Question

Nyanza textile ltd sells six brands of shower-proof jacket. The prices and the numbers sold in week are

Price 18,20,25,27

Number sold 8,6,5,2

Question.

Calculate the Pearson correlation coefficient for the data above. Interprete your results.

Answer 1

In order to compute the regression coefficients, the following table needs to be used:

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c} & X & Y & XY & X^2 & Y^2 \\ \hline & 18 & 8 & 144 & 324 & 64 \\ \hdashline & 20 & 6 & 120 & 400 & 36 \\ \hdashline & 25 & 5 & 125 & 625 & 25 \\ \hdashline & 27 & 2 & 54 & 729 & 4 \\ \hdashline Sum= & 90 & 21 & 443 & 2078 & 129 \\ \end{array}$$

$$\bar{X}=\dfrac{1}{n}\sum _{i}X_i=\dfrac{90}{4}=22.5$$

$$\bar{Y}=\dfrac{1}{n}\sum _{i}Y_i=\dfrac{21}{4}=5.25$$

$$SS_{XX}=\sum_iX_i^2-\dfrac{1}{n}(\sum _{i}X_i)^2$$ $$=2078-\dfrac{90^2}{4}=53$$

$$SS_{YY}=\sum_iY_i^2-\dfrac{1}{n}(\sum _{i}Y_i)^2$$ $$=129-\dfrac{(21)^2}{4}=18.75$$

$$SS_{XY}=\sum_iX_iY_i-\dfrac{1}{n}(\sum _{i}X_i)(\sum _{i}Y_i)$$ $$=443-\dfrac{90(21)}{4}=-29.5$$

$$r=\dfrac{SS_{XY}}{\sqrt{SS_{XX}SS_{YY}}}=\dfrac{-29.5}{\sqrt{53(18.75)}}$$

$$=-0.9358$$

Strong negative correlation