Page 328 - DMTH404_STATISTICS

Statistics

Notes
1
å ( X - X Y - Y ) )
i
)( i
n Cov (X Y,
= = .... (19)
1 2 s 2
Y -
å ( i Y ) Y
n
)( Y
( X
å
n X Y - å i å i )
i i
Also d = 2 2 .... (20)
( Y
å
n Y - å i )
i
This expression is useful for calculating the value of d. Another short-cut formula for the
calculation of d is given by
é ù
( u
)( v
h n u v - å i å i )
å
i i
d = ê 2 ú .... (21)
2
k ê n v - å ) ú
å
( v
ë i i û
X - A Y - B
where u = i and v = i
i
i
h k
Consider equation (19)
Cov (X Y, ) rs s Y s X
X
d = 2 = 2 = r× .... (22)
s s s
Y Y Y
Substituting the value of c from equation (15) into line of regression of X on Y we have
X = X dY dY or ( X - X = d Y - Y ) .... (23)
-
+
Ci i Ci ) ( i
s X
r
Y -
or ( X - X ) = × ( i Y ) .... (24)
Ci
s
Y
Remarks: It should be noted here that the two lines of regression are different because these
have been obtained in entirely two different ways. In case of regression of Y on X, it is assumed
2
that the values of X are given and the values of Y are estimated by minimising (Y - Y ) while
i Ci
in case of regression of X on Y, the values of Y are assumed to be given and the values of X are
estimated by minimising (X - X ) . Since these two lines have been estimated on the basis of
2
i Ci
different assumptions, they are not reversible, i.e., it is not possible to obtain one line from the
other by mere transfer of terms. There is, however, one situation when these two lines will
coincide. From the study of correlation we may recall that when r = ±1, there is perfect correlation
between the variables and all the points lie on a straight line. Therefore, both the lines of
regression coincide and hence they are also reversible in this case. By substituting r = ±1 in
equation (12) or (24) it can be shown that the lines of regression in both the cases become
æ Y - Y ö æ X - X ö
i
i
ç ÷ = ± ç ÷
è s Y ø è s X ø

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