Finding the intercept from the two averages

Question

A least-squares line predicts monthly spending ( $y$ ) from monthly income ( $x$ ), both in hundreds of dollars. You are told the slope is $b = 2$ , and the sample averages are

$\bar{x} = 10, \qquad \bar{y} = 35.$

What is the intercept of the line?

Accepted Answer

There is one fact that makes this quick: every least-squares line passes through the point of averages, (\bar{x}, \bar{y}). So the line must satisfy \bar{y} = a + b \, \bar{x}, where a is the intercept. Rearranging to solve for the intercept: a = \bar{y} - b \, \bar{x}. In words: start at the average of y, then back out the part the slope already accounts for at the average x. Substitute the numbers: a = 35 - 2 	imes 10 = 35 - 20 = 15. So the full line is \hat{y} = 15 + 2x. You can check it: at x = 10 it gives 15 + 20 = 35 = \bar{y}, exactly the point of averages, as it must. Takeaway. Becau…

Finding the intercept from the two averages

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