Intercept from averages, a gentle slope

Question

A line predicts a runner's resting heart rate (y) from weekly training miles (x). The slope is b = 0.5, and the averages are \bar{x} = 40, \qquad \bar{y} = 30. Find the intercept.

Accepted Answer

The least-squares line always goes through the point of averages (\bar{x}, \bar{y}), so a = \bar{y} - b \, \bar{x}. Substitute: a = 30 - 0.5 	imes 40 = 30 - 20 = 10. The intercept is 10, giving the line \hat{y} = 10 + 0.5 x. Even though the slope is a decimal, the method is unchanged: multiply slope by the average x, subtract from the average y. A quick check: at x = 40 the line gives 10 + 0.5 	imes 40 = 30 = \bar{y}, confirming it passes through the averages. Takeaway. a = \bar{y} - b\,\bar{x} works for any slope; 30 - 0.5 	imes 40 = 10.

Intercept from averages, a gentle slope

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