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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.
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"`)
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