Stat 104 – Lecture 21
Parameter – numerical
summary of the entire
population.
Population – all items
of interest.
Inference
Sample – a
few items from
the population.
Statistic – numerical
summary of the
sample.
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Poll on the Environment
• USA Today/Gallup Poll
conducted June 11 – 13, 2010
• “Would you favor or oppose
Congress passing new
legislation this year that would
regulate energy output from
private companies in an attempt
to reduce global warming?”
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Poll on the Environment
• Population – all adults in the
United States.
• Population Parameter –
proportion of all adults in the
U.S. who would favor
legislation.
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Stat 104 – Lecture 21
Poll on the Environment
• Sample – 1,014 adults in the
United States.
• Sample Statistic – proportion
of the 1,014 who favor new
legislation.
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Parameter – population proportion
0.53 ≤ p ≤ 0.59
Population – all adults
in the U.S.
Inference
Statistic – sample proportion
Sample – 1,014
pˆ = 0.56
adults polled.
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Poll on the Environment
• Description – 56% of the sample
are in favor of Congress passing
new legislation to regulate energy
output.
• Inference – between 53% and 59%
of the entire population of U.S.
adults would be in favor of
Congress passing new legislation
to regulate energy output.
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Stat 104 – Lecture 21
Margin of Error
• The approximate margin of
error is given by
1
× 100%
n
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68-95-99.7 Rule
• 95% of the time a sample
proportion, p̂ , will be within
2 standard deviations of the
population proportion, p .
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68-95-99.7 Rule
• 95% of the time the sample
proportion, p̂ , will be
between
p−2
p(1 − p)
p(1 − p)
and p + 2
n
n
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Stat 104 – Lecture 21
Standard Deviation
• Because p, the population
proportion is not known, the
standard deviation
SD( pˆ ) =
p(1 − p)
n
is also unknown.
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Standard Error
• Substitute p̂ as our estimate
(best guess) of p.
• The standard error of p̂ is:
SE ( pˆ ) =
pˆ (1 − pˆ )
n
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• About 95% of the time the
sample proportion, p̂ , will be
within
2SE( pˆ ) = 2
pˆ (1 − pˆ )
n
two standard errors of p.
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Stat 104 – Lecture 21
• About 95% of the time the
population proportion, p, will
be within
pˆ (1 − pˆ )
n
two standard errors of p̂ .
2SE ( pˆ ) = 2
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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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Example
pˆ = 0.56
0.56(0.44)
pˆ (1 − pˆ )
=
= 0.0156
1014
n
0.56 − 2(0.0156) to 0.56 + 2(0.0156)
0.529 to 0.591
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Stat 104 – Lecture 21
Interpretation
• We are 95% confident that
the population proportion of
all adults in the U.S. who
would favor new legislation
is between 52.9% and 59.1%.
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Interpretation
• Plausible values for the
population parameter p.
• 95% confidence in the
process that produced this
interval.
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95% Confidence
• If one were to repeatedly
sample at random 1014 adults
in the U.S. and compute a 95%
confidence interval for each
sample, 95% of the intervals
produced would contain the
population proportion p.
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Stat 104 – Lecture 21
Simulation
http://statweb.calpoly.edu/chance/
applets/Confsim/Confsim.html
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Margin of Error
• The approximate margin of
error is 2 standard errors with
95% confidence and pˆ = 0.5
2
0.5(1 − 0.5)
1
=
× 100%
n
n
21
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