Question

the regression equation of x and y is 3y-5x+108=0.if the mean value of y is 44 and the variance of x were 9/16 th of the variance of y .find the mean value of x ang correlation coefficient

Answer 1

$X\text{ is independent variable}, Y\text{ is a dependent variable}.\\ y=mx+b\\ m=\frac{\overline{xy}-\overline{x}\ \overline{y}}{\overline{x}^2-(\overline{x})^2}\\ b=\overline{y}-m\overline{x}\\ 3y-5x+108=0\\ y=\frac{5}{3}x-36\\ m=\frac{5}{3}\\ b=-36\\ -36=44-\frac{5}{3}\overline{x}\\ \overline{x}=48\\ m=\rho\frac{S_{Y}}{S_{X}}\text{ where }\rho \text{ is a correlation coefficient and }\\ S_{Y}, S_{X} \text{ are sample standard deviations of } X \text{ and } Y\\ \text{respectively}.\\ S_{X}=\frac{9}{16}S_{Y}\\ \frac{S_{Y}}{S_{X}}=\frac{16}{9}\\ \frac{5}{3}=\rho\frac{16}{9}\\ \rho=\frac{\frac{5}{3}}{\frac{16}{9}}=0.9375$