Subject: Applied Mathematics 4

Topic: Correlation

Difficulty: Medium

X: 78, 36, 98, 25, 75, 82, 90, 62, 65, 39

Y: 84, 51, 91, 60, 68, 62, 86, 58, 53, 47

Solution : $$ \begin{array}{c|c|c|c|c|c|c|} \hline \hline x & y & x-\bar{X} & y-\bar{Y} & (x-\bar{X})(y-\bar{Y}) & (x-\bar{X})^{2} & (y-\bar{Y})^2\\ \hline 78 & 2 y & 18 & 18 & 234 & 169 & 324 \\ \hline 36 & 57 & -29 & -15 & 435 & 841 & 225 \\ \hline 98 & 91 & 39 & 25 & 825 & 1089 & 625 \\ \hline 25 & 60 & 40 & -6 & 240 & 1600 & 36 \\ \hline 95 & 68 & 10 & 2 & 20 & 100 & 4 \\ \hline 82 & 62 & 17 & -4 & -68 & 289 & 16 \\ \hline90 & 86 & 25 & 20 & 500 & 63 & 400 \\ \hline 62 & 58 & -3 & -8 & 24 & 9 & 04 \\ \hline65 & 53 & 0 & -13 & 0 & 0 & 169 \\ \hline39 & 47 & -26 & -19 & 494 & 676 & 361 \\ \hline & & & & \sum=2704 & \sum=5398 & \sum=2224 \\ \hline \end{array} $$

$\operatorname{mean}(\bar{X})=\frac{\sum x}{N}=\frac{650}{10}=65$ and

$\operatorname{mean}(\bar{Y})=\frac{\sum y}{N}=\frac{660}{10}=66$

Case I : Equation of line of regression X on Y is,

$$ \begin{aligned} b x y=\frac{\Sigma(x-\bar{X})(y-\bar{Y})}{\Sigma(y-\bar{Y})^{2}} =\frac{2704}{2224} &=1.2158 \\ \therefore \quad(x-\bar{X}) &=b x y(y-\bar{Y}) \\ \therefore \quad(x-65) &=1.2158(y-66) \\ \therefore \quad x &=1.2158 y-15.2428 \end{aligned} $$

Case II : Equation of line of regression Y on X is,

$$ \begin{aligned} b y x=\frac{\Sigma(x-\bar{X})(y-\bar{Y})}{\Sigma(x-\bar{X})^{2}} =\frac{2704}{5398} &=0.5 \\ \therefore \quad(y-\bar{Y}) &=b y x(x-\bar{Y}) \\ \therefore \quad(y-66) &=0.5(x-65) \\ \therefore \quad y &=0.5 x+33.5 \end{aligned} $$