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OPRE 6301/SYSM 6303Quantitative Introduction to Risk and
Uncertainty in Business
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Chapter TenIntroduction to Estimation
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Two Types of Inference
Estimation
Hypothesis Testing
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Estimation
Objective:
to determine
the approximate value
of a population parameter
on the basis of
a sample statistic.
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Two Types of Estimators
Point Estimator
Interval Estimator
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Point Estimator
Draws inferences about a population by estimating the value of an unknown parameter
using a single value or point
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Point Estimator
Recall:
Point probabilities in continuous distributions were essentially equal to zero.
As the sample size used to determine the point estimator increases, we expect the point estimator to get closer to the population
parameter.
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Point Estimator
However, regardless of the sample size, a point estimator is still just a point value.
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Interval Estimator
Draws inferences about a population by estimating the value of an unknown parameter
using an interval.
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Interval Estimation
With an interval, we say:
With some ___% certainty, the population parameter of interest is between some lower
bound and some upper bound.
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Two Types of Estimators
Suppose we want to estimate the mean summer income of a class of business students.
For n=25 students,
or
The mean income is between
$380/week and $420/week.
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week/400$x
Qualities of Estimators
Unbiasedness
Consistency
Relative efficiency
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Unbiased Estimator
An estimator whose expected value is
equal to the parameter.
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Unbiased Estimator
The sample mean is an unbiased estimator
of the population mean
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xE
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Unbiased Estimator
The sample median is an unbiased
estimator of the population mean
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median sampleE
Consistency
An unbiased estimator is said to be
consistent if the difference between the
estimator and the parameter grows smaller
as the sample size grows larger.
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Consistency
The sample mean is a consistent estimator
of the population mean
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n
xV2
Consistency
The sample median is a consistent estimator
of the population mean
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n
V2
57.1median sample
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Relative Efficiency
If there are two unbiased estimators of a
parameter, the one whose variance is
smaller is said to be relatively efficient.
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Relative Efficiency
The sample mean and sample median
are both unbiased estimators of the
population mean.
However, the sample median has a greater
variance than the sample mean.
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Relative Efficiency
So we choose the sample mean since it is
relatively efficient when compared to the
sample median.
As it turns out, the sample mean is the
“best” estimator of the population mean.
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Estimating when is known
Recall:
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122 ZZZP
n
xZ
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Estimating when is known
Substitute into the probability equation
This equation provides us a confidence
interval estimate of
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122
nZx
nZxP
Estimating when is known
Substitute into the probability equation
Lower Confidence Limit:
Upper Confidence Limit:
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122
nZx
nZxP
nZx
2
nZx
2
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Estimating when is known
Substitute into the probability equation
The probability (1-) is the confidence level, which is a measure of how frequently the
interval will include .
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122
nZx
nZxP
Example: Doll Computers
The Doll Computer Company makes its own computers and delivers them directly to customers
who order them via Internet.
Demand during lead time is normally distributed. The Operations manager needs to know the mean to
compute the optimum inventory level.
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Example: Doll Computers
He observes 25 lead time periods. Use file Xm10-01
The manager would like a 95% confidence interval estimate of the mean demand during
lead time. Assume that the standard deviation is 75 computers.
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Example: Doll Computers
The estimation for the mean demand during lead time lies between 340.76 and 399.56.
This estimator is correct 95% of the time. That also means that 5% of the time the estimator will be
incorrect.
We often refer to the 95% figure as “19 times out of 20,” which emphasizes the long-run aspect of
the confidence level.
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Determining a sample size
In Chapter 5 we pointed out that sampling error is the difference between an estimator and a
parameter.
We can also define this difference as the error of estimation.
In this chapter this can be expressed as the difference between 𝑋 and µ.
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Determining sample size
With a little algebra we find the sample size to estimate a mean.
Where B: the bound on the error of estimation
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2
2/
B
zn
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Determining sample size
The manager has decided that the estimate of the mean demand during lead time should be
within 16 units.
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