estimation: confidence intervals based in part on chapter 6 general business 704

Estimation:Estimation:Confidence IntervalsConfidence Intervals

Estimation:Estimation:Confidence IntervalsConfidence Intervals

Based in part on Chapter 6Based in part on Chapter 6

General Business 704General Business 704General Business 704General Business 704

Objectives:Objectives:EstimationEstimationObjectives:Objectives:EstimationEstimation

Distinguish point & interval estimatesDistinguish point & interval estimates Explain interval estimatesExplain interval estimates Compute confidence interval estimates Compute confidence interval estimates

Population mean & proportionPopulation mean & proportion Population total & differencePopulation total & difference

Determine necessary sample sizeDetermine necessary sample size

Thinking ChallengeThinking ChallengeThinking ChallengeThinking Challenge

Suppose you’re Suppose you’re interested in the interested in the average amount of average amount of money that students money that students in this class (the in this class (the population) have in population) have in their possession. their possession. How would you find How would you find out?out?

Statistical MethodsStatistical MethodsStatistical MethodsStatistical Methods

StatisticalMethods

DescriptiveStatistics

InferentialStatistics

EstimationHypothesis

Testing

StatisticalMethods

DescriptiveStatistics

InferentialStatistics

EstimationHypothesis

Testing

Estimation ProcessEstimation ProcessEstimation ProcessEstimation Process

Mean, , is unknown

PopulationPopulation Random SampleRandom SampleI am 95%

confident that is between

40 & 60.

Mean X = 50

Sample

Population Parameter Population Parameter Estimates Estimates

Population Parameter Population Parameter Estimates Estimates

Estimate populationparameter...

with samplestatistic

Mean x

Proportion p ps

Variance 2 s2

Differences 1

2x1 -x2

Estimate populationparameter...

with samplestatistic

Mean x

Proportion p ps

Variance 2 s2

Differences 1

2x1 -x2

Estimation MethodsEstimation MethodsEstimation MethodsEstimation Methods

Estimation

PointEstimation

IntervalEstimation

ConfidenceInterval

Boot-strapping

Estimation

PointEstimation

IntervalEstimation

ConfidenceInterval

Boot-strapping

Point EstimationPoint EstimationPoint EstimationPoint Estimation

Provides single valueProvides single value Based on observations from 1 sampleBased on observations from 1 sample

Gives no information about how Gives no information about how close value is to the unknown close value is to the unknown population parameterpopulation parameter

Example: Sample meanExample: Sample meanX X = 3 is = 3 is point estimate of unknown point estimate of unknown population meanpopulation mean


Estimation

PointEstimation

IntervalEstimation

ConfidenceInterval

Boot-strapping

Estimation

PointEstimation

IntervalEstimation

ConfidenceInterval

Boot-strapping

Interval EstimationInterval EstimationInterval EstimationInterval Estimation

Provides range of values Provides range of values Based on observations from 1 sampleBased on observations from 1 sample

Gives information about closeness to Gives information about closeness to unknown population parameterunknown population parameter Stated in terms of probabilityStated in terms of probability

Example: Unknown population mean Example: Unknown population mean lies between 40 & 60 with 95% lies between 40 & 60 with 95% confidenceconfidence

Key Elements of Key Elements of Interval EstimationInterval EstimationKey Elements of Key Elements of

Interval EstimationInterval Estimation

Confidence Confidence intervalinterval

Sample statistic Sample statistic

(point estimate)(point estimate)

Confidence Confidence limit (lower)limit (lower)

Confidence Confidence limit (upper)limit (upper)

A A probabilityprobability that the population parameter that the population parameter falls somewhere within the interval.falls somewhere within the interval.

Confidence Limits Confidence Limits for Population Meanfor Population MeanConfidence Limits Confidence Limits

for Population Meanfor Population Mean

( )

( )

1

5

X Error

Error X X

ZX Error

Error Z

X Z

x x

x

x

(2) or

(3)

(4)

( )

( )

1

5

X Error

Error X X

ZX Error

Error Z

X Z

x x

x

x

(2) or

(3)

(4)

Parameter = Statistic ± Error

© 1984-1994 T/Maker Co.

