Paper 4: Fundamentals of Business Mathematics & Statistic

6.28 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Correlation and Regression

Regression coefficient of y on x or byx can be calculated by any of the following ways (Note that X & Y has

been interchanged in the above formulas)—

(1) when standard deviations are given-

YX

b r Y

X

= σ

σ

(2) when deviations are taken from actual mean

YX 2

b XY

= X

(3) when deviations are taken from assumed mean

( )

YX X Y X Y

2

2 X

X

b d d d d

d

d N

= −

−

6.4.8.1. FEATURES OF REGRESSION COEFFICIENTS

(i) Both of regression coefficients should have same sign i e., either positive or negative.

(2) Coefficient of correlation could be found out if regression coefficients are known; by

the formula

r b x b= XY YX

(3) Correlation coefficient would have the same sign as that of regression coefficients. ie., either positive

or negative.

(4) Since -1≤ r≤ 1 this implies both the regression coefficient cannot be greater than one.

For example if bXY = 2 and bYX = 1.5 then value of r = 2x1.5 3 1.732= = , which is not possible.

Example18 : Given:

N 5 X 20 Y 10= = =

∑(X - 20)^2 = 100 ∑(Y - 10)^2 = 60

∑(X-20) (Y- 10) =40

Find two regression equations.

Solution:

N = 5

X = 20

Y = 10

∑(X - 20)^2 = ∑(X - X)^2 = ∑X^2 = 100

∑(Y - 10)^2 = ∑(Y - Y)^2 = ∑Y^2 = 60