Confidence Intervals CHAPTER SIX. Confidence Intervals for the MEAN (Large Samples) Section 6.1.
Estimation: Confidence Intervals Based in part on Chapter 6 General Business 704.
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Transcript of 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
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 MethodsEstimation MethodsEstimation MethodsEstimation Methods
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
95% Samples95% Samples
99% Samples99% 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
Confidence Interval Confidence Interval EstimatesEstimates
ProportionMean
Unknown
ConfidenceIntervals
Variance
FinitePopulation
Known
ProportionMean
Unknown
ConfidenceIntervals
Variance
FinitePopulation
Known
Confidence Interval Confidence Interval EstimatesEstimates
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
Thinking ChallengeThinking ChallengeThinking ChallengeThinking Challenge
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
Confidence Interval Confidence Interval EstimatesEstimates
Confidence Interval Confidence Interval EstimatesEstimates
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
Thinking ChallengeThinking ChallengeThinking ChallengeThinking Challenge
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
Confidence Interval Confidence Interval EstimatesEstimates
Confidence Interval Confidence Interval EstimatesEstimates
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
Confidence Interval Confidence Interval EstimatesEstimates
Confidence Interval Confidence Interval EstimatesEstimates
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
Thinking ChallengeThinking ChallengeThinking ChallengeThinking Challenge
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 MethodsEstimation MethodsEstimation MethodsEstimation Methods
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
Thinking ChallengeThinking ChallengeThinking ChallengeThinking Challenge
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