# 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

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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?

9-5

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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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

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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