7 - 1 © 1998 prentice-hall, inc. chapter 7 inferences based on a single sample: estimation with...
TRANSCRIPT
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Chapter 7Chapter 7
Inferences Based on a Single Sample: Inferences Based on a Single Sample: Estimation with Confidence IntervalsEstimation with Confidence Intervals
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Learning ObjectivesLearning Objectives
1.1. State what is estimatedState what is estimated
2.2. Distinguish point & interval estimatesDistinguish point & interval estimates
3.3. Explain interval estimatesExplain interval estimates
4.4. Compute confidence interval estimates Compute confidence interval estimates for population mean & proportionfor population mean & proportion
5.5. Compute sample sizeCompute sample size
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Thinking ChallengeThinking Challenge
Suppose you’re Suppose you’re interested in estimating interested in estimating the average amount of the average amount of money that second-year money that second-year business students business students (population) have on (population) have on them. How would you them. How would you find out?find out?
AloneAlone GroupGroup Class Class
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Introduction Introduction to Estimationto Estimation
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Types of Types of Statistical Statistical
ApplicationsApplications
StatisticalMethods
DescriptiveStatistics
InferentialStatistics
EstimationHypothesis
Testing
StatisticalMethods
DescriptiveStatistics
InferentialStatistics
EstimationHypothesis
Testing
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Estimation ProcessEstimation Process
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Estimation ProcessEstimation Process
Mean, , is unknown
PopulationPopulation
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Estimation ProcessEstimation Process
Mean, , is unknown
PopulationPopulation Random SampleRandom Sample
Mean X= 50
Sample
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Estimation ProcessEstimation Process
Mean, , is unknown
PopulationPopulation Random SampleRandom SampleI am 95%
confident that is between
42 & 58.
Mean X= 50
Sample
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Unknown Population Unknown Population Parameters Are Parameters Are
Estimated Estimated
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Unknown Population Unknown Population Parameters Are Parameters Are
Estimated Estimated
Estimate populationEstimate populationparameter...parameter...
with samplewith samplestatisticstatistic
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Unknown Population Unknown Population Parameters Are Parameters Are
Estimated Estimated
Estimate populationEstimate populationparameter...parameter...
with samplewith samplestatisticstatistic
MeanMean xx
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Unknown Population Unknown Population Parameters Are Parameters Are
Estimated Estimated
Estimate populationEstimate populationparameter...parameter...
with samplewith samplestatisticstatistic
MeanMean xx
ProportionProportion pp pp̂̂
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Unknown Population Unknown Population Parameters Are Parameters Are
Estimated Estimated
Estimate populationEstimate populationparameter...parameter...
with samplewith samplestatisticstatistic
MeanMean xx
ProportionProportion pp pp̂̂
VarianceVariance 22 ss 22
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Unknown Population Unknown Population Parameters Are Parameters Are
Estimated Estimated
Estimate populationEstimate populationparameter...parameter...
