Page 333 - DMTH404_STATISTICS

Unit 23: Regression Analysis

(c) (i) To estimate blood pressure (Y) for a given age, X = 50 years, we shall use regression Notes
of Y on X
Y = 87.2 + 0.72×. 50 = 123.2
C
(ii) The estimate of blood pressure when age is 20 years

Y = 87.2 + 0.72×. 20 = 101.6
C
It should be noted here that this estimate is wrong because the blood pressure of a
normal person cannot be less than 110.

This result reflects the limitations of regression analysis with regard to estimation
or prediction. It is important to note that the prediction, based on regression line,
should be done only for those values of the variable that are not very far from the
range of the observed data, used to derive the line of regression. The prediction
from a regression line for a value of the variable that is far away from the observed
data is likely to give inconsistent results like the one obtained above.

Example 4:
A panel of judges P and Q graded seven dramatic performances by independently awarding
marks as follows :
Performance : 1 2 3 4 5 6 7
Marks by P : 46 42 44 40 43 41 45
Marks by Q : 40 38 36 35 39 37 41

The eighth performance which Judge Q could not attend, was awarded 37 marks by Judge P. If
Judge Q had also been present, how many marks would be expected to have been awarded by
him to eighth performance?
Solution.
Let us denote marks awarded by the Judge P as X and marks awarded by the Judge Q as Y. Since
we have to estimate marks that would have been awarded by Judge Q, we shall fit a line of
regression of Y on X to the given data.
Calculation table

X Y u = X - 43 v = X - 37 uv u 2 v 2
46 40 3 3 9 9 9
42 38 - 1 1 - 1 1 1
44 36 1 - 1 - 1 1 1
40 35 - 3 - 2 6 9 4
43 39 0 2 0 0 4
41 37 - 2 0 0 4 0
45 41 2 4 8 4 16
Total 0 7 21 28 35

From the table, we have
7
X = 43 and Y = 37 + = 38
7
-
´
n uv - å )( v ) 7 21 0
å
( u å
Further, b = 2 = = 0.75
-
2
´
( u
å
n u - å ) 7 28 0
Also a = Y bX = 38 0.75 43 = 5.75
´
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