Statistics 5021 Midterm 2
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There are 3 questions, with point values given in parentheses for each part of each question. Show
all work to receive credit. You are allowed two 8.5x11 inch sheets of paper with your notes written
on both sides. You are permitted to use a calculator.
Question
Points Earned
Points Possible
1
6
2
7
3
7
Total
20
1
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1. Suppose that a candidate in the upcoming 2012 presidential election is interested in estimating
the proportion
θ
of US voters that would vote for him or her. In particular, this candidate
wishes to obtain a conservative approximate 95% confidence interval for
θ
, with a margin of
error of at most 5%. Assume that the yettobe recorded responses:
X
1
, . . . , X
n
are a random
sample from Bern(
θ
).
Note that
z
0
.
975
= 1
.
959964
(a)
(1 point)
What sample size
n
is needed for this candidate to to obtain this conservative
approximate confidence interval (described above)? Note that the sample size must be
a positive integer.
Solution:
The required sample size, with
m
e
= 0
.
05 and
α
= 0
.
05.
n
≥
z
1

α/
2
0
.
5
m
e
2
=
1
.
959964
*
0
.
5
0
.
05
2
= 384
.
1459
Implying that 385 individuals must be included in the sample.
(b)
(1 point)
A sample of the size computed in part 1a was obtained and this candidate
computed the observed conservative approximate 95% confidence interval for
θ
to be:
(0
.
3708338
,
0
.
4707246)
Interpret this interval.
Solution:
We are approximately 95% confident that
θ
, the population proportion of
voters that will vote for this candidate is in the interval (0
.
3708338
,
0
.
4707246).
(c)
(2 points)
We wish to carry out the hypothesis test:
H
0
:
θ
= 0
.
5
H
a
:
θ <
0
.
5
at the
α
= 0
.
01 significance level.
Using the confidence interval given in part 1b and
the sample size computed in part 1a, compute the test statistic realization for this large
sample Ztest.
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 Spring '08
 Staff
 Statistics, Statistical hypothesis testing, test statistic realization

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