Transcript Ch7a

Chapter 7
Confidence Intervals and
Sample Sizes
7.2 Estimating a Proportion p
7.3 Estimating a Mean µ (σ known)
7.4 Estimating a Mean µ (σ unknown)
7.5 Estimating a Standard Deviation σ
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Example 1
In a recent poll, 70% or 1501 randomly selected adults
said they believed in global warming.
Q: What is the proportion of the adult population
that believe in global warming?
TRICK QUESTION!
We only know the sample proportion s,
We do not know the population proportion σ.
BUT…
The proportion of the sample (0.7) is our
best point estimate (i.e. best guess).
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Definition
Point Estimate
A single value (or point) used to approximate
a population parameter
Best Point
Estimate
Population
Parameter
Proportion
p
≈
p
Mean
µ
≈
x
Std. Dev.
σ
≈
s
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Definition
Confidence Interval : CI
The range (or interval) of values to estimate
the true value of a population parameter.
It is abbreviated as CI
In Example 1, the 95% confidence interval for the
population proportion p is CI = (0.677, 0.723)
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Definition
Confidence Level : 1 – α
The probability that the confidence interval
actually contains the population parameter.
The most common confidence levels used
are 90%, 95%, 99%
90% : α=0.1
95% : α=0.05
99% : α=0.01
In Example 1, the Confidence level is 95%
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Definition
Margin of Error : E
The maximum likely difference between
the observed value and true value of the
population parameter (with probability is 1–α)
The margin of error is used to determine a
confidence interval (of a proportion or mean)
In Example 1, the 95% margin of error for the population
proportion p is E = 0.023
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Example 1 Continued…
In a recent poll, 70% or 1501 randomly selected
adults said they believed in global warming.
Q: What is the proportion of the adult
population that believe in global warming?
A: 0.7 is the best point estimate of the proportion
of all adults who believe in global warming.
The 95% confidence interval of the population
proportion p is:
CI = (0.677, 0.723)
( with a margin of error E = 0.023 )
What does it mean, exactly?
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Interpreting a Confidence Interval
For the 95% confidence interval CI = (0.677, 0.723)
we say:
We are 95% confident that the interval from
0.677 to 0.723 actually does contain the true
value of the population proportion p.
This means that if we were to select many different
samples of size 1501 and construct the
corresponding confidence intervals, then 95% of
them would actually contain the value of the
population proportion p.
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!!! Caution !!!
Know the correct interpretation of a
confidence interval
It is incorrect to say
“ the probability that the population
parameter belongs to the confidence
interval is 95% ”
because the population parameter is not
a random variable, its value cannot change
The population is “set in stone”
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!!! Caution !!!
Do not confuse the two percentages
The proportion can be represented
by percents (like 70% in Example 1)
The confidence level may be represented
by percents (like 95% in Example 1)
Proportions can be any value from 0% to 100%
Confidence levels are usually 90%, 95%, or 99%
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Confidence Interval Formula
( y – E, y + E )
y = Best point estimate
E = Margin of Error
• Centered at the best point estimate
• Width is determined by E
The value of E depends the critical value of the CI
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Finding the Point Estimate and E
from a Confidence Interval
Point estimate : y
y = (upper confidence limit) + (lower confidence limit)
2
Margin of Error : E
E = (upper confidence limit) — (lower confidence limit)
2
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Definition
Critical Value
The number on the borderline separating
sample statistics that are likely to occur from
those that are unlikely to occur.
A critical value is dependent on a probability
distribution the parameter follows and the
confidence level (1 – α) .
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Normal Dist. Critical Values
For a population proportion p and mean µ
(σ known), the critical values are found using
z-scores on a standard normal distribution
The standard normal distribution is divided into
three regions: middle part has area 1 – α and
two tails (left and right) have area α/2 each:
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Normal Dist. Critical Values
The z-scores za/2 and –za/2 separate the values:
Likely values
( middle interval )
Unlikely values
( tails )
Use StatCrunch to calculate z-scores (see Ch. 6)
–za/2
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za/2
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Normal Dist. Critical Values
The value za/2 separates an area of a/2 in
the right tail of the z-dist.
The value –za/2 separates an area of a/2 in
the left tail of the z-dist.
