STAT
440
FINAL
EXAM
FALL,
1997
Instructions:
Work
any
three
problems.
Show
all
work
related
to
your
solutions.
You
may
use
calculators
or
a
3
x
5
card
with
formulas.
1.
In
a
population
of
N
elements,
positive
variables
x,-
and
y,'
are
de-
fined
for
each
element,
and
the
population
total
X
is
known.
It
is
proposed
to
estimate
the
population
total
Y
using
the
ratio
estimator
YR
=
X(y/x).
However,
the
sampling
plan
is
to
choose
a
sample
s
of
n
elements
with
prob-
ability
c^ies
x
i-
(a)
Show
how
to
find
c
so
that
YR
is
unbiased.
(b)
Show
that
s
can
be
selected
with
probability
c£,-
es
x,-
as
follows:
(i)
choose
a
first
element
with
probability
x,-/X;
(ii)
choose
n
—
I
additional
elements
by
simple
random
sampling
with-
out
replacement
from
the
remaining
N
—
1
elements.
2.
A
population
consists
of
N
clusters
with
M,-
elements
in
cluster
i.
The
total
number
of
elements
in
the
population,
MO,
is
unknown.
A
simple
random
sample
of
n
clusters
is
selected,
and
the
y
value
is
observed
on
each
element
of
the
selected
clusters.
The
estimator
\r
£—*l£s
eft
Y
R
=
^ rr
is
used
to
estimate
the
population
mean
per
element
N
Mi
where
y;
is
the
total
in
cluster
i.
Show
that
VarY#
is
approximately
equal
n
M
0
2
TV
-
1
and
propose
an
estimator
for
this
expression.
