Vanier 201-SN1-RE tutoring

201-SN1-RE: correlation, regression and the limits of a model

Use the illustrative, unitless observations (1, 2), (2, 3), (3, 7). Find the least-squares line, interpret the association, and explain why prediction outside the observed range needs caution. These three invented points are a teaching example, not a statistical study.

The means are $\bar x=2$ and $\bar y=4$. Subtract the means to get x-deviations −1, 0, 1 and y-deviations −2, −1, 3.

$$\begin{aligned}S_{xx}&=\sum(x-\bar x)^2\\&=1+0+1=2.\end{aligned}$$

$$\begin{aligned}S_{xy}&=\sum(x-\bar x)(y-\bar y)\\&=2+0+3=5.\end{aligned}$$

$$\begin{aligned}S_{yy}&=\sum(y-\bar y)^2\\&=4+1+9=14.\end{aligned}$$

The slope is $S_{xy}/S_{xx}=5/2=2.5$. The intercept is $\bar y-2.5\bar x=-1$, so the fitted line is $\hat y=2.5x-1$. The correlation is

$$\begin{aligned}r&=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}\\&=\frac5{\sqrt{28}}\approx0.945.\end{aligned}$$

For $x=2.5$, the line predicts $\hat y=5.25$. The observed points show a strong positive linear association, but association does not establish causation. A prediction outside the observed x-range $[1,3]$ is extrapolation; these three points give no basis for assuming the same relationship continues there.