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Question
The two regression equations are 5x − 6y + 90 = 0 and 15x − 8y − 130 = 0. Find `bar x, bar y`, r.
Solution
Given, the two regression equations are
5x - 6y + 90 = 0
i.e., 5x - 6y = - 90 ...(i)
and 15x - 8y - 130 = 0
i.e., 15x - 8y = 130 ...(ii)
By (i) × 3 – (ii), we get
15x - 18y = - 270
15x - 8y = 130
- + -
- 10y = - 400
∴ y = 40
Substituting y = 40 in (i), we get
5x - 6(40) = –90
∴ 5x - 240 = - 90
∴ 5x = - 90 + 240 = 150
∴ x = 30
Since the point of intersection of two regression lines is `(bar x, bar y)`,
∴ `bar x` = 30 and `bar y` = 40
Now, let 5x – 6y + 90 = 0 be the regression equation of Y on X.
∴ The equation becomes 6Y = 5X + 90
i.e., Y = `5/6 "X" + 90/6`
Comparing it with Y = bYX X + a, we get
∴ `"b"_"YX" = 5/6`
Now, other equation 15x – 8y – 130 = 0 be the regression equation of X on Y.
∴ The equation becomes 15X = 8Y + 130
i.e., X = `8/15 "Y" + 130/15`
Comparing it with X = bXY Y + a', we get
∴ `"b"_"XY" = 8/15`
∴ r = `+-sqrt("b"_"XY" * "b"_"YX")`
`= +- sqrt(8/15 * 5/6) = +- sqrt(4/9) = +- 2/3`
Since bYX and bXY both are positive,
r is positive.
∴ r = `2/3`
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