Chapter: Linear Regression· Mathematics & Statistics (Commerce) Part 2

For a bivariate data, the regression co-efficient of Y on X is 0.4 and the regression co-efficient of X on Y is 0.9. Find the value of variance of Y if variance of X is 9.

Q.5 (A)(ii) For a bivariate data, the regression co-efficient of Y on X is 0.4 and the regression co-efficient of X on Y is 0.9. Find the value of variance of Y if variance of X is 9.

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Answer: Var(Y) = 4

\(r^2 = b_{yx}\cdot b_{xy} = 0.4 \times 0.9 = 0.36\), so \(r = 0.6\) (positive, since both coefficients are positive).

\(\sigma_x = \sqrt{9} = 3\). Using \(b_{yx} = r\dfrac{\sigma_y}{\sigma_x}\): \(0.4 = 0.6 \times \dfrac{\sigma_y}{3} \Rightarrow \sigma_y = 2\).

Hence Var(Y) \(= \sigma_y^2 = 4\). (Check: \(b_{xy} = r\dfrac{\sigma_x}{\sigma_y} = 0.6 \times \dfrac{3}{2} = 0.9\).)

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For a bivariate data, the regression co-efficient of Y on X is 0.4 and the regression co-efficient of X on Y is 0.9. Find the value of variance of Y if variance of X is 9.

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