PROPERTIES OF REGRESSION LINES

Property 1 :

The regression coefficients remain unchanged due to a shift of origin but change due to a shift of scale.

This property states that if the original pair of variables is (x, y) and if they are changed to the pair (u, v) where

Property 2 :

The two lines of regression intersect at the point

where x and y are the variables under consideration. 

Property 3 :

The coefficient of correlation between two variables x and y in the simple geometric mean of the two regression coefficients. The sign of the correlation coefficient would be the common sign of the two regression coefficients.

This property says that if the two regression coefficients are denoted by b_yx and b_xy then the coefficient of correlation is given by

If both the regression coefficients are negative, r would be negative and if both are positive, r would assume a positive value.

Property 4 : 

The two lines of regression coincide i.e. become identical when r = –1 or 1 or in other words, there is a perfect negative or positive correlation between the two variables under discussion.

Property 5 :

The two lines of regression are perpendicular to each other when r = 0. 

Solved Problem

For the variables x and y, the regression equations are given as 7x – 3y – 18 = 0 and 4x – y – 11 = 0

(i) Find the arithmetic means of x and y.

(ii) Identify the regression equation of y on x.

(iii) Compute the correlation coefficient between x and y.

(iv) Given the variance of x is 9, find the SD of y.

Solution (i) :

By property, always the two lines of regression intersect at the point

(mean of 'x', mean of 'y')

Solving the given two regression equations, we get the point of intersection (3, 1).

Hence 

Arithmetic mean of 'x'  =  3

Arithmetic mean of 'y'  =  1

Solution (ii) :

Let us assume that 7x – 3y – 18 = 0 represents the regression line of y on x and 4x – y – 11 = 0 represents the regression line of x on y.

Now,

7x - 3y - 18  =  0 ------>  y  =  (-6) + (7/3)x

Therefore,  b_yx  =  7/3

4x - y - 11  =  0 ------>  x  =  11/4 + (1/4)y

Therefore,  b_xy  =  1/4

We can get the value of 'r', using the formula given below

Both b_yx and b_xy are positive, so 'r' is also positive. 

Using the above formula, the value of r is 0.7638.

Since r = 0.7638 which lies in the interval -1 ≤ r ≤ 1, our assumptions are correct. Thus, 7x – 3y – 18 = 0 truly represents the regression line of y on x.

Solution (iii) :

From solution (ii),

r  =  0.7638

Hence, correlation coefficient between x and y is 0.7638. 

Solution (iv) :

Given b_yx  =  r ⋅ Sᵧ/Sₓ

Variance of x  =  9 ------> S_x  =  3

Then, 

(7/3)  =  0.7638 ⋅ S_y/3

S_y  =  7 / 0.7638

Sᵧ  =  9.1647

So, the standard deviation of y is 9.1647.

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