chapter 09 - latest (1)

8/3/2019 Chapter 09 - Latest (1)

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The McGraw-Hill Companies, Inc. 2008McGraw-Hill/Irwin

Estimation and ConfidenceIntervals

Chapter 9


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GOALS

Define a point estimate. Define level of confidence. Construct a confidence interval for the population

mean when the population standard deviation isknown.

Construct a confidence interval for a populationmean when the population standard deviation isunknown.

Construct a confidence interval for a populationproportion. Determine the sample size for attribute and variable

sampling.


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Point and Interval Estimates

Apoint estimate is the statistic, computedfrom sample information, which is used to

estimate the population parameter. Aconfidence interval estimate is a range of

values constructed from sample data so thatthe population parameter is likely to occur

within that range at a specified probability.The specified probability is called the level ofconfidence.


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Factors Affecting Confidence IntervalEstimates

The factors that determine the widthof a confidence interval are:

1.The sample size, n.

2.The variability in the population, usually estimated by s.

3.The desired level of confidence.


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Interval Estimates - Interpretation

For a 95% confidence interval about 95% of the similarly constructedintervals will contain the parameter being estimated. Also 95% ofthe sample means for a specified sample size will lie within 1.96standard deviations of the hypothesized population


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Characteristics of the t-distribution

1. It is, like the zdistribution, a continuous distribution.

2. It is, like the zdistribution, bell-shaped andsymmetrical.

3. There is not one t distribution, but rathera family of tdistributions. All tdistributions have a mean of 0, buttheir standard deviations differ according to thesample size, n.

4. The t distribution is more spread out and flatter at thecenter than the standard normal distribution As thesample size increases, however, the tdistributionapproaches the standard normal distribution,


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Comparing the z and t Distributionswhen n is small


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Confidence Interval Estimates for theMean

Use Z-distribution

If the population

standard deviationis known or thesample is greaterthan 30.

Use t-distribution

If the population

standard deviationis unknown and thesample is less than30.


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When to Use the zortDistribution forConfidence Interval Computation


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Confidence Interval for the Mean Example using the t-distribution

A tire manufacturer wishes toinvestigate the tread life of itstires.Asample of 10 tiresdriven 50,000 miles revealed a

sample mean of 0.32 inch oftread remaining with a standarddeviation of 0.09 inch.Construct a 95 percentconfidence interval for thepopulation mean. Would it bereasonable for the

manufacturer to conclude thatafter 50,000 miles thepopulation mean amount oftread remaining is 0.30 inches?

n

stX

s

x

n

n 1,2/

unknown)is(sincedist.-t

theusingC.I.theCompute

09.0

32.0

10

:problemin theGiven

s

!

!

!

E

W


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Students t-distribution Table


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Confidence Interval Estimates for the Mean By Formula

$60.beounlikely tismeanpopulationthat theconclude

weHence,interval.confidencein thenotis$60ofvalueThe

$50.becouldmeanpopulationthat thereasonableisIt:Conclude

$53.57and$45.13areintervalconfidencetheofendpointsThe

22.435.4920

01.9093.235.49

20

01.935.49

unknown)is(sincedist.-ttheusing

C.I.theCompute

19,025.

120,2/05.

1,2/

s!

s!

s!

s!

s

t

n

stX

n

stX nE

W


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Confidence Interval for a PopulationProportion

The confidence interval for apopulation proportion is estimatedby:

n

pp

zp

)1(2/

s E


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Confidence Interval for a Population

Proportion- Example

The union representing the BottleBlowers ofAmerica (BBA) isconsidering a proposal to mergewith the Teamsters Union.According to BBA union bylaws,

at least three-fourths of theunion membership mustapprove any merger. A randomsample of2,000 current BBAmembers reveals 1,600 plan tovote for the merger proposal.What is the estimate of the

population proportion?Develop a 95 percent confidenceinterval for the populationproportion. Basing your decisionon this sample information, canyou conclude that the necessaryproportion of BBA membersfavor the merger? Why?

.membershipuniontheofpercent75than

greatervaluesincludesestimateintervalthebecause

passlikelywillproposalmergerThe:Conclude

)818.0,782.0(

018.80.2,000

)80.1(80.96.180.0

)1(C.I.

C.I.95%theCompute

80.02000

1,600

:proportionsamplethecomputeFirst,

2/

!

s!

s!

s!

!!!

n

ppzp

n

xp

E


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Selecting a Sample Size

There are 3 factors that determine thesize of a sample, none of which has

any direct relationship to the size ofthe population. They are:

The degree of confidence selected.

The maximum allowable error.

The variation in the population.


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Sample Size Determination for aVariable

To find the sample size for a variable:

sample)pilot(fromdeviationsamplethe-

confidenceoflevel

selectedthetoingcorrespondvalue-zthe-errorallowablethe-

:where

2

s

zE

Eszn

!


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A student in public administration wants todetermine the mean amount members ofcity councils in large cities earn permonth as remuneration for being acouncil member. The error in estimatingthe mean is to be less than $100 with a

95 percent level of confidence. Thestudent found a report by the Departmentof Labor that estimated the standarddeviation to be $1,000. What is therequired sample size?

Given in the problem: E, the maximum allowable error, is $100 The value ofzfor a 95 percent level of

confidence is 1.96, The estimate of the standard deviation is

$1,000.

385

16.384

)6.19(

100$

000,1$96.1

2

2

2

!

!

!

!

!

E

szn

Sample Size Determination for aVariable-Example


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A consumer group would like to estimate the mean monthly

electricity charge for a single family house in July within $5using a 99 percent level of confidence. Based on similarstudies the standard deviation is estimated to be $20.00. Howlarge a sample is required?

1075

)20)(58.2(2

!

!n

Sample Size Determination for aVariable- Another Example


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Sample Size forProportions

The formula for determining the samplesize in the case of a proportion is:

errorallowablemaximumthe-

levelconfidencedesiredfor thevalue-zthe-

usedis0.50otherwise,

source,someorstudypilotafromestimateis

:where

)1(

2

E

z

p

E

Zppn

!


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Another Example

TheAmerican Kennel Club wanted to estimate theproportion of children that have a dog as a pet. If theclub wanted the estimate to be within 3% of the

population proportion, how many children would theyneed to contact? Assume a 95% level of confidenceand that the club estimated that 30% of the childrenhave a dog as a pet.

89703.

96.1)70)(.30(.

2

!

!n


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Another Example

A study needs to estimate theproportion of cities that haveprivate refuse collectors. Theinvestigator wants the margin of

error to be within .10 of thepopulation proportion, thedesired level of confidence is90 percent, and no estimate isavailable for the populationproportion. What is the required

sample size?

cities69

0625.6810.

65.1)5.1)(5(.

2

!

!

!

n

n


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End of Chapter9

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