with samplewith samplestatisticstatistic
MeanMean xx
ProportionProportion pp pp̂̂
VarianceVariance 22 ss 22
DifferencesDifferences 11 - - 22 xx11 --xx22
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Estimation MethodsEstimation Methods
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Estimation MethodsEstimation Methods
Estimation
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Estimation MethodsEstimation Methods
Estimation
PointEstimation
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Estimation MethodsEstimation Methods
Estimation
PointEstimation
IntervalEstimation
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Estimation MethodsEstimation Methods
Estimation
PointEstimation
IntervalEstimation
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Point EstimationPoint Estimation
1.1. Provides single valueProvides single value Based on observations from 1 sampleBased on observations from 1 sample
2.2. Gives no information about how close Gives no information about how close value is to the unknown population value is to the unknown population parameterparameter
3.3. Example: Sample meanExample: Sample meanx x = 3 is point = 3 is point estimate of unknown population meanestimate of unknown population mean
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Interval EstimationInterval Estimation
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Estimation MethodsEstimation Methods
Estimation
PointEstimation
IntervalEstimation
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Interval EstimationInterval Estimation
1.1. Provides range of values Provides range of values Based on observations from 1 sampleBased on observations from 1 sample
2.2. Gives information about closeness to Gives information about closeness to unknown population parameterunknown population parameter Stated in terms of probabilityStated in terms of probability
Knowing exact closeness requires knowing unknown Knowing exact closeness requires knowing unknown population parameterpopulation parameter
3.3. Example: unknown population mean lies Example: unknown population mean lies between 50 & 70 with 95% confidencebetween 50 & 70 with 95% confidence
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Key Elements of Key Elements of Interval EstimationInterval Estimation
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Key Elements of Key Elements of Interval EstimationInterval Estimation
Sample statistic Sample statistic
(point estimate)(point estimate)
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Key Elements of Key Elements of Interval EstimationInterval Estimation
Confidence Confidence intervalinterval
Sample statistic Sample statistic
(point estimate)(point estimate)
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Key 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)
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Confidence Confidence intervalinterval
Key Elements of Key Elements of Interval EstimationInterval Estimation
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.
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Confidence 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.
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Many Samples Have Many Samples Have Same IntervalSame Interval
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Many Samples Have Many Samples Have Same IntervalSame Interval
xx__
XX
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Many Samples Have Many Samples Have Same IntervalSame Interval
xx__
XX
XX= = ± Z ± Zxx
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Many Samples Have Many Samples Have Same IntervalSame Interval
90% Samples90% Samples
xx__
XX
XX= = ± Z ± Zxx
+1.65+1.65xx-1.65-1.65xx
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Many Samples Have Many Samples Have Same IntervalSame Interval
90% Samples90% Samples
95% Samples95% Samples
+1.65+1.65xx
xx__
XX
+1.96+1.96xx
-1.65-1.65xx-1.96-1.96xx
XX= = ± Z ± Zxx
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Many Samples Have Many Samples Have Same IntervalSame Interval
90% Samples90% Samples
95% Samples95% Samples
99% Samples99% Samples
+1.645+1.645x x +2.58+2.58xx
xx__
XX
+1.96+1.96xx
-2.58-2.58xx -1.645-1.645xx
-1.96-1.96xx
XX= = ± Z ± Zxx
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
1.1. Probability that the unknown population Probability that the unknown population parameter falls within intervalparameter falls within interval
2.2. Denoted (1 - Denoted (1 - is probability that parameter is is probability that parameter is notnot
within intervalwithin interval
3.3. Typical values are 99%, 95%, 90%Typical values are 99%, 95%, 90%
Confidence Level Confidence Level
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Intervals & Intervals & Confidence Level Confidence Level
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Intervals & Intervals & Confidence Level Confidence Level
x =
1 - /2/2
X_
x_
x =
1 - /2/2
X_
x_Sampling Sampling
Distribution Distribution of Meanof Mean
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Intervals & Intervals & Confidence Level Confidence Level
x =
1 - /2/2
X_
x_
x =
1 - /2/2
X_
x_Sampling Sampling
Distribution Distribution of Meanof Mean
Confidence Confidence IntervalInterval
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Intervals & Intervals & Confidence Level Confidence Level
x =
1 - /2/2
X_
x_
x =
1 - /2/2
X_
x_Sampling Sampling
Distribution Distribution of Meanof Mean
Confidence Confidence IntervalInterval
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Intervals & Intervals & Confidence Level Confidence Level
x =
1 - /2/2
X_
x_
x =
1 - /2/2
X_
x_Sampling Sampling
Distribution Distribution of Meanof Mean
Intervals Intervals extend from extend from X - ZX - ZXX to to
X + ZX + ZXX
Confidence Confidence IntervalInterval
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Intervals & Intervals & Confidence Level Confidence Level
x =
1 - /2/2
X_
x_
x =
1 - /2/2
X_
x_Sampling Sampling
Distribution Distribution of Meanof Mean
Intervals Intervals extend from extend from X - ZX - ZXX to to
X + ZX + ZXX
Confidence Confidence IntervalInterval
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Intervals & Intervals & Confidence Level Confidence Level
x =
1 - /2/2
X_
x_
x =
1 - /2/2
X_
x_Sampling Sampling
Distribution Distribution of Meanof Mean
Intervals Intervals extend from extend from X - ZX - ZXX to to
X + ZX + ZXX
Confidence Confidence IntervalInterval
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Intervals & Intervals & Confidence Level Confidence Level
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
Confidence Confidence IntervalInterval
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Intervals & Intervals & Confidence Level Confidence Level
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 - )100 % )100 % of intervals of intervals contain contain . .
