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Question
For certain X and Y series, which are correlated the two lines of regression are 10y = 3x + 170 and 5x + 70 = 6y. Find the correlation coefficient between them. Find the mean values of X and Y.
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Solution
Let 10y = 3x + 170 be the regression equation of Y on X.
∴ The equation becomes 10y = 3x + 170
i.e., Y = `3/10 "X" + 170/10`
Comparing it with Y = bYX X + a, we get
`"b"_"YX" = 3/10`
Now other equation 5x + 70 = 6y be the regression equation of X on Y.
∴ The equation becomes 5x = 6y – 70
i.e., X = `6/5 "Y" - 70/5`
Comparing it with X = bXY Y + a', we get
`"b"_"XY" = 6/5`
∴ r = `+-sqrt("b"_"XY" * "b"_"YX") = +-sqrt(6/5 xx 3/10) = +- sqrt(9/25) +- 3/5`
Since bYX and bXY both are positive,
r is positive.
∴ r = `3/5` = 0.6
Now, two correlated lines of regression are
10y = 3x + 170
i.e., - 3x + 10y = 170 …(i)
and 5x + 70 = 6y
i.e., 5x - 6y = –70 …(ii)
By (i) × 5 + (ii) × 3, we get
- 15x + 50y = 850
+ 15x - 18y = - 210
32y = 640
∴ y = 20
Substituting y = 20 in equation (i), we get
- 3x +10(20) = 170
∴ - 3x + 200 = 170
∴ 3x = 200 - 170
∴ x = 10
Since the point of intersection of two regression lines is `(bar x, bar y)`,
`bar x` = mean value of X = 10, and
`bar y` = mean value of Y = 20.
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