Many Samples Have Many Samples Have Same IntervalSame Interval

Many Samples Have Many Samples Have Same IntervalSame Interval

90% Samples90% Samples



+1.65+1.65x x +2.58+2.58xx

xx__

XX

+1.96+1.96xx

-2.58-2.58xx -1.65-1.65xx

-1.96-1.96xx

XX= = ± Z ± Zxx

Probability that the unknown Probability that the unknown population parameter falls within population parameter falls within intervalinterval

Denoted (1 - Denoted (1 - is probability that parameter is is probability that parameter is notnot

within intervalwithin interval

Typical values are 99%, 95%, 90%Typical values are 99%, 95%, 90%

Level of ConfidenceLevel of ConfidenceLevel of ConfidenceLevel of Confidence

Intervals & Intervals & Level of ConfidenceLevel of Confidence

Intervals & Intervals & Level of ConfidenceLevel of Confidence

x =

1 - /2/2

X_

x_

x =

1 - /2/2

X_

x_Sampling Sampling

Distribution Distribution of Meanof Mean

Large number of intervalsLarge number of intervals

Intervals Intervals extend from extend from X - ZX - ZXX to to

X + ZX + ZXX

(1 - (1 - ) % of ) % of intervals intervals contain contain . .

% do not.% do not.

Factors Affecting Factors Affecting Interval WidthInterval Width

Factors Affecting Factors Affecting Interval WidthInterval Width

Data dispersionData dispersion Measured by Measured by

Sample sizeSample size X X = = / / nn

Level of confidence Level of confidence (1 - (1 - )) Affects ZAffects Z

Intervals extend from

X - ZX toX + ZX

© 1984-1994 T/Maker Co.

Confidence Interval Confidence Interval EstimatesEstimates


ProportionMean

Unknown

ConfidenceIntervals

Variance

FinitePopulation

Known

ProportionMean

Unknown

ConfidenceIntervals

Variance

FinitePopulation

Known

Confidence Interval Confidence Interval Mean (Mean ( Known) Known)

Confidence Interval Confidence Interval Mean (Mean ( Known) Known)

AssumptionsAssumptions Population standard deviation is knownPopulation standard deviation is known Population is normally distributedPopulation is normally distributed If not normal, can be approximated by If not normal, can be approximated by

normal distribution (normal distribution (nn 30) 30)

Confidence interval estimateConfidence interval estimate

X Zn

X Zn

/ /2 2X Zn

X Zn

/ /2 2

Note: 99% Z=2.58, 95% Z=1.96 , 90% Z=1.65 Note: 99% Z=2.58, 95% Z=1.96 , 90% Z=1.65

Estimation Example Estimation Example Mean (Mean ( Known) Known)

Estimation Example Estimation Example Mean (Mean ( Known) Known)

The mean of a random sample of The mean of a random sample of nn = 25 = 25 isisX = 50. Set up a 95% confidence X = 50. Set up a 95% confidence interval estimate for interval estimate for if if = 10. = 10.

X Zn

X Zn

/ /

. .

. .

2 2

50 1961025

50 1961025

46 08 53 92

X Zn

X Zn

/ /

. .

. .

2 2

50 1961025

50 1961025

46 08 53 92


You’re a Q/C inspector for You’re a Q/C inspector for Gallo. The Gallo. The for 2-liter for 2-liter bottles is bottles is .05.05 liters. A liters. A random sample of random sample of 100100 bottles showedbottles showedX =X = 1.991.99 liters. What is the liters. What is the 90%90% confidence interval confidence interval estimate of the true estimate of the true meanmean amount in 2-liter bottles? amount in 2-liter bottles?

2 liter

© 1984-1994 T/Maker Co.

Confidence Interval Confidence Interval Solution for GalloSolution for Gallo

Confidence Interval Confidence Interval Solution for GalloSolution for Gallo

X Zn

X Zn

/ /

. ..

. ..

. .

2 2

199 164505100

199 164505100

1982 1998

X Zn

X Zn

/ /

. ..

. ..

. .