% do % do not.not.
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Factors Affecting Factors Affecting Interval WidthInterval Width
1.1. Data dispersionData dispersion Measured by Measured by
2.2. Sample sizeSample size X X = = / / nn
3.3. Level of confidence Level of confidence (1 - (1 - )) Affects ZAffects Z
Intervals extend fromIntervals extend from
X - ZX - ZXX to toX + ZX + ZXX
© 1984-1994 T/Maker Co.
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Confidence Interval Confidence Interval EstimatesEstimates
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Confidence Interval Confidence Interval EstimatesEstimates
OnePopulation
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Confidence Interval Confidence Interval EstimatesEstimates
OnePopulation
Mean
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Confidence Interval Confidence Interval EstimatesEstimates
OnePopulation
Mean Proportion
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Confidence Interval Confidence Interval EstimatesEstimates
OnePopulation
ZDistribution
LargeLargeSampleSample
Mean Proportion
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Confidence Interval Confidence Interval EstimatesEstimates
OnePopulation
ZDistribution
tDistribution
LargeLargeSampleSample
Mean ProportionSmallSmall
SampleSample
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Confidence Interval Confidence Interval EstimatesEstimates
OnePopulation
ZDistribution
tDistribution
LargeSample
ZDistribution
Mean ProportionSmallSample
OnePopulation
ZDistribution
tDistribution
LargeSample
ZDistribution
Mean ProportionSmallSample
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Confidence Interval Confidence Interval Estimate Mean (Large Estimate Mean (Large
Sample)Sample)
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Confidence Interval Confidence Interval EstimatesEstimates
OnePopulation
ZDistribution
tDistribution
LargeLargeSampleSample
ZDistribution
Mean ProportionSmallSmall
SampleSample
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Confidence Interval Confidence Interval Mean (Large Sample)Mean (Large Sample)
1.1. AssumptionsAssumptions Sample size at least 30 (Sample size at least 30 (nn 30) 30) Random sample drawnRandom sample drawn If population standard deviation unknown, If population standard deviation unknown,
use sample standard deviationuse sample standard deviation
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Confidence Interval Confidence Interval Mean (Large Sample)Mean (Large Sample)
1.1. AssumptionsAssumptions Sample size at least 30 (Sample size at least 30 (nn 30) 30) Random sample drawnRandom sample drawn If population standard deviation unknown, If population standard deviation unknown,
use sample standard deviationuse sample standard deviation
2.2. Confidence interval estimateConfidence interval estimate
X Zn
X Zn
/ /2 2X Zn
X Zn
/ /2 2
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Estimation Example Estimation Example Mean (Large Mean (Large
Sample)Sample)The mean of a random sample of The mean of a random sample of nn = 36 = 36 isisX = 50. Set up a 95% confidence X = 50. Set up a 95% confidence interval estimate for interval estimate for if if = 12. = 12.
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Estimation Example Estimation Example Mean (Large Mean (Large
Sample)Sample)The mean of a random sample of The mean of a random sample of nn = 36 = 36 isisX = 50. Set up a 95% confidence X = 50. Set up a 95% confidence interval estimate for interval estimate for if if = 12. = 12.
X Zn
X Zn
/ /
. .
. .
