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Fit the line $y = a + b x$ by least squares to the points $(0, 1)$ , $(1, 3)$ , $(2, 4)$ and $(3, 8)$ . What is the slope $b$ ? Give a fraction in lowest terms.

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Hint

Centre both coordinates at their means; the slope is a ratio of $2$ sums.

Solution

The normal equations for $a$ and $b$ give the slope as $\frac{S _{x y}}{S _{xx}}$ , built from deviations from the means $x ˉ = 1.5$ and $y ˉ = 4$ .
$S_{x y} = (- 1.5) \times (- 3) + (- 0.5) \times (- 1) + 0.5 \times 0 + 1.5 \times 4 = 11$ .
$S_{xx} = 2.25 + 0.25 + 0.25 + 2.25 = 5$ .
So the slope is $b = \frac{11}{5}$ .

Answer: $\frac{11}{5}$

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