Page 194 - DMGT404 RESEARCH_METHODOLOGY

Research Methodology

Notes The necessary conditions for minima of S are
¶ S ¶ S ¶ S ¶ S
(i)  0 and (ii)  0, where and are the partial derivatives of S w.r.t. a and b
a ¶ b ¶ a ¶ b ¶
respectively.
¶ S n
-
2
Now a ¶ = - å 1 (Y - a bX i ) 0
i
i
n
n
n
-
-
Y -
or å ( i a bX i ) = å Y - na bå X  0
i
i
i 1 i 1 i 1
n
n
å i = na b+ å
or Y X i .... (1)
i 1 i 1
¶ S n
Y -
-
Also, b ¶ = 2å 1 ( i a bX i )( X- i ) 0
i
n
n
or - 2å (X Y - aX - bX i 2 ) = å (X Y - aX - bX 2 i ) 0
i
i
i
i
i
i
i 1 i 1
n
n
n
or å X Y - å X - å X 2 i = 0
a
b
i 1 i i i 1 i i 1
n
n
n
or å X Y i = aå X + å X i 2 .... (2)
b
i
i
i 1 i 1 i 1
Equations (1) and (2) are a system of two simultaneous equations in two unknowns a and b,
which can be solved for the values of these unknowns. These equations are also known as
normal equations for the estimation of a and b. Substituting these values of a and b in the
regression equation Y = a + b , we get the estimated line of regression of Y on X.
Ci Xi
Expressions for the Estimation of a and b.
Dividing both sides of the equation (1) by n, we have
å Y i  na + bå X i
+
n n n or Y  a bX .... (3)
This shows that the line of regression Y = a + bX passes through the point ( ,X Y ) .
Ci i
From equation (3), we have a Y - bX .... (4)
Substituting this value of a in equation (2), we have

SX Y = ( Y - bX ) X + b X 2
i i i i
= Y å X - å X + å X  nXY - . b nX + å X i 2
2
2
bX
b
b
i
i
i
å X Y - nXY = ( b å X - 2 )
2
or i i i nX
å X Y - nXY
i
i
or b = å X - nX 2 .... (5)
2
i
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