Question

h. The following sample observations were randomly selected.

X 4 5 3 6 10

Y 4 6 5 7 7

Determine the regression equation.

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c} & X & Y & XY & X^2 & Y^2\\ \hline & 4 & 4 & 16 & 16 & 16\\ & 5 & 6 & 30 & 25 & 36\\ & 3 & 5 & 15.8 & 9 & 25\\ & 6 & 7 & 42 & 36 & 49\\ & 10 & 7 & 70 & 100 & 49\\ Sum= & 28 & 29 & 173 & 186 & 175\\ \end{array}$$ $$\sum_iX_i=28, \sum_iY_i=29$$

$$\sum_iX_iY_i=173,\sum_iX_i=186, \sum_iY_i=175$$ $$\bar{X}=\dfrac{1}{n}\sum_iX_i=\dfrac{28}{5}=5.6$$

$$\bar{Y}=\dfrac{1}{n}\sum_iY_i=\dfrac{29}{5}=5.8$$

$$SS_{XX}=\sum_iX_i^2-\dfrac{1}{n}(\sum_iX_i)^2=100-\dfrac{(28)^2}{5}$$

$$=29.2$$

$$SS_{YY}=\sum_iY_i^2-\dfrac{1}{n}(\sum_iY_i)^2=175-\dfrac{(29)^2}{5}$$

$$=6.8$$

$$SS_{XY}=\sum_iX_iY_i-\dfrac{1}{n}(\sum_iX_i)(\sum_iY_i)$$

$$=173-\dfrac{28(29)}{5}=10.6$$

$$m=slope=\dfrac{SS_{XY}}{SS_{XX}}$$

$$=\dfrac{10.6}{29.2}=0.3630137$$

$$n=\bar{Y}-m\bar{X}$$

$$=5.8-0.3630137(5.6)$$

$$=3.7671233$$

Therefore, we find that the regression equation is:

$$Y=3.7671233+0.363013X$$

Correlation coefficient:

$$r=\dfrac{SS_{XY}}{\sqrt{SS_{XX}}\sqrt{SS_{YY}}}=\dfrac{10.6}{\sqrt{29.2}\sqrt{6.8}}$$

$$=0.752246$$

Strong positive correlation.