2 2

199 164505100

199 164505100

1982 1998



ProportionMean

Unknown

ConfidenceIntervals

Variance

FinitePopulation

s Known

ProportionMean

Unknown

ConfidenceIntervals

Variance

FinitePopulation

s Known

Confidence Interval Confidence Interval Mean ( Mean ( Unknown) Unknown)Confidence Interval Confidence Interval Mean ( Mean ( Unknown) Unknown)

AssumptionsAssumptions Population standard deviation is Population standard deviation is

unknownunknown Population must be normally distributedPopulation must be normally distributed

Use Student’s t distributionUse Student’s t distribution Confidence interval estimateConfidence interval estimate

X tSn

X tSnn n / , / ,2 1 2 1X t

Sn

X tSnn n / , / ,2 1 2 1

Student’s t DistributionStudent’s t DistributionStudent’s t DistributionStudent’s t Distribution

Zt

Zt

00

t (t (dfdf = 5) = 5)

Standard Standard normalnormal

t (t (dfdf = 13) = 13)

Bell-Bell-shapedshaped

SymmetricSymmetric

‘‘Fatter’ tailsFatter’ tails

Note: As d.f. approach 120, Z and t become very similarNote: As d.f. approach 120, Z and t become very similar

Upper Tail Area

df .25 .10 .05

1 1.000 3.078 6.314

2 0.817 1.886 2.920

3 0.765 1.638 2.353

Upper Tail Area

df .25 .10 .05

1 1.000 3.078 6.314

2 0.817 1.886 2.920

3 0.765 1.638 2.353

t0 t0

Student’s Student’s tt Table TableStudent’s Student’s tt Table Table

Assume:Assume:nn = 3 = 3dfdf = = nn - 1 = 2 - 1 = 2 = .10= .10/2 =.05/2 =.05

2.9202.920t valuest values

/ 2/ 2

.05.05

Degrees of FreedomDegrees of FreedomDegrees of FreedomDegrees of Freedom

Number of observations that are free Number of observations that are free to vary after sample statistic has been to vary after sample statistic has been calculatedcalculated

ExampleExample Sum of 3 numbers is 6Sum of 3 numbers is 6

XX1 1 = 1 (or any number)= 1 (or any number)

XX2 2 = 2 (or any number)= 2 (or any number)

XX3 3 = 3 = 3 (cannot vary)(cannot vary)

Sum = 6Sum = 6

degrees of freedom = n -1 = 3 -1= 2

Estimation Example Estimation Example Mean (Mean ( Unknown) Unknown)

Estimation Example Estimation Example Mean (Mean ( Unknown) Unknown)

A random sample of A random sample of nn = 25 has = 25 hasX = 50 X = 50 & & SS = 8. Set up a 95% confidence = 8. Set up a 95% confidence interval estimate for interval estimate for ..

X tSn

X tSnn n

/ , / ,

. .

. .

2 1 2 1

50 2 0639825

50 2 0639825

46 69 53 30

X tSn

X tSnn n

/ , / ,

. .

. .

2 1 2 1

50 2 0639825

50 2 0639825

46 69 53 30


You’re a time study You’re a time study analyst in manufacturing. analyst in manufacturing. You’ve recorded the You’ve recorded the following task times (min.): following task times (min.): 3.6, 4.2, 4.0, 3.5, 3.8, 3.13.6, 4.2, 4.0, 3.5, 3.8, 3.1..

What is the What is the 90%90% confidence interval confidence interval estimate of the population estimate of the population meanmean task time? task time?

Confidence Interval Confidence Interval Solution for Time StudySolution for Time Study

Confidence Interval Confidence Interval Solution for Time StudySolution for Time Study

X = 3.7X = 3.7

SS = 3.8987 = 3.8987

nn = 6, df = = 6, df = nn - 1 = 6 - 1 = 5 - 1 = 6 - 1 = 5

SS / / nn = 3.8987 / = 3.8987 / 6 = 1.5926 = 1.592

tt.05,5.05,5 = 2.0150 = 2.0150

3.7 - (2.015)(1.592) 3.7 - (2.015)(1.592) 3.7 + (2.015)3.7 + (2.015)(1.592) (1.592)

