Answer to Question #282141 in Statistics and Probability for Ryan

Question

Compute the correlation coefficient of the data below.

*number of years vs. selling price

X 1 3 5 7 9

y 27 23 25 20 15

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c} & X & Y & XY & X^2 & Y^2 \\ \hline & 1 & 27 & 27 & 1 & 729 \\ \hdashline & 3 & 23 & 69 & 9 & 529 \\ \hdashline & 5 & 25 & 125 & 25 & 625 \\ \hdashline & 7 & 20 & 140 & 49 & 400 \\ \hdashline & 9 & 15 & 135 & 81 & 225 \\ \hdashline Sum= & 25 & 110 & 496 & 165 & 2508 \\ \hdashline \end{array}$$

$$\bar{X}=\dfrac{1}{n}\displaystyle\sum_{i=1}^nX_i=\dfrac{25}{5}=5$$

$$\bar{Y}=\dfrac{1}{n}\displaystyle\sum_{i=1}^nY_i=\dfrac{110}{5}=22$$

$$SS_{XX}=\displaystyle\sum_{i=1}^nX_i^2-\dfrac{1}{n}(\displaystyle\sum_{i=1}^nX_i)^2=165-\dfrac{25^2}{5}=40$$

$$SS_{YY}=\displaystyle\sum_{i=1}^nY_i^2-\dfrac{1}{n}(\displaystyle\sum_{i=1}^nY_i)^2=2508-\dfrac{110^2}{5}=88$$

$$SS_{XY}=\displaystyle\sum_{i=1}^nX_iY_i-\dfrac{1}{n}(\displaystyle\sum_{i=1}^nX_i)(\displaystyle\sum_{i=1}^nY_i)$$

$$=496-\dfrac{25(110)}{5}=-54$$

Correlation coefficient:

$$r=\dfrac{SS_{XY}}{\sqrt{SS_{XX}}\sqrt{SS_{YY}}}=\dfrac{-54}{\sqrt{40}\sqrt{88}}$$

$$\approx -0.91017$$

$$-0.7<r<-11.0$$

$r=-0.91017,$ strong negative correlation.