Given x = 2y + 4 and y = kx + 6 are the lines of regression of x on y and y on x respective.

Find the value of k, if value of r is 0.5.

CONCEPT:

Correlation coefficient is the geometric mean between regression coefficients i.e.\({\rm{r}} = \pm \sqrt {{{\rm{b}}_{{\rm{yx}}}}{{\rm{b}}_{{\rm{xy}}}}} \)

CALCULATIONS:

Given equations are x = 2y + 4 and y = kx + 6.

\({{\rm{b}}_{{\rm{yx}}}} = k\) And \({{\rm{b}}_{{\rm{xy}}}} = 2\)

\({\rm{r}} = \pm \sqrt {{{\rm{b}}_{{\rm{yx}}}}{{\rm{b}}_{{\rm{xy}}}}} \)

\(\frac{1}{2} = \pm \sqrt {{{\rm{b}}_{{\rm{yx}}}} \times 2} \)           (On squaring both sides)

\(2{{\rm{b}}_{{\rm{yx}}}} = \frac{1}{4}\)                      ⇒ \({{\rm{b}}_{{\rm{yx}}}} = \frac{1}{8}\)

∴ \({{\rm{b}}_{{\rm{yx}}}} = k = \frac{1}{8}\)