Stat312: Sample Final Exam
Moo K. Chung, Yulin Zhang
[email protected], [email protected]
May 16, 2003
1. Let
X
1
,
···
,X
n
be a random sample from Bernoulli distribution with parameter
p
.
(a) Find the maximum likelihood estimator of
(1

p
)
2
(20pts).
(c) Is your estimator in (a) unbiased? If it is biased, ﬁnd an unbiased estimator of
(1

p
)
2
(20pts).
Solution.
(a) First ﬁnd the MLE for
p
. Note that the Bernoulli probability function is
P
(
X
i
=
x
) =
p
x
(1

p
)
1

x
for
x
= 0
,
1
(5pts). The likelihood function is then
L
(
x
1
,
···
,x
n
;
p
) =
p
∑
n
i
=1
x
i
(1

p
)
∑
n
i
=1
1

x
i
(5pts). By solving
∂
log
L/∂p
= 0
, we get
ˆ
p
= ¯
x
(5pts). From the invariance principle,
\
(1

p
)
2
= (1

¯
x
)
2
(5pts). (b)
E
(1

¯
X
)
2
=
1

2
E
¯
X
+
E
¯
X
2
. Note that
E
¯
X
2
=
Var
¯
X

(
E
¯
X
)
2
=
p
(1

p
)
/n

p
2
. So
E
(1

¯
X
)
2
= 1

2
p
+
p
(1

p
)
/n

p
2
.
Hence it is biased (5pts). We only need to ﬁnd an unbiased estimator of
p
2
. See Midterm I, where it is given by
¯
X

S
2
,
where
S
is the sample variance. So the unbiased estimator of
(1

p
)
2
is
1

2
¯
X
+
¯
X

S
2
= 1

¯
X

S
2
(15pts).
2. Gross sales before and after a training program is given by
Salesperson : 1 2 3 4 5 6
Sales before: 90 83 105 97 110 78
Sales after : 97 80 110 93 123 84
Determine if the training program is effective at
α
= 0
.
1
. Carefully do your analysis stating all the relevant assumptions
(30 pts).
Solution.
This problem can be also solved using regression analysis. The points will be given in three categories.
Model speciﬁcation
(10 pts). We assume that the gross sale
X
i
of
i
th sales person before the training to follow
normal, i.e.
X
i
∼
N
(
μ
b
,σ
2
b
)
. We also assume that the gross sale
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 Spring '04
 Chung
 Statistics, Bernoulli

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