OPRE 6301-SYSM 6303 Chapter 10 Slides_students

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  • 10/21/2015

    1

    OPRE 6301/SYSM 6303Quantitative Introduction to Risk and

    Uncertainty in Business

    10-1

    Chapter TenIntroduction to Estimation

    10-2

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    Two Types of Inference

    Estimation

    Hypothesis Testing

    10-3

    Estimation

    Objective:

    to determine

    the approximate value

    of a population parameter

    on the basis of

    a sample statistic.

    10-4

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    Two Types of Estimators

    Point Estimator

    Interval Estimator

    10-6

    Point Estimator

    Draws inferences about a population by estimating the value of an unknown parameter

    using a single value or point

    10-7

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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.

    10-9

    Point Estimator

    However, regardless of the sample size, a point estimator is still just a point value.

    10-10

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    Interval Estimator

    Draws inferences about a population by estimating the value of an unknown parameter

    using an interval.

    10-11

    Interval Estimation

    With an interval, we say:

    With some ___% certainty, the population parameter of interest is between some lower

    bound and some upper bound.

    10-12

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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.

    10-15

    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.

    10-17

    Unbiased Estimator

    The sample mean is an unbiased estimator

    of the population mean

    10-18

    xE

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    Unbiased Estimator

    The sample median is an unbiased

    estimator of the population mean

    10-19

    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.

    10-20

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    Consistency

    The sample mean is a consistent estimator

    of the population mean

    10-21

    n

    xV2

    Consistency

    The sample median is a consistent estimator

    of the population mean

    10-22

    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.

    10-23

    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.

    10-25

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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.

    10-27

    Estimating when is known

    Recall:

    10-29

    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

    10-33

    122

    nZx

    nZxP

    Estimating when is known

    Substitute into the probability equation

    Lower Confidence Limit:

    Upper Confidence Limit:

    10-35

    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 .

    10-36

    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.

    10-37

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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.

    10-38

    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.

    10-39

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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 .

    10-40

    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

    10-41

    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.

    10-42