0.492 0.492 6.908 6.908



ProportionMean

Unknown

ConfidenceIntervals

Variance

FinitePopulation

Known

ProportionMean

Unknown

ConfidenceIntervals

Variance

FinitePopulation

Known

Estimation for Estimation for Finite PopulationsFinite Populations

Estimation for Estimation for Finite PopulationsFinite Populations

AssumptionsAssumptions Sample is large relative to populationSample is large relative to population

n n / / N N > .05 > .05

Use finite population correction factorUse finite population correction factor

Confidence interval (mean, Confidence interval (mean, unknown)unknown)

X tSn

N nN

X tSn

N nNn n

/ , / ,2 1 2 11 1

X tSn

N nN

X tSn

N nNn n

/ , / ,2 1 2 11 1



ProportionMean

Unknown

ConfidenceIntervals

Variance

FinitePopulation

Known

ProportionMean

Unknown

ConfidenceIntervals

Variance

FinitePopulation

Known

Confidence Interval Confidence Interval Proportion Proportion

Confidence Interval Confidence Interval Proportion Proportion

AssumptionsAssumptions Two categorical outcomesTwo categorical outcomes Population follows binomial distributionPopulation follows binomial distribution Normal approximation can be usedNormal approximation can be used

n·n·pp 5 & 5 & nn·(1 - ·(1 - pp) ) 5 5

Confidence interval estimateConfidence interval estimate

p Zp p

np p Z

p pns

s ss

s s

( ) ( )1 1

p Zp p

np p Z

p pns

s ss

s s

( ) ( )1 1

Estimation Example Estimation Example ProportionProportion

Estimation Example Estimation Example ProportionProportion

A random sample of 400 graduates A random sample of 400 graduates showed 32 went to grad school. Set showed 32 went to grad school. Set up a 95% confidence interval estimate up a 95% confidence interval estimate for for pp..

p Zp p

np p Z

p pn

p

p

ss s

ss s

/ /( ) ( )

. .. ( . )

. .. ( . )

. .

2 21 1

08 19608 1 08

40008 196

08 1 08400

053 107

p Zp p

np p Z

p pn

p

p

ss s

ss s

/ /( ) ( )

. .. ( . )

. .. ( . )

. .

2 21 1

08 19608 1 08

40008 196

08 1 08400

053 107


You’re a production You’re a production manager for a newspaper. manager for a newspaper. You want to find the % You want to find the % defective. Of defective. Of 200200 newspapers, newspapers, 3535 had had defects. What is the defects. What is the 90%90% confidence interval confidence interval estimate of the population estimate of the population proportionproportion defective? defective?

Confidence Interval Confidence Interval Solution for DefectsSolution for DefectsConfidence Interval Confidence Interval Solution for DefectsSolution for Defects

p Zp p

np p Z

p pn

p

p

ss s

ss s

/ /( ) ( )

. .. (. )

. .. (. )

. .

2 21 1

175 1645175 825

200175 1645

175 825200

1308 2192

p Zp p

np p Z

p pn

p

p

ss s

ss s

/ /( ) ( )

. .. (. )

. .. (. )

. .

2 21 1

175 1645175 825

200175 1645

175 825200

1308 2192

nn··pp 5 5nn·(1 - ·(1 - pp) ) 5 5


Estimation

PointEstimation

IntervalEstimation

ConfidenceInterval

Boot-strapping

Estimation

PointEstimation

IntervalEstimation

ConfidenceInterval

Boot-strapping

Bootstrapping MethodBootstrapping MethodBootstrapping MethodBootstrapping Method

Used if population is not normalUsed if population is not normal Requires significant computer powerRequires significant computer power StepsSteps

Take initial sampleTake initial sample Sample repeatedly from initial sampleSample repeatedly from initial sample Compute sample statisticCompute sample statistic Form resampling distributionForm resampling distribution Limits are values that cut off smallest & Limits are values that cut off smallest &

largest largest /2 %/2 %

Finding Sample Sizes For Finding Sample Sizes For Estimating Estimating

Finding Sample Sizes For Finding Sample Sizes For Estimating Estimating

I don’t want to sample too much or too little!

