Page 225 - DMGT209_QUANTITATIVE_TECHNIQUES_II

Quantitative Techniques-II

Notes n n

or  Y i = na b  X i .... (1)
i 1 i 1


S ¶ n

Also, = 2  Y  a bX i  X i   0
i
¶ b
i 1

n n
or  2  X Y  aX  bX 2 i  =  X Y  aX  bX 2 i   0
i
i
i
i
i
i
i 1 i 1


n n n
or  X Y  a  X  b  X 2 i = 0
i
i
i


i 1 i 1 i 1

n n n
a
or  X Y i =  X  b  X 2 i .... (2)
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
 Y i na b  X i
=  or Y  a bX .... (3)

n n n

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 i  b X i 2
i i
2
2

= Y  X  bX  X  b  X  nXY b.nX  b  X 2 i
i
i
i
2
or  X Y  nXY =  b  X  nX 2 
i
i
i
 X Y  nXY
i
i
or b = 2 2 .... (5)
 X  nX
i
Also,  X Y  nXY =   X  X  Y  Y 
i
i
i
i
2
and  X  nX 2 =   X  X  2
i
i
  X  X  Y  Y 
i
i
 b = 2 .... (6)
  X  X 
i
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