# statistik - chap08

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2002 Prentice-Hall, Inc. Chap 8-1

Chapter 8

Confidence Interval Estimation

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2002 Prentice-Hall, Inc. Chap 8-2

Chapter Topics

Estimation process

Point estimates

Interval estimates Confidence interval estimation for the mean

( known)

Determining sample size

Confidence interval estimation for the mean( unknown)

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2002 Prentice-Hall, Inc. Chap 8-3

Chapter Topics

Confidence interval estimation for theproportion

Confidence interval estimation for

population total Confidence interval estimation for total

difference in the population

Estimation and sample size determination

for finite population Confidence interval estimation and ethical

considerations

(continued)

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2002 Prentice-Hall, Inc. Chap 8-4

Estimation Process

Mean, , isunknown

Population Random Sample

MeanX = 50

Sample

I am 95%

confident that

is between 40& 60.

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2002 Prentice-Hall, Inc. Chap 8-5

Point Estimates

Estimate PopulationParameters

with SampleStatistics

Mean

Proportion

Variance

Difference

p

2

1 2

X

SP

2S

1 2X X

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2002 Prentice-Hall, Inc. Chap 8-6

Interval Estimates

Provides range of values

Takes into consideration variation in sample

statistics from sample to sample Is based on observation from one sample

unknown population parameters Is stated in terms of level of confidence

Never 100% certain

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Confidence Interval Estimates

Mean

Unknown

ConfidenceIntervals

Proportion

Known

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Confidence Interval for( Known)

Assumptions

Population standard deviation is known

Population is normally distributed If population is not normal, use large

sample

Confidence interval estimate

/ 2 / 2 X Z X Z n n

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Elements ofConfidence Interval Estimation

Level of confidence

Confidence in which the interval will contain

the unknown population parameter Precision (range)

Closeness to the unknown parameter

Cost Cost required to obtain a sample of size n

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Level of Confidence

Denoted by

A relative frequency interpretation

In the long run, of all the confidenceintervals that can be constructed will contain theunknown parameter

A specific interval will either contain or not

contain the parameter No probability involved in a specific interval

100 1 %

100 1 %

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Interval and Level of Confidence

Confidence Intervals

Intervals

extend from

to of intervalsconstructedcontain ;

donot.

_Sampling Distribution of the Mean

XX Z

X

/ 2/ 2

XX

1

XX Z

1 100%

100 %

/ 2 XZ

/ 2 XZ

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Factors AffectingInterval Width (Precision)

Data variation

Measured by

Sample size

Level of confidence

Intervals Extend from

1984-1994 T/Maker Co.

X- Z to X+ Zxx

Xn

100 1 %

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Determining Sample Size (Cost)

Too Big:

Requirestoo many

resources

Too small:

Wont dothe job

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Determining SampleSize for Mean

What sample size is needed to be 90% confident of

being correct within 5? A pilot study suggested that

the standard deviation is 45.

Round Up

2 22 22 2

1.645 45219.2 220

Error 5

Zn

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Determining SampleSize for Mean in PHStat

PHStat | sample size | determination for themean

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Assumptions

Population standard deviation is unknown

Population is normally distributed If population is not normal, use large sample

Use students t distribution

Confidence interval estimate

Confidence Interval for( Unknown)

/ 2, 1 / 2, 1n n

S S X t X t

n n

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2002 Prentice-Hall, Inc. Chap 8-17

Students tDistribution

Z

t0

t(df= 5)

t(df= 13)Bell-ShapedSymmetric

Fatter

Tails

Standard

Normal

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2002 Prentice-Hall, Inc. Chap 8-18

Degrees of Freedom (df)

Number of observations that are free tovary after sample mean has been calculated

Example: Mean of 3 numbers is 2

degrees of freedom= n -1

= 3 -1

= 2

1

2

3

1 (or any number)2 (or any number)

3 (cannot vary)

XX

X

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2002 Prentice-Hall, Inc. Chap 8-19

Students t Table

Upper Tail Area

df .25 .10 .05

1 1.000 3.078 6.314

20.817 1.886

2.920

3 0.765 1.638 2.353

t0 2.920tValues

Let: n = 3

df = n - 1 = 2

= .10

/2 =.05

/ 2 = .05

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2002 Prentice-Hall, Inc. Chap 8-20

Example

/ 2, 1 / 2, 1

8 850 2.0639 50 2.063925 25

46.69 53.30

n n

S S X t X t

n n

A random sample of 25 has 50 and 8.

