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Question
From the two regression equations, find r, `bar x and bar y`. 4y = 9x + 15 and 25x = 4y + 17
Solution
Given regression equations are
4y = 9x + 15
i.e., - 9x + 4y = 15 ....(i)
and 25x = 4y + 17
i.e., 25x - 4y = 17 ...(ii)
Adding equations (i) and (ii), we get
- 9x + 4y = 15
25x - 4y = 17
16x = 32
∴ x = 2
Substituting x = 2 in (i), we get
- 9(2) + 4y = 15
∴ - 18 + 4y = 15
∴ 4y = 33
∴ y = 8.25
Since the point of intersection of two regression lines is `(bar x, bar y), bar x = 2 and bar y = 8.25`
Let 4y = 9x + 15 be the regression equation of Y on X.
∴ The equation becomes Y = `9/2 "X" + 15/4`
Comparing it with Y = bYX X + a, we get
`"b"_"YX" = 9/4 = 2.25`
Now, the other equation, i.e., 25x = 4y + 17 is the regression equation of X on Y.
∴ The equation becomes X = `4/25 "Y" + 17/25`
Comparing it with X = bXY Y + a', we get
`"b"_"XY" = 4/25 = 0.16`
r = `+-sqrt("b"_"XY" * "b"_"YX")`
`= +- sqrt (0.16 xx 2.25)`
`= +- sqrt0.36 = +- 0.6`
Since bYX and bXY are positive,
r is also positive.
∴ r = 0.6
∴ `bar x = 2 and bar y = 8.25` and r = 0.6
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