Question

Solve for r and interpret the result

X 2,4,6,7,10

Y 8,10,12,6,16

Answer 1

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

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c} & X & Y & XY & X^2 & Y^2 \\ \hline & 2 & 8 & 16 & 4 & 64 \\ \hdashline & 4 & 10 & 40 & 16 & 100 \\ \hdashline & 6 & 12 & 72 & 36 & 144 \\ \hdashline & 7 & 6 & 42 & 49 & 36 \\ \hdashline & 10 & 16 & 160 & 100 & 256 \\ \hdashline Sum= & 29 & 52 & 330 & 205 & 600 \\ \hdashline \end{array}$$

$$\bar{X}=\dfrac{1}{n}\sum _{i}X_i=\dfrac{29}{5}=5.8$$

$$\bar{Y}=\dfrac{1}{n}\sum _{i}Y_i=\dfrac{52}{5}=10.4$$

$$SS_{XX}=\sum_iX_i^2-\dfrac{1}{n}(\sum _{i}X_i)^2$$ $$=205-\dfrac{29^2}{5}=36.8$$

$$SS_{YY}=\sum_iY_i^2-\dfrac{1}{n}(\sum _{i}Y_i)^2$$ $$=600-\dfrac{(52)^2}{5}=59.2$$

$$SS_{XY}=\sum_iX_iY_i-\dfrac{1}{n}(\sum _{i}X_i)(\sum _{i}Y_i)$$ $$=330-\dfrac{29(52)}{5}=28.4$$

$$r=\dfrac{SS_{XY}}{\sqrt{SS_{XX}SS_{YY}}}=\dfrac{28.4}{\sqrt{36.8(59.2)}}$$

$$=0.608462$$

$$0.4<r<0.7$$

Moderate positive correlation

$$m=slope=\dfrac{SS_{XY}}{SS_{XX}}=\dfrac{28.4}{36.8}=0.7717$$ $$n=\bar{Y}-m\bar{X}=10.4-\dfrac{28.4}{36.8}(5.8)=5.9239$$

The regression equation is:

$$y=5.9239+0.7717x$$