2

22

)3(

(2)

(1)

Error

Zn

nZZError

ErrorXZ

x

xx

2

22

)3(

(2)

(1)

Error

Zn

nZZError

ErrorXZ

x

xx

Sample Size ExampleSample Size ExampleSample Size ExampleSample Size Example

What sample size is needed to be 90% What sample size is needed to be 90% confident of being correct within confident of being correct within 5? 5? A pilot study suggested that the A pilot study suggested that the standard deviation is 45.standard deviation is 45.

nZ

Error

2 2

2

2 2

2

1645 45

5219 2 220

..

a fa fafn

Z

Error

2 2

2

2 2

2

1645 45

5219 2 220

..

a fa faf


You work in Human You work in Human Resources at Merrill Lynch. Resources at Merrill Lynch. You plan to survey employees You plan to survey employees to find their average medical to find their average medical expenses. You want to be expenses. You want to be 95%95% confident that the confident that the sample sample meanmean is within is within ± $50± $50. . A pilot study showed that A pilot study showed that was about was about $400$400. What . What samplesample sizesize do you use? do you use?

Sample Size SolutionSample Size SolutionMedical ExpensesMedical Expenses

Sample Size SolutionSample Size SolutionMedical ExpensesMedical Expenses

nZ

Error

2 2

2

2 2

2

196 400

50

245 86 246

.

.

a fa faf

nZ

Error

2 2

2

2 2

2

196 400

50

245 86 246

.

.

a fa faf

Finding Sample Sizes For Finding Sample Sizes For Estimating ProportionsEstimating Proportions

Finding Sample Sizes For Finding Sample Sizes For Estimating ProportionsEstimating Proportions

I don’t want to sample too much or too little! 2

2 )1(

Error

ppZn

2

2 )1(

Error

ppZn

Remember•Error is acceptable error•Z is based on confidence level chosen•p is the true proportion of “success”

•Never under-estimate p•When in doubt, use p=.5

Remember•Error is acceptable error•Z is based on confidence level chosen•p is the true proportion of “success”

•Never under-estimate p•When in doubt, use p=.5

Sample Size Example Sample Size Example for Estimating pfor Estimating p

Sample Size Example Sample Size Example for Estimating pfor Estimating p

What sample size is needed to be 90% What sample size is needed to be 90% confident (Z=1.645) of being correct confident (Z=1.645) of being correct within proportion of .04 when using within proportion of .04 when using p=.5 (since no useful estimate of p is p=.5 (since no useful estimate of p is available)?available)?

42382.42204.

)5)(.5(.645.1)1(2

2

2

2

Error

ppZn

42382.42204.

)5)(.5(.645.1)1(2

2

2

2

Error

ppZn

Estimation of Population Estimation of Population TotalTotal

Estimation of Population Estimation of Population TotalTotal

In auditing, population total is more In auditing, population total is more important than meanimportant than mean Total = Total = NNXX

Confidence interval (population total)Confidence interval (population total)

Degrees of freedom = Degrees of freedom = nn - 1 - 1

NX tSn

N nN

Total NX tSn

N nN

1 1

NX tSn

N nN

Total NX tSn

N nN

1 1

Estimation Estimation of Differencesof DifferencesEstimation Estimation

of Differencesof Differences

Used to estimate the magnitude of Used to estimate the magnitude of errorserrors

StepsSteps Determine sample sizeDetermine sample size Compute average difference,Compute average difference,DD Compute standard deviation of Compute standard deviation of

differencesdifferences Set up confidence interval estimateSet up confidence interval estimate

Estimation of Differences Estimation of Differences EquationsEquations

Estimation of Differences Estimation of Differences EquationsEquations

D

D

ns

D nD

n

ND NtS

nN nN

ND NtS

nN nN

ii

n

D

ii

n

DD

D

1

2

1

1

1 1

D

D

ns

D nD

n

ND NtS

nN nN

ND NtS

nN nN

ii

n

D

ii

n

DD

D

1

2

1

1

1 1

Mean Difference:Mean Difference: Standard Deviation:Standard Deviation:

Interval Estimate:Interval Estimate:

Objectives:Objectives:EstimationEstimationObjectives:Objectives:EstimationEstimation

Distinguish point & interval estimatesDistinguish point & interval estimates Explain interval estimatesExplain interval estimates Compute confidence interval estimates Compute confidence interval estimates

Population mean & proportionPopulation mean & proportion Population total & differencePopulation total & difference

Determine necessary sample sizeDetermine necessary sample size

estimation: confidence intervals based in part on chapter 6 general business 704

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