OPRE 6301-SYSM 6303 Chapter 12 Slides_students_S16

8/17/2019 OPRE 6301-SYSM 6303 Chapter 12 Slides_students_S16

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OPRE 6301/SYSM 6303Quantitative Introduction to Risk and

Uncertainty in Business

12-1

Chapter TwelveInference about a Population

12-2


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Student t Distribution

11-3


In addition to mean, m , and standard deviation, s ,

the Student t distribution also requires degrees offreedom, n , be specified.

11-4


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11-5

Pt t P A

)( ,n


11-6


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11-7

Estimating m whens is unknown

When s is unknown, we must estimate the

standard deviation with its point estimator s.

11-8


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When s is unknown, we must estimate thestandard deviation with its point estimator s.

When we use the point estimator s, we must

use the Student t distribution so we do not

overestimate our confidence in the results.

11-9


Substitute into the probability equation

11-10

m n n

1

,2,2n

st x

n

st x P


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This equation provides us a confidence

interval estimate of m when the population

standard deviation is unknown.

11-11

m n n

1

,2,2n

st x

n

st x P



Lower Confidence Limit:

11-12

n

st x n ,2

m n n

1

,2,2n

st x

n

st x P


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Lower Confidence Limit:

Upper Confidence Limit:

11-13

m n n

1

,2,2n

st x

n

st x P

n

st x

n ,2

n

st x

n ,2



The probability (1- ) is the confidence level,

which is a measure of how frequently the

interval will include m .

11-14

m n n

1

,2,2n

st x

n

st x P


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Let’s investigate the interval estimate using thedata from Example 10.1. Doll Computers

11-15

Testing m when s is unknown

Test statistic

which is Student t-distributedwith n=n-1 degrees of freedom

12-16

ns

x t

m


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Testing m when s is unknown

Let’s investigate Example 12.1

12-17

Testing a population proportion

Sample proportion

Recall the sampling distribution

12-18

n

x p ˆ

n p p

pPz

/)1(

ˆ


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Test statistic

which is approximately normal whennp and n(1-p) are greater than 5.

12-19

n p p

pPz

/)1(

ˆ


Confidence Interval

12-20

n p pz p /)1(ˆ 2


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Let’s investigate Example 12.5

12-21


Sample size to estimate the proportion

12-22

n p pzB /)1(2


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Sample size to estimate the proportion

By rearranging the formula algebraically

12-23

n p pzB /)ˆ1(ˆ2

2

2 )ˆ1(ˆ

B

p pz

n


So, if we want our error to be no larger than 0.03

12-24

2

03.0

)ˆ1(ˆ96.1

p pn


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Problem: we need an estimate of the proportion

Method 1

It is the most conservative value we can choose

12-25

5.0ˆ p



Method 1

It is the most conservative value we can choose

12-26

5.0ˆ p

068,167.3203.0

)5.01(5.096.1 22

n


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Method 2

Set the proportion to the best estimate you have

12-27



Method 2

Set the proportion to the best estimate you have

12-28

68313.2603.0

)2.01(2.096.1 22

n

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OPRE 6301-SYSM 6303 Chapter 12 Slides_students_S16