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Question
The equation of the line of regression of y on x is y = `2/9` x and x on y is x = `"y"/2 + 7/6`.
Find (i) r, (ii) `sigma_"y"^2 if sigma_"x"^2 = 4`
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Solution
Given, regression equation of Y on X is
y = `2/9`x
i.e., Y = `2/9`X
Comparing with Y = bYX X + a, we get
`"b"_"YX" = 2/9`
and regression equation of X on Y is
`"x" = "y"/2 + 7/6`
i.e., X = `1/2 "Y" + 7/6`
Comparing it with X = bXYY + a', we get
`"b"_"XY" = 1/2`
(i) r = `+-sqrt("b"_"XY" * "b"_"YX")`
`= +- sqrt(1/2 * 2/9) = +- sqrt(1/9) = +- 1/3`
Since bYX and bXY both are positive,
r is positive.
∴ r = `1/3`
(ii) Given, `sigma_"X"^2 = 4`
∴ σX = 2
we know that, `"b"_"YX" = "r" sigma_"Y"/sigma_"X"`
∴ `sigma_"Y" = ("b"_"YX" xx sigma_"X")/"r" = (2/9 xx 2)/(1/3) = (4 xx 3)/9 = 4/3`
∴ `sigma_"Y"^2 = 16/9`
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