Page 336 - DMTH404_STATISTICS

Statistics

Notes (c) The coefficient of correlation

×
r = b d = 0.37 2.2 = 0.90
´

Example 8:
Find the means of X and Y variables and the coefficient of correlation between them from the
following two regression equations :

3Y - 2X - 10 = 0
2Y - X - 50 = 0
Solution.
(a) The means of X and Y
d
We know that both the lines of regression intersect at the point X ,Yi . The simultaneous
solution of the given equations will give the mean values of X and Y as
X =130 and Y = 90 respectively.
(b) Correlation Coefficient

Let us assume that the first equation be regression of Y on X. Rewriting this equation as 3Y
2 10
= 2X + 10 or = X + .
Y
3 3
2
 The corresponding regression coefficient, b =
3
Further, assuming the second equation as regression of X on Y, we can rewrite this equation
as X = 2Y - 50.
 The regression coefficient, d = 2

2 4
Since b.d = 2 × = > 1, therefore, our assumptions regarding the two regression lines
3 3
are wrong.
Now we reverse these assumptions and assume that the first equation is regression of X on
Y and second the regression of Y on X.

3
5
 The first equation can be written as 2X = 3Y - 10 or = Y - , so that the corresponding
X
2
3
regression coefficient is d = . Further, the second equation can be written as 2Y = X + 50
2
1 1
or Y = X 25 , so that the corresponding regression coefficient is b = . Since b.d
+
2 2
3 1 3
= ´ = < 1 , our assumption is correct.
2 2 4
3 3
2
Also r = b.d =  r = =0.87
4 4

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