Answer to Question #348786 in Statistics and Probability for John Lloyd

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.