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Question
Regression equations of two series are 2x − y − 15 = 0 and 3x − 4y + 25 = 0. Find `bar x, bar y` and regression coefficients. Also find coefficients of correlation. [Given `sqrt0.375` = 0.61]
Solution
Given regression equation of two series are
2x - y – 15 = 0
i.e., 2x – y = 15 …(i)
and 3x - 4y + 25 = 0
i.e., 3x – 4y = –25 …(ii)
By (i) × – (ii), we get
8x - 4y = 60
3x - 4y = - 25
- + +
5x = 85
∴ x = 17
Substituting x = 17 in in equation (i), we have
2(17) - y = 15
∴ 34 - y = 15
∴ y = 34 - 15 = 19
Since the point of intersection of two regression lines is `(bar x, bar y)`,
`bar x = 17 and bar y = 19`
Now,
Let 2x - y – 15 = 0 be the regression equation of X on Y.
∴ The equation becomes 2x = y + 15
i.e., X = `"Y"/2 + 15/2`
Comparing it with X = bXY Y + a, we get
∴ `"b"_"XY" = 1/2`
Now, other equation 3x - 4y + 25 = 0 be the regression equation of Y on X.
∴ The equation becomes 4y = 3x + 25
i.e., Y = `3/4 "X" + 25/4`
Comparing it with Y = bYX X + a', we get
∴ `"b"_"YX" = 3/4`
∴ r = `+-sqrt("b"_"XY" * "b"_"YX")`
`= +- sqrt(1/2 * 3/4) = +- sqrt0.375 = +- 0.61`
Since bYX and bXY both are positive,
r is positive.
∴ r = 0.61
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