Page 325 - DMTH404_STATISTICS

Unit 23: Regression Analysis

Notes
¶ S n
Also, = 2å ( i a bX i )( X- i ) = 0
-
Y -
b ¶ i 1
=
n 2 n 2
or - å ( X Y - aX - bX i ) = å ( X Y - aX - bX i ) = 0
2
i i
i
i i
i
=
=
i 1 i 1
n n n 2
a
or å X Y - å X - å X = 0
b
i i
i
i
=
i 1 i 1 i 1
=
=
n n n
a
b
or å X Y = å X + å X 2 .... (2)
i i 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 + bX , we get the estimated line of regression of Y on X.
Ci i
Expressions for the Estimation of a and b.
Dividing both sides of the equation (1) by n, we have
å Y i = na + b X i or .... (3)
å
+
n n n Y = a bX
d
This shows that the line of regression Y = a + bX passes through the point X ,Yi .
Ci i
From equation (3), we have a =Y - bX .... (4)
Substituting this value of a in equation (2), we have
2
å X Y = ( Y bX- )å X + å i
b X
i
i i
2 2 2
.
Y X -
b X =
-
b X
= å i bX X + å i nXY b nX + å i
å
i
2 2
or å X Y - nXY = ( b å X - nX )
i i
i
i i
or b = å X Y - nXY .... (5)
2
å X - nX 2
i
Also, å X Y - nXY = å ( X - X Y - Y ) (See Chapter 12)
i i
i
)( i
2
2
and å X - nX = å ( X - X ) 2
i
i
å ( X - X Y - Y )
i
)( i
 b = 2 .... (6)
å ( X - X )
i
or b = å x y .... (7)
i i
å x 2 i
where x and y are deviations of values from their arithmetic mean.
i i
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