Question

The table below shows the time in hours (𝑥) spent by six (6) students in playing

Clash of Clansand the scores of these students got on a math test. Solve for

Pearson Product Correlation Coeffcient. Construct a Scatter Plot.

𝑥 1 2 3 4 5 6

𝑦 30 25 25 10 15 5

Answer 1

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

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c} & X & Y & XY & X^2 & Y^2 \\ \hline & 1 & 30 & 30 & 1 & 900 \\ \hdashline & 2 & 25 & 50 & 4 & 625 \\ \hdashline & 3 & 25 & 75 & 9 & 625 \\ \hdashline & 4 & 10 & 40 & 16 & 100 \\ \hdashline & 5 & 15 & 75 & 25 & 225 \\ \hdashline & 6 & 5 & 30 & 36 & 25 \\ \hdashline Sum= & 21 & 110 & 300 & 91 & 2500 \\ \hdashline \end{array}$$ $$\bar{X}=\dfrac{1}{n}\sum _{i}X_i=\dfrac{21}{6}=\dfrac{7}{2}$$

$$\bar{Y}=\dfrac{1}{n}\sum _{i}Y_i=\dfrac{110}{6}=\dfrac{55}{3}$$

$$SS_{XX}=\sum_iX_i^2-\dfrac{1}{n}(\sum _{i}X_i)^2$$ $$=91-\dfrac{21^2}{6}=\dfrac{35}{2}$$

$$SS_{YY}=\sum_iY_i^2-\dfrac{1}{n}(\sum _{i}Y_i)^2$$ $$=2500-\dfrac{110^2}{6}=\dfrac{1450}{3}$$

$$SS_{XY}=\sum_iX_iY_i-\dfrac{1}{n}(\sum _{i}X_i)(\sum _{i}Y_i)$$ $$=300-\dfrac{21(110)}{6}=-85$$

$$r=\dfrac{SS_{XY}}{\sqrt{SS_{XX}SS_{YY}}}=\dfrac{-85}{\sqrt{\dfrac{35}{2}(\dfrac{1450}{3})}}$$ $$=-0.9242$$

Strong negative correlation.