For 5 pairs (x, y) you know Σx = 20, Σy = 50, Σxy = 270 and Σx² = 120. What is the least-squares slope of y on x?

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Slope from Sums

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For $5$ pairs $(x, y)$ you know $\sum x = 20$ , $\sum y = 50$ , $\sum x y = 270$ and $\sum x^{2} = 120$ . What is the least-squares slope of $y$ on $x$ ?

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Hint

The slope is the centred cross-product divided by the centred sum of squares.

Solution

The centred cross-product is $S_{x y} = 270 - 20 \times \frac{50}{5} = 70$ .
The centred sum of squares is $S_{xx} = 120 - \frac{2 0 ^{2}}{5} = 40$ .
The slope is $\frac{70}{40}$ , which reduces to $\frac{7}{4}$ .

Answer: $\frac{7}{4}$

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