Solution of AMS312_2010 Practice Midterm 1
AMS312.01
Practice Midterm Exam #1
Spring, 2010
Instructions: This is a close book exam. Anyone who cheats in the exam shall receive a
grade of F. Please provide complete solutions for full credit. Good luck!
(***Note: the
real midterm will have only 3 questions. Here I provided more to help you review.)
1. A biologist wishes to estimate the effect of an antibiotic on the growth of a particular
bacterium by examining the mean amount of bacteria present per plate of culture when a
fixed amount of the antibiotic is applied. Previous experimentation with the antibiotic on
this type bacterium indicates that the standard deviation of the amount of bacteria present
per plate is approximately 12
2
cm
. (*Assume the population distribution is normal if
necessary.)
(1)
Please derive the general formula for sample size calculation in this type of scenario
based on a maximum error of
E
at a confidence level of 100(1α) %.
(2)
Plug the information given in this particular problem to determine the number of
plates necessary to estimate the mean amount of bacteria present within 6
2
cm
with a
probability of 99%.
Solution:
This is a sample size determination problem for the inference of
μ
based on
the maximum error E.
(1) P.Q.
~
(0,1)
/
X
Z
N
n
μ
σ

=
(

)
1
P
X
E
μ
α

≤
=

(
)
1
E
X
E
P
n
n
n
μ
α
σ
σ
σ


≤
≤
=

2
2
/
2
/
/
=
⇒
=
E
Z
n
Z
n
E
σ
σ
α
α
(2)We just need to plug in the numbers:
E=6,
1
0.99
0.01
α
=

=
,
12
σ
=
,
0.05
2
2.575
Z
Z
α
=
=
,
2
2.575*12
(
)
27
6
n
=
=
2. Let
. .
.
2
1
,
,
~
(
,
)
i i d
n
X
X
N
μ σ
K
, be a random sample from the normal population. Please
prove that
(1)
)
,
(
~
2
n
N
X
σ
μ
.
(2)
~
(0,1)
/
X
Z
N
n
μ
σ

=
.
ProofM
(1)
2
2
1
1
2
1
( )
(
)
(
)
(
)
(
)
(
(
))
i
i
i
t
t
t
n
t
X
X
X
X
t
t
tX
n
n
n
n
n
n
X
i
M
t
E e
E e
E e
E e
E e
e
σ
μ
+
=
∑
∑
=
=
=
=
=
=
∏
1
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Solution of AMS312_2010 Practice Midterm 1
Hence
)
,
(
~
2
n
N
X
σ
μ
(2)
2
2
2
/
/
/
/
/
1
1
2
/
/
/
/
2
( )
(
)
(
)
(
)
(
)
/
X
t
t
t
t
t
X
X
n
n
n
n
n
Z
t
t
t
t
t
n
n
n
n
n
X
M
t
E e
E e
e
E e
t
e
M
e
e
e
n
μ
μ
μ
σ
σ
σ
σ
σ
σ
μ
μ
μ
σ
σ
σ
σ
σ



+


+
=
=
=
=
=
=
Hence
~
(0,1)
/
X
Z
N
n
μ
σ

=
3. University officials are planning to audit 1586 new appointments to estimate the
proportion
p
who have been incorrectly processed by the Payroll Department.
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 Spring '08
 Zhu,W
 Normal Distribution, µ, Jerry sample, AMS312_2010 Practice Midterm

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