Question

The two lines of regressions are x + 2y – 5 = 0 and 2x + 3y – 8 = 0 and the variance of x is 12. Find the variance of y and the coefficient of correlation.

Answer 1

Let y = `-1/2"x"+5/2` be the regression line of y on x

and x =`-3/2"y" +8/2` be the regression line of x on y

Now, b_yx=`-1/2   "b"_("yx")  = -3/2`

`sqrt("b"_("yx")."b"_("xy")) = sqrt(( -1)/2.(-3)/2)`

`=sqrt(3/4)      =(-sqrt3)/2 <1`

r =`(-sqrt3)/2`

Now,   `sigma_"x"=sqrt12=2sqrt3`

We have:    `"b"_("yx") = "r"  sigma_"y"/sigma_"x"`

`-1/2=-sqrt3/2.sigma_"y"/(2sqrt3)`

⇒  `sigma_"y"=2`

∴ Variance of y =4

coefficient of correlation = `(-sqrt3)/2`    ...(same sign as `"b"_("yx") and "b"_("yx"`)