Page 196 - DMGT404 RESEARCH_METHODOLOGY

Research Methodology

Notes Substituting these values in equation (6), we have
hkå ( i u v - ) v k å ( i u v - ) v
u -
u -
)( i
)( i
b = h å ( i ) u 2 = h å ( i ) u 2
u -
u -
2
é nå u v - (å )(å ) ù
k ê i i u i v i ú
= h ê nå u - (å ) 2 ú .... (10)
2
ë i u i û
(Note: if h = k they will cancel each other)
Consider equation (8), b = Cov ( ,X Y )
s X 2
r s s s

Writing Cov(X, Y) = r  s s , we have b = X Y = r  Y
X Y s 2 X s X
The line of regression of Y on X, i.e Y = a + bX can also be written as
Ci i
or Y = Y - bX + bX or Y - Y = ( b X - X ) .... (11)
Ci i Ci i
s
or Y - Y ) = r  Y (X - X ) .... (12)
s i
( Ci
X
Line of Regression of X on Y

The general form of the line of regression of X on Y is X = c + dY , where X denotes the
Ci i Ci
predicted or calculated or estimated value of X for a given value of Y = Y and c and d are
i
constants. d is known as the regression coefficient of regression of X on Y.
In this case, we have to calculate the value of c and d so that
2
S = (X – X ) is minimised.
i Ci
As in the previous section, the normal equations for the estimation of c and d are
X = nc + dY .... (13)
i i
and X Y = cY + dY 2 .... (14)
i i i i
Figure 9.5

Y
X c bY i
+
=
ci
Y
i
Y X
ci i
c
O X

Dividing both sides of equation (13) by n, we have X = c dY .
+

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