19/2/2013
1
Ratio Estimation
1
•
Use of auxiliary variable is common in
statistical analysis
•
Auxiliary variable can improve the efficiency of
the estimate of the main interest
2
Example
•
Wholesale price paid for oranges in large
shipments is based on the sugar content of
the load.
•
The exact sugar content can be estimated by
taking a random sample of
n
oranges.
•
Let
y
i
be the sugar content of the
i
th selected
orange. Then,
would estimate the total
sugar content for the load. Unfortunately, it is
time consuming and costly to determine
N
.
3
y
N
•
We could estimate the total by using the fact
that the sugar content of an individual orange
Y
is closely related to its weight
X
•
The ratio of the total sugar content to the
total weight of the truck load is equal to the
ratio of the mean sugar content per orange to
4
the mean weight per orange.
•
Note that
and thus
X
Y
T
T
X
Y
=
Y
where
and
are, respectively the population
totals
of variables
Y
and
X
.
X
Y
T
X
T
=
Y
T
X
T
5
•
We can estimate population means of
Y
and
X
by
the sample mean of
Y
and
X
from the sample.
Also we can measure the total weight of the
oranges on the trucks. Thus we have
X
Y
T
x
y
t
=
6

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