Question

Find x2 in the set of given below

x 12 14 10 12 14 15 12 11 11 10

y 45 60 25 25 70 45 50 50 60 70

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c} & X & Y & XY & X^2 & Y^2\\ \hline & 12 & 45 & 540 & 144 & 2025\\ & 14 & 60 & 840 & 196 & 3600\\ & 10 & 25 & 250.8 & 100 & 625\\ & 12 & 25 & 300 & 144 & 625\\ & 14 & 70 & 980 & 196 & 4900\\ & 15 & 45 & 675 & 225 & 2025\\ & 12 & 50 & 600 & 144 & 2500\\ & 11 & 50 & 550 & 121 & 2500\\ & 11 & 60 & 660 & 121 & 3600\\ & 10 & 70 & 700.6 & 100 & 4900\\ Sum= & 121 & 500 & 6095 & 1491 & 27300\\ \end{array}$$ $$\sum_iX_i=121, \sum_iY_i=500$$

$$\sum_iX_iY_i=6095,\sum_iX_i=1491, \sum_iY_i=27300$$

$$\bar{X}=\dfrac{1}{n}\sum_iX_i=\dfrac{121}{10}=12.1$$

$$\bar{Y}=\dfrac{1}{n}\sum_iY_i=\dfrac{500}{10}=50$$

$$SS_{XX}=\sum_iX_i^2-\dfrac{1}{n}(\sum_iX_i)^2=1491-\dfrac{(121)^2}{10}$$

$$=26.9$$

$$SS_{YY}=\sum_iY_i^2-\dfrac{1}{n}(\sum_iY_i)^2=27300-\dfrac{(500)^2}{10}$$

$$=2300$$

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

$$=6095-\dfrac{121(500)}{10}=45$$

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

$$=\dfrac{45}{26.9}=1.672862$$

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

$$=50-1.672862(12.1)$$

$$=29.758364$$

Therefore, we find that the regression equation is:

$$Y=29.758364+1.672862X$$

Correlation coefficient:

$$r=\dfrac{SS_{XY}}{\sqrt{SS_{XX}}\sqrt{SS_{YY}}}=\dfrac{45}{\sqrt{26.9}\sqrt{2300}}$$

$$=0.180914$$

No correlation.