Slope from correlation, rainfall and crop yield

Question

An agronomist relates seasonal rainfall in inches (x) to crop yield in bushels per acre (y), with r = 0.9, \qquad sx = 5, \qquad sy = 15. What slope predicts yield from rainfall?

Accepted Answer

Slope from correlation and the two spreads: b = r \cdot \frac{sy}{sx} = 0.9 \cdot \frac{15}{5} = 0.9 	imes 3 = 2.7. The very strong correlation (0.9) keeps most of the size, and the spread ratio 15/5 = 3 scales it up into yield units, giving about 2.7 bushels per acre for each extra inch of rain. Sanity check the direction and magnitude: positive correlation gives a positive slope (more rain, more yield in this range), and because yield varies three times as widely as rainfall, one inch of rain maps to a several-bushel change. Both match intuition. Takeaway. b = r \, sy/sx = 0.9 	imes (15/5…

Slope from correlation, rainfall and crop yield

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