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CHAPTER 9
ESTIMATION
Outline
• Estimation
– Point estimator
– Interval estimator
– Unbiased estimator
• Confidence interval estimator of population mean
when the population variance is known
• Selecting the sample size
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ESTIMATION
• Point estimator: A point estimator draws inferences about a
population by estimating the value of an unknown
parameter using a single value or point.
• Interval estimator: An interval estimator draws inferences
about a population by estimating the value of an unknown
parameter using an interval.
• Example: A manager of a plant making cellular telephones
wants to estimate the time to assemble the telephone. A
sample of 30 assemblies show a mean time of 400
seconds. The sample mean time of 400 seconds may be
considered a point estimate of the population mean.
Chapter 9 will provide a method for estimating an interval.
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ESTIMATION
• Unbiased estimator: an unbiased estimator of a population
parameter is an estimator whose expected value is equal to
that parameter.
• In Chapter 4, the sample variance is defined as follows:
N
s2 
2
(
x

x
)
 i
i 1
n 1
• The use of n-1 in the denominator is necessary to get an
unbiased estimator of variance. The use of n in the
denominator produces a smaller value of variance.
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CONFIDENCE INTERVAL ESTIMATOR OF
POPULATION MEAN WHEN THE
POPULATION VARIANCE IS KNOWN
• For some confidence level 1-, sample size n, sample
mean, x and the sample standard deviation,  the
confidence interval estimator of mean,  is as follows:


 

x  z / 2
also written as  x  z / 2
, x  z / 2

n
n
n

• Recall from Chapter 7 that z / 2 is that value of z for which
area on the right is /2
• Lower confidence limit (LCL)
• Upper confidence limit (UCL)
x  z / 2
x  z / 2

n

n
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Area=1-
x 
f(x)
CONFIDENCE
INTERVAL

n
Area
Area
=0.5-/2 =0.5-/2
Area=/2
-z/2
x  z / 2

n
Area=/2
z=0
x
z/2
x  z / 2

n5
Area=0.82
f(x)
AREAS FOR
THE 82%
CONFIDENCE
INTERVAL
x 
Area
=0.41
x  z / 2

n
n
Area
=0..41
Area=.09
-z/2

Area=.09
z=0
x
z/2
x  z / 2

n6
Area=0.82
f(x)
x 
Area
=0.41
AREAS AND z
AND x VALUES
FOR THE 82%
CONFIDENCE
INTERVAL
Area=.09
-1.34 z=0
x  1.34
n
n
Area
=0..41
Area=.09


x
1.34
x  1.34

n7
CONFIDENCE INTERVAL ESTIMATOR OF
POPULATION MEAN WHEN THE
POPULATION VARIANCE IS KNOWN
• Interpretation:
– There is (1-) probability that the sample mean will be
equal to a value such that the interval (LCL, UCL) will
include the population mean
– If the same procedure is used to obtain a confidence
interval estimate of the population mean for a sufficiently
large number of k times, the interval (LCL, UCL) is
expected to include the population mean (1-)k times See Table 9.2 on p. 310 for an example
• Wrong interpretation: There is (1-) probability that the
population mean lies between LCL and UCL. Population
mean is fixed, not uncertain/probabilistic.
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CONFIDENCE INTERVAL ESTIMATOR OF
POPULATION MEAN WHEN THE
POPULATION VARIANCE IS KNOWN
• Interpretation of the 95% confidence interval:
– There is 0.95 probability that the sample mean will be
equal to a value such that the interval (LCL, UCL) will
include the population mean
– If the same procedure is used to obtain a confidence
interval estimate of the population mean for a sufficiently
large number of k times, the interval (LCL, UCL) is
expected to include the population mean 0.95k times See Table 9.2 on p. 310 for an example
• Wrong interpretation: There is 0.95 probability that the
population mean lies between LCL and UCL. Population
mean is fixed, not uncertain/probabilistic.
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CONFIDENCE INTERVAL
Example 1 (Text 9.3): The following data represent a random
sample of 10 observations from a normal population whose
standard deviation is 2. Estimate the population mean with
90% confidence: 7,3,9,11,5,4,8,3,10,9
f(x)
x 

n
x
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SELECTING SAMPLE SIZE
• A narrow confidence interval is more desirable.
• For a given a confidence level, a narrow confidence interval
can be obtained by increasing the sample size.
• Bound on error of estimation: If the confidence interval has
the form of x  B then, B is the bound on the error of
estimation.
• For a given confidence level (1-), bound on the error of
estimation B and the population standard deviation  the
sample size necessary to estimate population mean,  is
 z / 2 
n

 B 
2
An approximation for  :  = Range/4
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SELECTING SAMPLE SIZE
Example 2 (Text 9.11): Determine the sample size that is
required to estimate a population mean to within 0.2%
units with 90% confidence when the standard deviation is
1.0.
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READING AND EXERCISES
• Reading: pp. 303-322
• Exercises: 9.2, 9.4, 9.6, 9.12, 9.14
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