Rainfall and Paddy Yield: Prediction or Causation?
A block development office studied 10 villages in a single paddy season. Rainfall x, measured in centimetres, ranged from 60 to 140, with mean 100 and standard deviation 20. Paddy yield y, in quintals per acre, had mean 48 and standard deviation 9. The correlation between rainfall and yield was r = 0.82. A least-squares line of y on x was fitted as y = 11.1 + 0.369x. One official wants to use this line to predict yield for a new hill village that receives 300 cm of rainfall. Another official says the high correlation proves that rainfall causes higher yield, so the office should recommend only rainfall-related irrigation works. A farmer with 120 cm of rainfall asks what yield to expect. The office also must report how much of the variation in yield is explained by the linear relationship and must decide whether the prediction for the hill village is trustworthy.
1. Interpret r, r² and the slope in context. What does the intercept mean?
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r = 0.82 is a strong positive linear association: villages with higher rainfall tended to have higher yield in this sample. r² = 0.82² = 0.6724, so about 67.2% of the sample variation in yield is accounted for by the linear regression on rainfall; 32.8% is due to other factors and random variation. Slope 0.369: within 60–140 cm, each additional cm of rainfall is associated with about 0.37 quintal/acre more yield on average. Intercept 11.1 is the fitted yield at x = 0; since 0 cm is far outside the observed range and unrealistic here, it should not be interpreted as a real yield.
2. Predict yield at 120 cm and at 300 cm. Which prediction is defensible and why?
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At x = 120: y = 11.1 + 0.369(120) = 11.1 + 44.28 = 55.38 quintals/acre. At x = 300: y = 11.1 + 0.369(300) = 11.1 + 110.7 = 121.8 quintals/acre. The 120 cm prediction is interpolation inside 60–140 and is reasonably supported. The 300 cm prediction is extrapolation far beyond the data; the linear relationship need not continue, and 121.8 may exceed what the crop can produce. It is not defensible without evidence from high-rainfall villages.
3. Does r = 0.82 prove that rainfall causes higher yield? Give a reasoned answer.
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No. A correlation of 0.82 shows association in these 10 villages, not causation. Higher rainfall may coexist with better soil, more fertilizer, earlier sowing or better drainage; any of these could raise yield. A randomized or controlled comparison, or a model including those variables, would be needed before claiming cause. The office can say rainfall is a useful predictor within the observed range, not that it alone causes yield.