Page 341 - DMTH404_STATISTICS

Unit 23: Regression Analysis

Notes
Example 13:
Comment on the following statements :

(i) The two regression coefficients of bivariate data are 0.7 and 1.4.
(ii) A correlation coefficient r = 0.8, between the two variables X and Y, implies a relationship
twice as close as r = 0.4.

Solution.
2
(i) This statement implies that r = 0.7 × 1.4 = 0.98, i.e., a linear regression fitted to the data
would explain 98% of the variations in the dependent variable.
(ii) The given statement is wrong. Since r = 0.8 implies that a regression fitted to the data
would explain 64% of the variations in the dependent variable while r = 0.4 implies that
the proportion of such variations is only 16%. Thus, r = 0.8 implies a relation that is four
times as close as r = 0.4.

Example 14:
The correlation coefficient between two variables is found to be 0.8. Explain the meaning of this
statement.
Solution.
The given statement implies that :
(i) Two variables are highly correlated.

(ii) There is positive association between them, i.e., an increase in value of one is accompanied
by the increase in value of the other and vice-versa.
(iii) A linear regression fitted to the data would explain 64% of the variations in the dependent
variable.

23.4 Mean of the Estimated Values

We may recall that Y and X are the estimated values from the regressions of Y on X and X on
C C
Y respectively.
d
Consider the regression equation Y - Y =b X - Xi.
Ci
i
Taking sum over all the observations, we get
å ( Ci Y ) = bå ( X - X ) = 0
Y -
i
å Y Ci
Þ å Y - nY = 0 or = Y = Y .... (1)
Ci
C
n
Similarly, it can be shown that X = X .
C
This implies that the mean of the estimated values is also equal to the mean of the observed
values.

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