Question

In a partially destroyed laboratory record of an analysis of regression data, the following data are legible:

Variance of X = 9
Regression equations:
8x − 10y + 66 = 0
and 40x − 18y = 214.
Find on the basis of above information

1. The mean values of X and Y.
2. Correlation coefficient between X and Y.
3. Standard deviation of Y.

Answer 1

Given, `sigma_"X"^2 = 9`

∴ σ_X = 3

(i) The two regression equations are

8x - 10y + 66 = 0

i.e., 8x - 10y = - 66      ...(i)

and 40x - 18y = 214    ....(ii)

By 5 × (i) - (ii), we get

40x - 50y = - 330

40x - 18y = 214
(-)    (+)      (-)   
- 32y = - 544

∴ y = `544/32 = 17`

Substituting y = 17 in (i), we get

8x - 10 × 17 = - 66

∴ 8x - 170 = - 66

∴ 8x = - 66 + 170

∴ 8x = 104

∴ x = `104/8 = 13`

Since the point of intersection of two regression lines is `(bar x, bar y)`,

`bar x` = mean value of X = 13, and

`bar y` = mean value of X = 17.

(ii) Let 8x - 10y + 66 = 0 be the regression equation of Y on X.

∴ The equation becomes 10Y = 8X + 66

i.e., Y = `8/10 "X" + 66/10`

i.e., Y = `4/5 "X" + 33/5`

Comparing it with Y = b_YX X + a, we get

`"b"_"YX" = 4/5`

Now, the other equation, i.e., 40x - 18y = 214 is the regression equation of X on Y.

∴ The equation becomes X = `18/40 "Y" + 214/40`

i.e., X = `9/20 "Y" + 107/20`

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

`"b"_"XY" = 9/20`

r = `+- sqrt("b"_"XY" * "b"_"YX")`

∴ r = `+- sqrt(9/20 xx 4/5) = +- sqrt(9/25) = +- 3/5 = +- 0.6`

Since b_YX and b_XY both are positive,

r is also positive.

∴ r = 0.6

(iii) `"b"_"YX" = "r" sigma_"Y"/sigma_"X"`

∴ `4/5 = 0.6 xx sigma_"Y"/3`

∴ `4/5 = sigma_"Y"/5`

∴ `sigma_"Y" = 4`