 450 -------------------------------------------------------- 54 48 102 531 1116 494 2188 1034 ``` Substituting: ```text 54 = 6a + 48b₁ + 102b₂ 531 = 48a + 494b₁ + 1034b₂ 1116 = 102a + 1034b₁ + 2188b₂ ``` Solving the three equations simultaneously (elimination or Cramer's rule): a = 1.7061, b₁ = 0.7376, b₂ = 0.0820 **Fitted regression: y = 1.706 + 0.738x₁ + 0.082x₂** #### Estimate for x₁ = 13 mm, x₂ = 9 kg y = 1.7061 + 0.7376(13) + 0.0820(9) = **12.03** thousand tonnes. **Answer: y = 1.706 + 0.738x₁ + 0.082x₂; estimated yield ≈ 12.03 thousand tonnes (about 12032 tonnes).** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How many normal equations are solved for y = a + b1x1 + b2x2? A. Two B. Four C. Three, for a, b1 and b2 D. One Answer: C. There is one normal equation for each of the three unknowns a, b1, b2. (Key point: Normal equations) 2. The fitted plane is: A. y = 0.082 + 0.738x1 + 1.706x2 B. y = 1.706 + 0.738x1 + 0.082x2 C. y = 1.706 + 0.082x1 + 0.738x2 D. y = 54 + 48x1 + 102x2 Answer: B. Solving gives a = 1.7061, b1 = 0.7376 (rainfall), b2 = 0.0820 (fertilizer). (Key point: Fitted equation) 3. Estimated yield for rainfall 13 mm and fertilizer 9 kg is about: A. 9.6 B. 13.5 C. 12.03 thousand tonnes D. 1.7 Answer: C. 1.7061 + 0.7376(13) + 0.0820(9) = 12.03. 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