Intercept with a decimal slope, commute times

Question

A line predicts a morning commute time in minutes (y) from the number of miles driven (x). The slope is b = 1.25, and the averages are \bar{x} = 8, \qquad \bar{y} = 22. Find the intercept.

Accepted Answer

Apply the point-of-averages relationship: a = \bar{y} - b \, \bar{x} = 22 - 1.25 	imes 8 = 22 - 10 = 12. The intercept is 12, so the line is \hat{y} = 12 + 1.25 x. In words, there is a fixed 12 minutes of the commute (parking, walking, waiting at lights) that does not depend on distance, and each mile adds 1.25 minutes. Check the anchor point: at x = 8, 12 + 1.25 	imes 8 = 12 + 10 = 22 = \bar{y}, exactly the average commute. The decimal slope does not change anything; 1.25 	imes 8 = 10 is the only arithmetic to watch. Takeaway. Decimal slopes behave the same way: a = 22 - 1.25 	imes 8 = 1…

Intercept with a decimal slope, commute times

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