Build and predict when the intercept goes negative

Question

A study relates hours of coaching (x) to a skills-test score (y). The summaries are \bar{x} = 15, \quad \bar{y} = 45, \quad sx = 3, \quad sy = 12, \quad r = 0.9. Predict the score after x = 20 hours of coaching.

Accepted Answer

Build the line from the summaries, then predict. Slope: b = r \cdot \frac{sy}{sx} = 0.9 \cdot \frac{12}{3} = 0.9 	imes 4 = 3.6. Each extra hour of coaching is associated with about 3.6 more points. Intercept: a = \bar{y} - b \, \bar{x} = 45 - 3.6 	imes 15 = 45 - 54 = -9. The intercept is negative, giving \hat{y} = -9 + 3.6x. A negative intercept is fine: it just means the line crosses zero at some positive x, and it does not need a real-world meaning (zero coaching hours sits below the data range, so the intercept is only a mathematical anchor). Prediction at x = 20: \hat{y} = -9 + 3.6 	im…

Build and predict when the intercept goes negative

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