STAT 409
Examples for 09/28/2009
Fall 2009
The
sample
proportion
:
n
x
p
=
ˆ
where
x
is the number of elements in the sample found to belong
to the category of interest (the number of "successes"), and
n
is
the sample size.
E(
P
ˆ
) =
p
,
Var(
P
ˆ
) =
( 29
n
p
p

⋅
1
,
SD(
P
ˆ
) =
( 29
1
n
p
p

⋅
.
A
largesample
confidence interval
for
the
population
proportion
p
is
n
p
p
z
p

±
⋅
⋅
ˆ
ˆ
ˆ
1
α
2
.
1.
Just prior to an important election, in a random sample of 749 voters, 397
preferred Candidate Y over Candidate Z.
Construct a 90% confidence
interval for the overall proportion of voters who prefer Candidate Y over
Candidate Z.
X = 397.
n
= 749.
749
397
X
p
ˆ
=
=
n
=
0.53
.
The confidence interval :
n
)
p
ˆ
1
(
p
ˆ
z
p
ˆ
2

⋅
⋅
±
α
.
90% confidence level
α
= 0.10
α
2
=
0.05
.
2
z
α
=
1.645
.
749
)
47
0
)(
53
0
(
645
1
53
0
.
.
.
.
⋅
±
0.53
±
0.03
( 0.50
, 0.56
)
Uppertail probability
0.25
0.20
0.15
0.10
0.05
0.025
0.02
0.01
0.005
0.0025
0.001
0.0005
z
0.674
0.841
1.036
1.282
1.645
1.960
2.054
2.326
2.576
2.807
3.091
3.291
50%
60%
70%
80%
90%
95%
96%
98%
99%
99.5%
99.8%
99.9%
Confidence level
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An article on secretaries' salaries in the
Wall Street
Journal
reports:
"Threefourth of surveyed secretaries said they make less than $25,000
a year."
Suppose that the
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 Spring '11
 Stepanov
 α, Confidence Level

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