2 2
50 1961236
50 1961236
46 08 53 92
X Zn
X Zn
/ /
. .
. .
2 2
50 1961236
50 1961236
46 08 53 92
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Thinking 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 = 1.99 X = 1.99 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.
2 liter
AloneAlone GroupGroup Class Class
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Confidence Interval Confidence Interval Solution*Solution*
X Zn
X Zn
/ /
. ..
. ..
. .
2 2
199 164505100
199 164505100
1982 1998
X Zn
X Zn
/ /
. ..
. ..
. .
2 2
199 164505100
199 164505100
1982 1998
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Confidence Interval Confidence Interval Estimate Mean (Small Estimate Mean (Small
Sample)Sample)
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Confidence Interval Confidence Interval EstimatesEstimates
OnePopulation
ZDistribution
tDistribution
LargeLargeSampleSample
ZDistribution
Mean ProportionSmallSmall
SampleSample
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Confidence Interval Confidence Interval Mean (Small Sample) Mean (Small Sample)
1.1. AssumptionsAssumptions Sample size less than 30 (Sample size less than 30 (nn < 30) < 30) Population normally distributedPopulation normally distributed Population standard deviation unknownPopulation standard deviation unknown
2.2. Use Student’s t distributionUse Student’s t distribution
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Confidence Interval Confidence Interval Mean (Small Sample) Mean (Small Sample)
1.1. AssumptionsAssumptions Sample size less than 30 (Sample size less than 30 (nn < 30) < 30) Population normally distributedPopulation normally distributed Population standard deviation unknownPopulation standard deviation unknown
2.2. Use Student’s t distributionUse Student’s t distribution
3.3. Confidence interval estimateConfidence interval estimate
X tSn
X tSnn n / , / ,2 1 2 1X t
Sn
X tSnn n / , / ,2 1 2 1
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Student’s t Student’s t DistributionDistribution
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ZZ
Student’s t Student’s t DistributionDistribution
00
Standard Standard NormalNormal
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
ZZtt
Student’s t Student’s t DistributionDistribution
00
Standard Standard NormalNormal
t (t (dfdf = 13) = 13)Bell-ShapedBell-Shaped
SymmetricSymmetric
‘‘Fatter’ TailsFatter’ Tails
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
ZZtt
Student’s t Student’s t DistributionDistribution
00
t (t (dfdf = 5) = 5)
Standard Standard NormalNormal
t (t (dfdf = 13) = 13)Bell-ShapedBell-Shaped
SymmetricSymmetric
‘‘Fatter’ TailsFatter’ Tails
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Student’s Student’s tt Table Table
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v t.10 t.05 t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
v t.10 t.05 t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
Student’s Student’s tt Table Table
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Student’s Student’s tt Table Table
t valuest values
v t.10 t.05 t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
v t.10 t.05 t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Student’s Student’s tt Table Table
t valuest valuest0 t0
/ 2/ 2
/ 2/ 2
v t.10 t.05 t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
v t.10 t.05 t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Student’s Student’s tt Table Table
t valuest valuest0 t0
/ 2/ 2
/ 2/ 2
Assume:Assume:nn = 3 = 3dfdf = = nn - 1 = 2 - 1 = 2 = .10= .10/2 =.05/2 =.05
v t.10 t.05 t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
v t.10 t.05 t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
v t.10 t.05 t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
v t.10 t.05 t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
Student’s Student’s tt Table Table
t valuest valuest0 t0
/ 2/ 2
/ 2/ 2
Assume:Assume:nn = 3 = 3dfdf = = nn - 1 = 2 - 1 = 2 = .10= .10/2 =.05/2 =.05
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
v t.10 t.05 t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
v t.10 t.05 t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
Student’s Student’s tt Table Table
t valuest valuest0 t0
/ 2/ 2Assume:Assume:nn = 3 = 3dfdf = = nn - 1 = 2 - 1 = 2 = .10= .10/2 =.05/2 =.05
.05.05
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
v t.10 t.05 t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
v t.10 t.05 t.025
1 3.078 6.314 12.706
2 1.886 2.920 4.303
3 1.638 2.353 3.182
t0 t0
Student’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
