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    OPRE 6301/SYSM 6303Quantitative Introduction to Risk and

    Uncertainty in Business

    9-1

    Chapter NineSampling Distributions

    9-2

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    Statistical Inference

    Converts data to information

    We can estimate population parameters by

    collecting sample data and calculating thecorresponding sample statistics.

    We expect our estimates to be close,

    but how close?

    9-5

    Sampling Distributionof the MeanLet’s investigate the throwing of a fair die

    Let x = the # of spots on one throw

    9-9

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    Sampling Distributionof the Mean

    Let’s now investigate the throwing of two fair dice

    For each die, we note the value of x.

    We will also calculate .

    This is equivalent to sampling from the samedistribution of x two time – i.e. n=2.

    9-12

     x 

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    Sampling Distributionof the Mean

    We can now create a distribution of .

    9-16

     x 

     x We call this

    the sampling distribution of .

    Sampling Distributionof the MeanWe can calculate the parameters of the

    distribution of .

    9-19

     x 

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    Sampling Distributionof the Mean

    Compare the distribution of x …

    … with the distribution of .

    9-21

     x 

    Sampling Distributionof the MeanNote also that

    9-22

          x 

    n x 

    2

    2       

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    Sampling Distributionof the Mean

    Let’s investigate how the sampling distribution

    of the mean changes as weincrease the sample size, n.

    9-23

    9-26

    n=5

    n=10

    n=25

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    Sampling Distributionof the Mean

    These relationships define thesampling distribution of .

    9-31

          x 

    n x 

    2

    2       

     x 

    nn x 

         

    2

    We refer to this as the

    “standard error”

    Central Limit Theorem

    The sampling distribution of the mean of arandom sample drawn from any population isapproximately normal for a sufficiently large

    sample size.

    The larger the sample size, the more closely thesampling distribution of x will resemble a normal

    distribution.

    9-33

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    Sampling Distributionof the Mean

    Let’s look at Example 9.1

    The foreman of a bottling plant has observed that

    the amount of soda in each 32-ounce bottle isactually a normally distributed random variable

    with a mean of 32.2 ounces and a standarddeviation of 0.3 ounces.

    9-34

    Sampling Distributionof the MeanIf a customer buys one bottle, what is the

    probability that the bottlewill contain more than 32 ounces?

    9-39

    2.32    3.0 

      7486.02514.0167.01

    67.0

    3.0

    2.323232

     

      

       

     Z P

     Z P

     X P X P

     

     

      ?32    X P

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    Sampling Distributionof the Mean

    If a customer buys a carton of four bottles, what isthe probability that the mean amount of the four

    bottles will be greater than 32 ounces?

    9-44

    2.32 x     15.04

    3.0

     x  

      9082.00918.0133.11

    33.1

    15.0

    2.323232

     

     

     

       

     Z P

     Z P

     X P X P

     x 

     x 

     

     

      ?32    X P

    Sampling Distributionof the Mean

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    Sampling Distributionof the Sample Proportion

    Let’s define the sample proportion of apopulation to be the number of successes in a

    sample of n.

    9-49

    n

     X P ˆ

    Sampling Distributionof the Sample Proportionis approximately normally distributed provided

    np and n(1-p) are greater than or equal to 0.5.

    9-54

      pPE    ˆ

      n

     p pPV  p

      1ˆ   2ˆ

     

     n p p p

        1ˆ

     

    We refer to this as the

    “standard error of theproportion”

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    Sampling Distributionof the Sample Proportion

    Let’s investigate Example 9.2

    In the last election, a state representative

    received 52% of the votes cast.

    One year after the election, the representativeorganized a survey that asked a random sample

    of 300 people whether they would vote for him inthe next election.

    9-56

    Sampling Distributionof the Sample ProportionIf we assume that his popularity has not changed,what is the probability that more than half of the

    sample would vote for him?

    9-61

    300n   52.0 p   ?50.0ˆ   PP

      7549.02451.0169.01

    69.0

    0288.

    52.050.0

    1

    ˆ50.0ˆ

     

     

     

       

     Z P

     Z P

    n p p

     pPPPP

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    Statistical Inference

    9-64