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Statistics

Notes Solution.

(a) The means of X and Y
d
Since the lines of regression pass through the point X ,Yi , the simultaneous solution of
the given regression equations would give the mean values of X and Y as X =13,Y =17

(b) The coefficient of determination
We assume that 8X - 10Y + 66 = 0 is the regression of Y on X and 40X - 18Y = 214 is the
8
regression of X on Y. Thus, the respective regression coefficients b and d are given by
10
18
and .
40

8 18
 The coefficient of determination r = b.d = ´ = 0.36
2
10 40
(c) The standard error of the estimate of Y

2 s Y
r
We know that s Y X = s Y 1- r . To find s we use the relation b = × .
.
Y
s
X
9 3 b.s X 8 5
2
Also r = \ r = Thus, s = = ´ ´ 3 = 4
Y
25 5 r 10 3
-
s = 4 1 0.36 = 3.2
Hence, Y X.
23.6 Summary of Formulae
I. Regression of Y on X

-
å
( X å
1. Regression coefficient b = å XY nX Y = n XY - å )( Y )
2
2
å X - nX 2 n X - å ) 2
( X
å
Cov (X Y, ) s
Also b = = r × Y
s 2 s
X X
2. Change of scale and origin
å
X A Y B k é n uv - å )( v ) ù
-
( u å
-
If u = and v = , then b = ê . ú
2
h h h ê n u - å ) 2 ú
( u
å
ë û
3. Constant term a =Y - bX
4. Alternative form of regression equation
s Y
Y - Y = ( X X- ) or Y - Y = × ( X X- )
r
C C
s
X
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