The subscript a/2 is simply a reminder that the zscore separates an area of a/2 in the tail.
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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  za 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  za 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
za / 2
E=
p
ˆ qˆ
n
(solve for n by algebra)
n=
( Za / 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=
( za / 2 )2 pˆ qˆ
E2
When no estimate of p
ˆ is known:
use p
ˆ = qˆ = 0.5
2
(
)
n=
za / 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  za 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=
( za / 2 )2 pˆ qˆ
E2
When no estimate of p is known (use p = q = 0.5)
n=
( za / 2)2 0.25
E2
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Section 7.3
Estimating a Population mean µ
(σ known)
Objective
Find the confidence interval for a population
mean µ when σ is known
Determine the sample size needed to estimate
a population mean µ when σ is known
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Best Point Estimation
The best point estimate for a population
mean µ (σ known) is the sample mean x
Best point estimate : x
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Notation
 = population mean
 = population standard deviation
x = sample mean
n = number of sample values
E = margin of error
za/2 = z-score separating an area of α/2 in the
right tail of the standard normal
distribution
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Requirements
(1) The population standard deviation σ is known
(2) One or both of the following:
The population is normally distributed
or
n > 30
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Margin of Error
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Confidence Interval
( x – E, x + E )
where
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Definition
The two values x – E and x + E are
called confidence interval limits.
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Round-Off Rules for Confidence
Intervals Used to Estimate µ
1. When using the original set of data, round the
confidence interval limits to one more decimal
place than used in original set of data.
2. When the original set of data is unknown and
only the summary statistics (n, x, s) are used,
round the confidence interval limits to the same
number of decimal places used for the
sample mean.
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Example
Find the 90% confidence interval for the population mean If the
population standard deviation is known to be 10 and a sample of
size 42 has a mean of 38.4
Direct Computation
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Example
Find the 90% confidence interval for the population mean If the
population standard deviation is known to be 10 and a sample of
size 42 has a mean of 38.4
Using StatCrunch
Stat → Z statistics → One Sample → with Summary
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Example
Find the 90% confidence interval for the population mean If the
population standard deviation is known to be 10 and a sample of
size 42 has a mean of 38.4
Using StatCrunch
Enter Parameters
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Example
Find the 90% confidence interval for the population mean If the
population standard deviation is known to be 10 and a sample of
size 42 has a mean of 38.4
Using StatCrunch
Click Next
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Example
Find the 90% confidence interval for the population mean If the
population standard deviation is known to be 10 and a sample of
size 42 has a mean of 38.4
Using StatCrunch
Select ‘Confidence Interval’
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Example
Find the 90% confidence interval for the population mean If the
population standard deviation is known to be 10 and a sample of
size 42 has a mean of 38.4
Using StatCrunch
Enter Confidence Level, then click ‘Calculate’
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Example
Find the 90% confidence interval for the population mean If the
population standard deviation is known to be 10 and a sample of
size 42 has a mean of 38.4
Using StatCrunch
Standard Error
Lower Limit
Upper Limit
From the output, we find the Confidence interval is
CI = (35.862, 40.938)
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Sample Size for Estimating a
Population Mean
 = population mean
σ = population standard deviation
x = sample mean
E = desired margin of error
zα/2 = z score separating an area of a/2 in the right tail of
the standard normal distribution
n=
(za/2)  
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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
We want to estimate the mean IQ score for the population of
statistics students. How many statistics students must be
randomly selected for IQ tests if we want 95% confidence that
the sample mean is within 3 IQ points of the population mean?
What we know:
a /2 = 0.025
a = 0.05
n =
1.96 • 15
 = 15
2 = 96.04 = 97
3
z a/ 2 = 1.96
(using StatCrunch)
E=3
With a simple random sample of only 97
statistics students, we will be 95%
confident that the sample mean is within
3 IQ points of the true population mean .
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Summary
Confidence Interval of a Mean µ
(σ known)
σ = population standard deviation
x = sample mean
n = number sample values
1 – α = Confidence Level
( x – E, x + E )
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Summary
Sample Size for Estimating a Mean µ
(σ known)
E = desired margin of error
σ = population standard deviation
x = sample mean
1 – α = Confidence Level
n=
(za/2)  
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