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Multiple regression: yield on rainfall, fertilizer

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.

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.

Yield46791315
Rainfall34681215
Fertilizer41014202430

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

      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:

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).

✓ Answer checked by Claude Sonnet 5.5

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