Fit a simple regression line

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Problem

Fit the ordinary least-squares line $\hat y=\hat\beta_0+\hat\beta_1x$ to the three points $(0,1),(1,2),(2,2)$ . Give the slope, intercept, prediction at $x=3$ , and the three residuals in data order.

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Here $\bar x=1$ , $\bar y=5/3$ ,

S_{xx}=(-1)^2+0^2+1^2=2,

and

S_{xy}=(-1)(-2/3)+0(1/3)+1(1/3)=1

Hence $\hat\beta_1=S_{xy}/S_{xx}=1/2$ and $\hat\beta_0=\bar y-\hat\beta_1\bar x=7/6$ . The prediction at $3$ is $7/6+3/2=8/3$ . The fitted values are $(7/6,5/3,13/6)$ , so the residuals $y-\hat y$ are

(-1/6,1/3,-1/6)

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Fit a simple regression line

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