# In a partially destroyed record, the following data are available: variance of X = 25, Regression equation of Y on X is 5y − x = 22 and regression equation of X on Y is 64x − 45y = 22 - Mathematics and Statistics

Sum

In a partially destroyed record, the following data are available: variance of X = 25, Regression equation of Y on X is 5y − x = 22 and regression equation of X on Y is 64x − 45y = 22 Find

1. Mean values of X and Y
2. Standard deviation of Y
3. Coefficient of correlation between X and Y.

#### Solution

Given, sigma_"X"^2 = 25

∴ sigma_"X" = 5

Regression equation of Y on X is

5y – x = 22

Regression equation of X on Y is

64x - 45y = 22

(i) Consider, the two regression equation

- x + 5y = 22       ....(i)

64x - 45y = 22    ....(ii)

By (i) × + (ii), we get

- 9x + 45y = 198
+ 64x - 45y = 22
55x       = 220

∴ x = 4

Substituting x = 4 in (i), we get

- 4 + 5y = 22

∴ 5y = 22 +  4

∴ y = 26/5 = 5.2

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

bar x = mean value of X = 4 and

bar y = mean value of Y = 5.2

(ii) To find standard deviation of Y we should first find the coefficient of correlation between X and Y.

Regression equation of Y on X is

5y - x = 22

i.e., 5Y = X + 22

i.e., Y = "X"/5 + 22/5

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

"b"_"YX" = 1/5

Now, regression equation of X on Y is

64x - 45y = 22

i.e., 64X - 45Y = 22

i.e., 64X = 45Y + 22

i.e., X = "45Y"/64 + 22/64

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

"b"_"XY" = 45/64

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

= +- sqrt((1/5)(45/64)) = +- sqrt(9/64) = +- 3/8

Since bYX and bXY are positive,

r is also positive.

∴ r = 3/8

Now, "b"_"YX" = "r" sigma_"Y"/sigma_"X"

∴ 1/5 = 3/8 xx sigma_"Y"/5

∴ sigma_"Y" = 1/5 xx 8/3 xx 5

∴ sigma_"Y"= Standard deviation of Y = 8/3

(iii) The correlation coefficient of X and Y is

r = 8/3 = 0.375

Concept: Properties of Regression Coefficients
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