Intercept from averages, steep slope small mean

Question

A line predicts a bakery's daily profit in dollars (y) from the number of catering orders (x). The slope is b = 4, and the averages are \bar{x} = 3, \qquad \bar{y} = 20. Find the intercept.

Accepted Answer

The least-squares line passes through (\bar{x}, \bar{y}), so a = \bar{y} - b \, \bar{x} = 20 - 4 	imes 3 = 20 - 12 = 8. The intercept is 8, giving \hat{y} = 8 + 4x. Read plainly, on an average day the bakery would still make about \8 profit with no catering orders, and each catering order adds about \4. Verify with the point of averages: at x = 3, 8 + 4 	imes 3 = 8 + 12 = 20 = \bar{y}. Good. Takeaway. a = \bar{y} - b\,\bar{x} = 20 - 4 	imes 3 = 8; always confirm the line hits (\bar{x}, \bar{y}).

Intercept from averages, steep slope small mean

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