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Home  >>  JEEMAIN and NEET  >>  Mathematics  >>  Class11  >>  Statistics

If two variables $x, y$ are related by $ax+by+c=0$ with $ab < 0$ then correlation coefficient between $x$ and $y$ is given by

$\begin {array} {1 1} (A)\;1 & \quad (B)\;-1 \\ (C)\;0 & \quad (D)\;None\: of \: these \end {array}$

1 Answer

$ax+by+c=0$
$ \Rightarrow a\overline x + b \overline y +c=0$
$ \Rightarrow a(x- \overline x)+b(y- \overline y)=0$
$ x- \overline x= \large\frac{-b}{a} (y- \overline y)$
Now $ cov (x,y)= \large\frac{1}{n} \Sigma (x- \overline x) \: (y- \overline y)$
$ = -\large\frac{b}{a} \: \large\frac{1}{n} \Sigma (y-\overline y)^2$
$ -\large\frac{b}{a} \sigma_y^2$
Also $ \sigma_x^2 = \large\frac{1}{n} \Sigma (x - \overline x)^2$
$ = \large\frac{b^2}{a^2} \bigg( \large\frac{1}{n} \Sigma ( y - \overline y ) ^2 \bigg)$
$ = \large\frac{b^2}{a^2} \sigma_y^2$
$ \therefore r = \large\frac{cov (x,y)}{\sigma_x \sigma_y}$
$ = \large\frac{-\large\frac{b}{a} \sigma_y^2}{\sqrt{\large\frac{b^2}{a^2} \sigma_y^2. \sigma_y}}$
$ = \large\frac{-\large\frac{b}{a}}{|\large\frac{b}{a}|}$ $ = \large\frac{-\large\frac{b}{a}}{-\large\frac{b}{a}}$
$ = 1$
Ans : (A)
answered Jan 31, 2014 by thanvigandhi_1
 

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