Question

If the two regression lines for a bivariate data are 2x = y + 15 (x on y) and 4y = 3x + 25 (y on x), find

1. `bar x`,
2. `bar y`,
3. b_YX
4. b_XY
5. r [Given `sqrt0.375` = 0.61]

Answer 1

Given,

regression equation of X on Y is 2x = y + 15,

i.e., 2x - y = 15           …(i)

regression equation of Y on X is 4y = 3x + 25,

i.e., - 3x + 4y = 25       …(ii)

By (i) ×  4 + (ii) we get

    8x - 4y = 60
+ - 3x + 4y = 25 
   5x    = 85

∴ x = 17

Substituting the value of x = 17 in (i), we get

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)`,

(i) `bar x` = 17

(ii) `bar y` = 19

(iii) Regression equation of Y on X is 4y = 3x + 25

i.e., 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`

(iv) Regression equation of X on Y is 2x = y + 15

i.e., 2X = Y + 15

i.e., X = `"Y"/2 + 15/2`

Comparing it with X = b_XY Y + a' we get,

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

(v) 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 also positive.

∴ r = 0.61