Q. 7. In a partially destroyed laboratory record of an analysis of correlation data, only the following results are legible: (Dec. 2000)
Variance of x = 9
Regression equations:
8x - 10y + 66 = 0 of y on x
40x - 18y - 214 = 0 of x on y

Find out
(i) the mean values of x and y
(ii) the correlation coefficient between x and y

Solution.

(i) Mean values of x and y

8x - 10y = -66 ........(1)
40x - 18y = 214 ........(2)

Multiplying equation (1) by 5 and subtracting (2) from (1), we get

or y = 17

Substituting the value of y in equation (1)
8x - 10 X 17 = -66
or x = 13

Therefore, mean values of x and y are 13 and 17 respectively.

(ii) Correlation coefficient between x and y

From equation (1)
x = -66/8 + (10/8)y
Therefore, regression coefficient of x on y (b_xy) = 10/8 = 1.25

From equation (2)
y = -214/18 + (40/18)x
Therefore, regression coefficient of y on x (b_yx) = 40/18 = 2.22

Since both the regression coefficients are exceeding 1, our assumption is wrong. Hence, the equation (1) is the equation of y on x.

From equation (1)
-10y = -8x - 66
or y = (8/10)x + 6.6
Therefore, b_yx = 8/10 = 0.8

From equation (2)
b_xy = 18/40 = 0.45

r² = (b_xy X b_yx)
r² = (0.45 X 0.8)
or r = ± 0.6
Since both the regression coefficients are positive, we take r = + 0.6
Therefore, correlation coefficient between x and y is 0.6.

Q. 8. Fit a straight line to the data given by the following table: (Dec. 2002)

Solution.

Equation of straight line:

y = a + bx
Here, n = 6

or b = 9.83
a = [(1/n) X (å z - bå y)]
or a = (1/6) X [(190 - (9.83 X 30)] = -17.48
Therefore, y = -17.48 + 9.83x