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10/21/2015

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OPRE 6301/SYSM 6303Quantitative Introduction to Risk and

Uncertainty in Business

10-1

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 and .

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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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22/

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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