Question

The equations of two regression lines are 8x – 10y + 66 = 0 and 40x – 18y = 214. Find

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

Answer 1

(a) We know that the co-ordinates of the point of intersection of the two lines are `barx` and `bary`, the means of x and y

The regression equations are 8x – 10y + 66 = 0 and 40x – 18y = 214

Solving these equations simultaneously.

We get

40x – 50y = –330
40x – 18y = 214
(–)    (+)      (–)   
–32y = –544

y = 17

put y = 17 in 8x – 10y = –66

i.e. 8x – 10(17) = –66

8x = –66 + 170

8x = 104

x = 13

Hence, the means of X and Y are

`barx` = 13 and `bary` = 17

(b) 

Now, to find correlation coefficient we have to find the regression coefficients b_YX and b_XY. For this we have to choose one of the lines as that of line of regression of Y on X and the other line of regression of X on Y.

Let 8x – 10y + 66 = 0 be the line of regression

 Y on X

i.e. 10y = 8x + 66

y = `8/10x+66/10`

b_YX = `8/10`

= `4/5`

Then the other equation is that of line of regression of X on Y is

40x = 18y + 214

x = `18/40y + 214/40`

b_XY = `18/40`

∴ b_XY = `9/20`

Now, we know that

r² = b_XY .b_YX

= `9/20xx4/5`

= `9/25`

r = ± `3/5`

∴ r = ± 0.6

The correlation coefficient has the sign as that of b_YX and b_XY

∴ r = 0.6

= `3/5`