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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. - Mathematics

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Sum

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.

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Solution

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, byx=`-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"`)

Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
  Is there an error in this question or solution?
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