Set up a 95% confidence interval estimate for

n X S

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2002 Prentice-Hall, Inc. Chap 8-21

PHStat | confidence interval | estimate for themean, sigma unknown

Confidence Interval for( Unknown) in PHStat

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2002 Prentice-Hall, Inc. Chap 8-22

Confidence IntervalEstimate for Proportion

Assumptions

Two categorical outcomes

Population follows binomial distribution Normal approximation can be used if

and

Confidence interval estimate

5np 1 5n p

/ 2 / 2

1 1S S S SS S

p p p p p Z p p Z

n n

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2002 Prentice-Hall, Inc. Chap 8-23

Example

A random sample of 400 voters showed 32

preferred candidate A. Set up a 95% confidence

interval estimate for p.

/ /

1 1

.08 1 .08 .08 1 .08.08 1.96 .08 1.96400 400

.053 .107

s s s s

s s

p p p p p Z p p Z

n n

p

p

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2002 Prentice-Hall, Inc. Chap 8-24

Confidence Interval Estimate forProportion in PHStat

PHStat | confidence interval | estimate for theproportion

D t i i S l Si f

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2002 Prentice-Hall, Inc. Chap 8-25

Determining Sample Size for

Proportion

Out of a population of 1,000, we randomly

selected 100, of which 30 were defective. What

sample size is needed to be within 5% with

90% confidence?

Round Up

2 2

2 2

1 1.645 0.3 0.7

Error 0.05227.3 228

Z p pn

f

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2002 Prentice-Hall, Inc. Chap 8-26

Determining Sample Size forProportion in PHStat

PHStat | sample size | determination for theproportion

C fid I t l f

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2002 Prentice-Hall, Inc. Chap 8-27

Confidence Interval for

Population Total Amount

Point estimate

Confidence interval estimate

NX

/ 2, 1

1n

N nS NX N t

Nn

C fid l f

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2002 Prentice-Hall, Inc. Chap 8-28

Confidence Interval forPopulation Total: Example

An auditor is faced with apopulation of 1000 vouchersand wants to estimate thetotal value of the population.

A sample of 50 vouchers isselected with averagevoucher amount of\$1076.39, standard deviation

of \$273.62. Set up the 95%confidence interval estimateof the total amount for thepopulation of vouchers.

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2002 Prentice-Hall, Inc. Chap 8-29

Example Solution

/ 2, 1

1000 50 \$1076.39 \$273.62

1

273.62 1000 501000 1076.39 1000 2.0096

1000 1100

1,076,390 75,830.85

n

N n X S

N nS NX N t

Nn

The 95% confidence interval for the population totalamount of the vouchers is between 1,000,559.15, and

1,152,220.85

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2002 Prentice-Hall, Inc. Chap 8-30

Example Solution in PHStat

PHStat | confidence intervals | estimate forthe population total

Excel spreadsheet for the voucher example

C fid I t l f T t l

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2002 Prentice-Hall, Inc. Chap 8-31

Confidence Interval for TotalDifference in the Population

Point estimate

Where is the sample average

difference Confidence interval estimate

Where

ND 1

n

i

i

D

Dn

/ 2, 1

1

Dn

N nS ND N t

Nn

2

1

1

n

i

i

D

D D

Sn

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2002 Prentice-Hall, Inc. Chap 8-32

Estimation for Finite Population

Samples are selected without replacement

Confidence interval for the mean ( unknown)

Confidence interval for proportion

/ 2, 11

nN nSX tNn

/ 2

1

1

S S

S

p p N np Z

n N

S l Si D t i ti

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2002 Prentice-Hall, Inc. Chap 8-33

Sample Size Determinationfor Finite Population

Samples are selected without replacement

When estimating the mean

When estimating the proportion

2 2

/ 20 2

Zne

2/ 20 2

1 Z p pne

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2002 Prentice-Hall, Inc. Chap 8-34

Ethical Considerations

Report confidence interval (reflect sampling

error) along with the point estimate

Report the level of confidence Report the sample size

Provide an interpretation

of the confidence interval estimate

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2002 Prentice-Hall, Inc. Chap 8-35

Chapter Summary

Illustrated estimation process

Discussed point estimates

Addressed interval estimates Discussed confidence interval estimation

for the mean ( known)

Addressed determining sample size Discussed confidence interval estimation

for the mean ( unknown)

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Chapter Summary

Discussed confidence interval estimation forthe proportion