The equations of two regression lines are

2x + 3y − 6 = 0

and 2x + 2y − 12 = 0 Find

- Correlation coefficient
- `sigma_"X"/sigma_"Y"`

#### Solution

The given regression equations are

2x + 3y – 6 = 0 and 2x + 2y – 12 = 0

**(i) **Let 2x + 3y – 6 = 0 be the regression equation of Y on X

∴ The equation becomes 3Y = – 2X + 6

i.e., Y = `(-2)/3 "X" + 6/3`

Comparing it with Y = b_{YX} X + a, we get

`"b"_"YX" = - 2/3`

Now, the other equation, i.e., 2x + 2y – 12 = 0 is the regression equation of X on Y.

∴ The equation becomes 2X = –2Y + 12

i.e., X = `- 2/2 "Y" + 12/2`

Comparing it with X = b_{XY} Y + a' we get

`"b"_"XY" = - 2/2 = - 1`

∴ r = `+-sqrt("b"_"XY" * "b"_"YX")`

`= +- sqrt(-1 * (- 2/3)) = +-sqrt(2/3) = +- 0.82`

since b_{XY} and b_{YX} are negative,

r is also negative.

∴ r = - 0.82

**(ii) **`"b"_"XY" = "r" sigma_"X"/sigma_"Y"`

∴ `- 1 = - 0.82 xx sigma_"X"/sigma_"Y"`

∴ `sigma_"X"/sigma_"Y" = (- 1)/- 0.82`

∴ `sigma_"X"/sigma_"Y"` = 1.22