If two regression equations are 3x + 2y=26 and 6x + y=31 the ratio variance of y variance of x is

The equations of two lines of Regression are 3x + 2y = 26 , 6x + y = 31 Find

(i) Mean of x

(ii) Coefficient of correlation between x & y

(iii) σ$_y$ if σ$_x$ = 3

1 Answer

(i) Means $\bar{x}$ and $\bar{y}$ satisfy the equation of lines of regression.

$ \therefore 3\bar{x} + 2\bar{y} = 26 \\ 6\bar{x} + \bar{y} = 31 $

Solving, we get, $\bar{x}$ = 4 and $\bar{y}$ = 7

(ii) $ 6x + y = 31 \\ x = -\frac{1}{6}y + \frac{31}{6} \hspace{0.5cm} [b_{xy} = -\frac{1}{6}] \\ y = - \frac{3}{2}x + \frac{26}{2} \\ y = - \frac{3}{2}x + 13 \hspace{0.5cm} [b_{yx} = -\frac{3}{2}] $

$ \therefore r^2 = b_{yx} \times b_{xy} = \frac{1}{4} \\ r = \frac{1}{2}, -\frac{1}{2} \implies r = 0.5, -0.5 $

Since b$_{xy}$ and b$_{yx}$ are both negative, r is negative. Therefore, r = -0.5

(iii) $ b_{yx} = r \frac{\sigma_y}{\sigma_x} \\ -\frac{3}{2} = -\frac{1}{2} (\frac{\sigma_y}{3}) \\ \sigma_y = 9 $

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