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Regression of rice yield on tillers

Short answer

Regression line: ŷ = 4200.38 + 4.826x (r = 0.811). At tillers x = 210, predicted grain yield ≈ 5214 kg/ha.

The international rice research institute in the Philippines wants to relate the grain yield of rice varieties, y, to the tiller number, x. They conducted experiments for some rice varieties and tillers. Below are the results obtained for the rice variety Milfor 6.

Grain Yield, kg/ha4,8625,2445,1285,0525,2985,410
Tillers, no./m²160175192195238240
i) Fit a linear regression line to predict grain yield with respect to Tillers.
ii) Estimate the grain yield if Tillers value is 210.

Let x = tillers (no./m²), the independent variable, and y = grain yield (kg/ha), the dependent variable. n = 6.

i) Fitting the regression line

         x         y        xy        x²        y²
       160      4862    777920     25600  23639044
       175      5244    917700     30625  27499536
       192      5128    984576     36864  26296384
       195      5052    985140     38025  25522704
       238      5298   1260924     56644  28068804
       240      5410   1298400     57600  29268100
--------------------------------------------------
      1200     30994   6224660    245358 160294572
  • x̄ = 1200/6 = 200.000, ȳ = 30994/6 = 5165.667
  • Sxx = Σx² − (Σx)²/n = 5358.000
  • Sxy = Σxy − (Σx)(Σy)/n = 25860.000

b = Sxy/Sxx = 25860.000/5358.000 = 4.8264 a = ȳ − b·x̄ = 5165.667 − (4.8264)(200.000) = 4200.381

Regression line: ŷ = 4200.38 + 4.826 x (r = 0.811)

ii) Estimate at x = 210

ŷ = 4200.381 + 4.8264 × 210 = 5213.9 kg/ha

Answer: grain yield ≈ 5214 kg/ha when tillers = 210/m².

✓ Answer checked by Claude Sonnet 5.5

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