THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT 1302 PROBABILITY AND STATISTICS II
Assignment 4 (Sketch Solution)
Section A
Hypotheses & likelihood ratios
A1. Let
X
be the no. of marked fish among a sample of 2,000 caught from the lake and
N
be the
total no. of fish in the lake.
(a) (i)
H
0
:
N
≤
21000 vs
H
1
:
N >
21000.
(ii) Note that the likelihood is
‘
(
N
) =
(
1000
100
)(
N

1000
1900
)
/
(
N
2000
)
. Consider
‘
(
N
+1)
/‘
(
N
) = 1

100(
N

19999)(
N
+1)

1
(
N

2899)

1
>
1
,
2900
≤
N
≤
19998
,
= 1
,
N
= 19999
,
<
1
,
N
≥
20000
.
Thus
‘
(
N
) increases as
N
increases up to 19,999, stays constant at
N
= 19999 and 20000,
and then decreases as
N
increases beyond 20,000.
Under
H
1
,
‘
(
N
) is maximized at
N
=
ˆ
N
1
= 21000.
Under
H
0
,
‘
(
N
) is maximized at
N
=
ˆ
N
0
= 20000 (or 19999). The likelihood ratio is
‘
(
ˆ
N
1
)
/‘
(
ˆ
N
0
) = 0
.
872511.
(b) (i)
H
0
:
N >
21000 vs
H
1
:
N
≤
21000.
(ii) Under
H
0
,
‘
(
N
) is maximized at
N
=
ˆ
N
0
= 21000. Under
H
1
,
‘
(
N
) is maximized at
N
=
ˆ
N
1
= 20000 (or 19999). The likelihood ratio is
‘
(
ˆ
N
1
)
/‘
(
ˆ
N
0
) = 1
.
146118.
A2. Let
X
s
, X
m
, X
n
be nos. substantially, mildly and not improved by the ordinary treatment re
spectively. Define
Y
s
, Y
m
, Y
n
for the new treatment similarly. Then
(
X
s
, X
m
, X
n
)
∼
Multinomial(54;
p
s
, p
m
, p
n
)
and
(
Y
s
, Y
m
, Y
n
)
∼
Multinomial(54;
q
s
, q
m
, q
n
)
.
(i)
H
0
:
p
s
=
q
s
, p
m
=
q
m
vs
H
1
:
p
s
< q
s
, p
m
< q
m
.
(ii) The likelihood is
‘
(
p, q
)
∝
p
12
s
p
6
m
p
36
n
q
18
s
q
12
m
q
24
n
. Under
H
1
,
‘
is max’d at
p
s
= 12
/
54,
p
m
=
6
/
54,
p
n
= 36
/
54,
q
s
= 18
/
54,
q
m
= 12
/
54 and
q
n
= 24
/
54. Under
H
0
,
‘
is max’d at
p
s
=
q
s
=
30
/
108,
p
m
=
q
m
= 18
/
108 and
p
n
=
q
n
= 60
/
108. The likeihood ratio is 16
.
9724.
A3. Let
X
be the no. of heads. Then
X
∼
Binomial(100
, p
),
p
∈
[0
,
1].
(i)
H
0
:
p
= 1
/
2 vs
H
1
:
p
6
= 1
/
2.
(ii) The likelihood is
‘
(
p
)
∝
p
60
(1

p
)
40
, maximized at
p
= ˆ
p
= 0
.
6.
The likelihood ratio is
‘
(ˆ
p
)
/‘
(1
/
2) = 7
.
48987.
1