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Best Point Estimation
Section 7.3
The best point estimate for a population
mean µ (σ known) is the sample mean x
Estimating a Population mean µ
(σ known)
Best point estimate : x
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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Notation
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Requirements
 = population mean
(1) The population standard deviation σ is known
 = population standard deviation
(2) One or both of the following:
The population is normally distributed
x = sample mean
or
n = number of sample values
n > 30
E = margin of error
z/2 = z-score separating an area of α/2 in the
right tail of the standard normal
distribution
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Margin of Error
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Confidence Interval
( x – E, x + E )
where
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Round-Off Rules for Confidence
Intervals Used to Estimate µ
Definition
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.
The two values x – E and x + E are
called confidence interval limits.
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
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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
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
Using StatCrunch
Stat → Z statistics → One Sample → with Summary
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Example
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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
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
Using StatCrunch
Enter Parameters
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Click Next
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Example
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
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
Using StatCrunch
Enter Confidence Level, then click ‘Calculate’
Select ‘Confidence Interval’
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Example
Sample Size for Estimating a
Population Mean
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
 = population mean
σ = population standard deviation
Using StatCrunch
Standard Error
x = sample mean
Lower Limit
E = desired margin of error
Upper Limit
zα/2 = z score separating an area of /2 in the right tail of
the standard normal distribution
n=
From the output, we find the Confidence interval is
CI = (35.862, 40.938)
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(z/2)  
E
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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:
If the computed sample size n is not
a whole number, round the value of n
up to the next larger whole number.
 /2 = 0.025
 = 0.05
n =
(using StatCrunch)
round up to 311
round up to 296
round up to 114
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E=3
1.96 • 15
 = 15
2 = 96.04 = 97
3
z / 2 = 1.96
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Example
Round-Off Rule for Determining
Sample Size
Examples:
n = 310.67
n = 295.23
n = 113.01
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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
Summary
Confidence Interval of a Mean µ
(σ known)
Sample Size for Estimating a Mean µ
(σ known)
σ = population standard deviation
E = desired margin of error
x = sample mean
σ = population standard deviation
n = number sample values
x = sample mean
1 – α = Confidence Level
𝑬 = 𝒛𝜶
𝟐
𝒔
𝒏
1 – α = Confidence Level
n=
(z/2)  
( x – E, x + E )
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2
E
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