Question

The two regression equations are 5x − 6y + 90 = 0 and 15x − 8y − 130 = 0. Find `bar x, bar y`, r.

Answer 1

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

∴ 5x = 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 = b_YX 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 = b_XY 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 b_YX and b_XY both are positive, r is positive.

∴ r = `2/3`