Transcript Ch7-2

Section 7.2
Estimating a Population Proportion
Objective
Find the confidence interval for a population
proportion p
Determine the sample size needed to estimate
a population proportion p
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Definitions
The best point estimate for a population
proportion p is the sample proportion p
Best point estimate : p
The margin of error E is the maximum
likely difference between the observed
value and true value of the population
proportion p (with probability is 1–α)
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Margin of Error for Proportions
E  z 2
ˆˆ
pq
n
E = margin of error
p = sample proportion
q=1–p
n = number sample values
1 – α = Confidence Level
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Confidence Interval for a
Population Proportion p
( pˆ – E, pˆ + E )
where
E  z 2
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ˆˆ
pq
n
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Finding the Point Estimate and E
from a Confidence Interval
Point estimate of p:
p = (upper confidence limit) + (lower confidence limit)
2
Margin of Error:
E = (upper confidence limit) — (lower confidence limit)
2
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Round-Off Rule for
Confidence Interval Estimates of p
Round the confidence interval limits
for p to
three significant digits.
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Example 1
Find the 95%confidence interval for the population
proportion If a sample of size 100 has a proportion 0.67
Direct Computation
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Example 1
Find the 95%confidence interval for the population
proportion If a sample of size 100 has a proportion 0.67
Using StatCrunch
Stat → Proportions → One Sample → with Summary
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Example 1
Find the 95%confidence interval for the population
proportion If a sample of size 100 has a proportion 0.67
Using StatCrunch
Enter Values
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Example 1
Find the 95%confidence interval for the population
proportion If a sample of size 100 has a proportion 0.67
Using StatCrunch
Click ‘Next’
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Example 1
Find the 95%confidence interval for the population
proportion If a sample of size 100 has a proportion 0.67
Using StatCrunch
Select ‘Confidence Interval’
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Example 1
Find the 95%confidence interval for the population
proportion If a sample of size 100 has a proportion 0.67
Using StatCrunch
Enter Confidence Level, then click ‘Calculate’
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Example 1
Find the 95%confidence interval for the population
proportion If a sample of size 100 has a proportion 0.67
Using StatCrunch
Standard Error
Lower Limit
Upper Limit
From the output, we find the Confidence interval is
CI = (0.578, 0.762)
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Sample Size
Suppose we want to collect sample data in
order to estimate some population
proportion. The question is how many
sample items must be obtained?
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Determining Sample Size
z / 2
E=
p
ˆ qˆ
n
(solve for n by algebra)
n=
( Z / 2)2 p
ˆ ˆq
E2
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Sample Size for Estimating
Proportion p
ˆ
When an estimate of p is known:
n=
( z / 2 )2 pˆ qˆ
E2
When no estimate of p
ˆ is known:
use p
ˆ = qˆ = 0.5
2
(
)
n=
z / 2
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0.25
E2
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Round-Off Rule for Determining
Sample Size
If the computed sample size n is not
a whole number, round the value of n
up to the next larger whole number.
Examples:
n = 310.67
n = 295.23
n = 113.01
round up to 311
round up to 296
round up to 114
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Example 2
A manager for E-Bay wants to determine the current
percentage of U.S. adults who now use the Internet.
How many adults must be surveyed in order to be
95% confident that the sample percentage is in error
by no more than three percentage points when…
(a) In 2006, 73% of adults used the Internet.
(b) No known possible value of the proportion.
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Example 2
(a) Given:
Given a sample has proportion of 0.73,
To be 95% confident that our sample proportion
is within three percentage points of the true
proportion, we need at least 842 adults.
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Example 2
(b) Given:
For any sample,
To be 95% confident that our sample proportion
is within three percentage points of the true
proportion, we need at least 1068 adults.
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Summary
Confidence Interval of a Proportion
E = margin of error
p = sample proportion
n = number sample values
1 – α = Confidence Level
E  z 2
ˆˆ
pq
n
( p – E, p + E )
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Summary
Sample Size for Estimating a Proportion
When an estimate of p is known:
n=
( z / 2 )2 pˆ qˆ
E2
When no estimate of p is known (use p = q = 0.5)
n=
( z / 2)2 0.25
E2
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