OPRE 6301-SYSM 6303 Chapter 09 Slides_students
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8/17/2019 OPRE 6301-SYSM 6303 Chapter 09 Slides_students
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OPRE 6301/SYSM 6303Quantitative Introduction to Risk and
Uncertainty in Business
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Chapter NineSampling Distributions
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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?
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Sampling Distributionof the MeanLet’s investigate the throwing of a fair die
Let x = the # of spots on one throw
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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.
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x
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Sampling Distributionof the Mean
We can now create a distribution of .
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x
x We call this
the sampling distribution of .
Sampling Distributionof the MeanWe can calculate the parameters of the
distribution of .
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x
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Sampling Distributionof the Mean
Compare the distribution of x …
… with the distribution of .
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x
Sampling Distributionof the MeanNote also that
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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.
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n=5
n=10
n=25
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Sampling Distributionof the Mean
These relationships define thesampling distribution of .
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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.
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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.
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Sampling Distributionof the MeanIf a customer buys one bottle, what is the
probability that the bottlewill contain more than 32 ounces?
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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?
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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.
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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.
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P̂
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.
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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?
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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
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