Question

A scatter diagram includes the data points (x=3,y=8),(x=5,y=18),(x=7,y=30),and (x=9,y=32). Two regression lines are proposed: (1)yˆ=5+3x, and (2) yˆ=22+4x. Using the least- squares criterion, which of these regression lines is the better fit to the data?

Answer 1

The first line $\hat{y}=5+3x$

$$\def\arraystretch{1.5} \begin{array}{c:c:c} x & y & \hat{y} \\ \hline 3 & 8 & 14 \\ \hdashline 5 & 18 & 20 \\ \hdashline 7 & 30 & 26 \\ \hdashline 9 & 32 & 32 \\ \end{array}$$

The sum of the squared deviations between observed and predicted y values is

$$(8-14)^2+(18-20)^2+(30-26)^2+(32-32)^2=56$$

The second line $\hat{y}=-2+4x$

$$\def\arraystretch{1.5} \begin{array}{c:c:c} x & y & \hat{y} \\ \hline 3 & 8 & 10 \\ \hdashline 5 & 18 & 18 \\ \hdashline 7 & 30 & 26 \\ \hdashline 9 & 32 & 34 \\ \end{array}$$

The sum of the squared deviations between observed and predicted y values is

$$(8-10)^2+(18-18)^2+(30-26)^2+(32-34)^2=24$$

Since $56>24,$ then according to the least-squares criterion, the second line $\hat{y}=-2+4x$ is a better fit to the data than the first line $\hat{y}=5+3x.$