Stat 101 – Lecture 24
• About 95% of the time the sample
proportion, p̂ , will be within
pˆ (1 − pˆ )
n
two standard errors of p.
2SE( pˆ ) = 2
1
• About 95% of the time the sample
proportion, p, will be within
pˆ (1 − pˆ )
n
two standard errors of p̂ .
2SE ( pˆ ) = 2
2
Confidence Interval for p
• We are 95% confident that p will
fall between
pˆ − 2
pˆ (1− pˆ )
pˆ (1− pˆ )
and pˆ + 2
n
n
3
Stat 101 – Lecture 24
Example
p̂ = 0.82
p̂( 1 − p̂ )
0.82( 0.18 )
=
= 0.013
n
900
0.82 − 2( 0.013 ) to 0.82 + 2( 0.013 )
0.794 to 0.846
4
Interpretation
• We are 95% confident that the
population proportion of all
registered voters in the U.S. who
believe global warming exists is
between 79.4% and 84.6%.
5
Interpretation
• Plausible values for the population
parameter p.
• 95% confidence in the process that
produced this interval.
6
Stat 101 – Lecture 24
95% Confidence
• If one were to repeatedly sample at
random 900 registered voters and
compute a 95% confidence interval
for each sample, 95% of the
intervals produced would contain
the population proportion p.
7
Simulation
http://statweb.calpoly.edu/chance/ap
plets/Confsim/Confsim.html
8
9
Stat 101 – Lecture 24
Margin of Error
2 SE ( pˆ ) = 2
pˆ (1 − pˆ )
n
Is called the Margin of Error (ME).
This is the furthest p̂ can be from
p, with 95% confidence.
10
Margin of Error
• What if we want to be 99.7%
confident?
pˆ (1 − pˆ )
n
ME = 3SE ( pˆ ) = 3
11
Margin of Error
ME = z * SE ( pˆ ) = z *
Confidence
z*
80%
90%
95%
pˆ (1 − pˆ )
n
98%
99%
1.282 1.645 2 or 1.96 2.326 2.576
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Stat 101 – Lecture 24
Another Example
• FOX News/Opinion Dynamics Poll,
Jan. 30-31, 2007. Asked of the 737
respondents who believed global
warming exists.
• Do you believe global warming is
caused by normal climate patterns or
by people’s behavior, such as driving
and burning too much fossil fuel like
coal and oil?
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Another Example
n=737 randomly selected adults who
believe global warming exists.
Global warming due too …
Normal
Patterns
People’s
Behavior
Both
Unsure
14%
41%
38%
8%
14
Another Example
• 90% confidence interval for p, the population
proportion who believe global warming is due
to people’s behavior.
p̂(1 − p̂ )
= 0.0181 z* = 1.645
n
0.41 − 1.645( 0.0181) to 0.41 + 1.645( 0.0181)
0.41− 0.030 to 0.41+ 0.030
0.38 to 0.44
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p̂ = 0.41 SE( p̂ ) =
Stat 101 – Lecture 24
What Sample Size?
• Conservative Formula
– The sample size to be 95% confident
that p̂, the sample proportion, will
be within ME of the population
proportion, p.
1
ME 2
n=
16
Example
• Suppose we want to be 95%
confident that our sample
proportion will be within 0.02 of
the population proportion.
n=
1
ME 2
⇒ n = (0.021 )
2
= 2 ,500
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Sample Size
• More general formula for sample
size.
(z *)
n=
pˆ (1 − pˆ )
ME2
2
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