4.3 The Least-Squares Regression Line

Notes

The least-squares regression line $\hat{y} = a + bx$ is the line that makes the sum of squared vertical distances to the points as small as possible.

• Slope: $\displaystyle b = r\,\frac{s_y}{s_x}$. Intercept: $a = \bar{y} - b\bar{x}$. The line always passes through $(\bar{x}, \bar{y})$.
• Slope interpretation: for each 1-unit increase in $x$, the predicted $y$ changes by $b$ units.
• Intercept interpretation: the predicted $y$ when $x = 0$ (it may not make sense in context).
• Example: $r = 0.8$, $s_x = 5$, $s_y = 10$, $\bar{x} = 20$, $\bar{y} = 50$ give $\displaystyle b = 0.8 \cdot \frac{10}{5} = 1.6$ and $a = 50 - 1.6(20) = 18$.