Page 343 - DMTH404_STATISTICS

Unit 23: Regression Analysis

å(Y – Y )Y = å[(Y – Y ) –b(X – X )][ Y + b(X – X )] Notes
i ci ci i i i
= Y å(Y – Y ) – bY å(X – X ) + bå(X – X )(Y – Y ) – b å(X – X ) 2
2
i i i i i
2
2
2
= 0 – 0 + b å(X – X ) – b å(X – X ) = 0
2
i i
23.5.1 Standard Error of the Estimate
The standard error of the estimate of regression is given by the positive square root of the
variance of e values.
i
The standard error of the estimate of regression of Y on X or simply the standard error of the
estimate of Y is given as, S Y X = s Y 1- r 2 .
.

Similarly, S = s 1- r 2 is the standard error of the estimate X.
X Y X
.
According to the theory of estimation, to be discussed in Chapter 21, an unbiased estimate of the
variance of e values is given by
i
å e i 2 n å e 2 i n 2 2
2
s = = × = s × ( 1- r )
.
Y X Y
n 2 n 2 n n 2
-
-
-
 The standard errors of the estimate of Y and that of X are written as
n 2 n 2
s Y X = s Y ( 1- r ) and s X Y = s X ( 1- r ) respectively.
.
.
(n 2- ) (n 2- )
Example 15:
From the following data, compute (i) the coefficient of correlation between X and Y, (ii) the
standard error of the estimate of Y :
2
2
-
-

å x = 24 å y = 42 å xy = 30 N = 10, where x = X X and y = Y Y .
Solution.
The coefficient of correlation between X and Y is given by
å xy 30
r = = = 0.94
å x 2 å y 2 24 42

The standard error of the estimate of Y is given by (n < 30)

( 1- r 2 )å y 2 ( 1 0.94- 2 ) 42´
s Y X = = = 0.79
.
n 2 8
-

Example 16: For 100 items, it is given that the regression equations of Y on X and X on Y
are 8X – 10Y + 66 = 0 and 40X – 18Y = 214 respectively. Compute the arithmetic means of X and
Y and the coefficient of determination. If the standard deviation of X is given to be 3, compute
the standard error of the estimate of Y.

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