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]

Answer 1

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 = b_XY 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 = b_YX X + a', we get

∴ `b_YX = 3/4`

∴ r = `+-sqrt(b_XY * b_YX)`

`= +- sqrt(1/2 * 3/4)`

= `+- sqrt0.375`

= `+- 0.61`

Since b_YX and b_XY both are positive,

r is positive.

∴ r = 0.61