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
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Stat 101 – Lecture 24
Example
pˆ = 0.63
pˆ (1 − pˆ )
0.63(0.37)
=
= 0.016
n
900
0.63 − 2(0.016) to 0.63 + 2(0.016)
0.598 to 0.662
4
Interpretation
• We are 95% confident that the
population proportion of registered
voters in the U.S. who are
concerned about the spread of bird
flu in the U.S. is between 59.8%
and 66.2%.
5
Interpretation
• Plausible values for the population
parameter p.
• 95% confidence in the process that
produced this interval.
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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
2SE ( 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
• CNN/USA Today/Gallup Poll. Oct. 1316, 2005. Adults nationwide.
• As you may know, Harriet Miers is the
person nominated to serve on the
Supreme Court. Would you like to see
the Senate vote in favor of Miers
serving on the Supreme Court, or not?"
13
Another Example
n=485 randomly selected adults
In favor. Not in favor. Unsure
44%
36%
20%
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Another Example
• 90% confidence interval for p
pˆ (1 − pˆ )
= 0.0225 z* = 1.645
n
0.44 − 1.645(0.0225) to 0.44 + 1.645(0.0225)
pˆ = 0.44 SE( pˆ ) =
0.44 − 0.037 to 0.44 + 0.037
0.403 to 0.477
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Stat 101 – Lecture 24
What Sample Size?
• Maximum 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.
1
1
n≤
n≤
= 2,500
2
(0.02 )2
ME
⇒
17
Sample Size
• More general formula for sample
size.
(z *)
n=
pˆ (1 − pˆ )
ME2
2
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