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Degrees of Freedom Degrees of Freedom ((dfdf))
1.1. Number of observations that are free to Number of observations that are free to vary after sample statistic has been vary after sample statistic has been calculatedcalculated
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Degrees of Freedom Degrees of Freedom ((dfdf))
1.1. Number of observations that are free to Number of observations that are free to vary after sample statistic has been vary after sample statistic has been calculatedcalculated
2.2. Example:Example:Sum of 3 numbers is 6Sum of 3 numbers is 6
XX1 1 = =
XX2 2 = =
XX3 3 ==
Sum = 6Sum = 6
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Degrees of Freedom Degrees of Freedom ((dfdf))
1.1. Number of observations that are free to Number of observations that are free to vary after sample statistic has been vary after sample statistic has been calculatedcalculated
2.2. Example:Example:Sum of 3 numbers is 6Sum of 3 numbers is 6
XX1 1 = 1 (Or any number)= 1 (Or any number)
XX2 2 = =
XX3 3 ==
Sum = 6Sum = 6
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Degrees of Freedom Degrees of Freedom ((dfdf))
1.1. Number of observations that are free to Number of observations that are free to vary after sample statistic has been vary after sample statistic has been calculatedcalculated
2.2. Example:Example: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 ==
Sum = 6Sum = 6
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Degrees of Freedom Degrees of Freedom ((dfdf))
1.1. Number of observations that are free to Number of observations that are free to vary after sample statistic has been vary after sample statistic has been calculatedcalculated
2.2. Example:Example: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 = = 33 (Cannot vary) (Cannot vary)
Sum = 6Sum = 6
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Degrees of Freedom Degrees of Freedom ((dfdf))
1.1. Number of observations that are free to Number of observations that are free to vary after sample statistic has been vary after sample statistic has been calculatedcalculated
2.2. Example:Example: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 = = 33 (Cannot vary) (Cannot vary)
Sum = 6Sum = 6
degrees of freedom degrees of freedom = = nn -1 -1 = 3 -1= 3 -1= 2= 2
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Estimation Example Estimation Example Mean (Small Mean (Small
Sample)Sample)A random sample of A random sample of nn = 25 has = 25 hasxx = 50 & = 50 & ss = 8. Set up a 95% confidence interval = 8. Set up a 95% confidence interval estimate for estimate for ..
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Estimation Example Estimation Example Mean (Small Mean (Small
Sample)Sample)A random sample of A random sample of nn = 25 has = 25 hasxx = 50 & = 50 & ss = 8. Set up a 95% confidence interval = 8. Set up a 95% confidence interval estimate for 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
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Thinking 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?
AloneAlone GroupGroup Class Class
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Confidence Interval Confidence Interval Solution*Solution*
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
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Confidence Interval Confidence Interval Estimate of ProportionEstimate of Proportion
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Data TypesData Types
Data
Quantitative Qualitative
Discrete Continuous
Data
Quantitative Qualitative
Discrete Continuous
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Qualitative DataQualitative Data
1.1. Qualitative random variables yield Qualitative random variables yield responses that classifyresponses that classify e.g., gender (male, female)e.g., gender (male, female)
2.2. Measurement reflects # in categoryMeasurement reflects # in category
3.3. Nominal or ordinal scaleNominal or ordinal scale
4.4. ExamplesExamples Do you own savings bonds? Do you own savings bonds? Do you live on-campus or off-campus?Do you live on-campus or off-campus?
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
ProportionsProportions
1.1. Involve qualitative variablesInvolve qualitative variables
2.2. Fraction or % of population in a categoryFraction or % of population in a category
3.3. If two qualitative outcomes, binomial If two qualitative outcomes, binomial distributiondistribution Possess or don’t possess characteristicPossess or don’t possess characteristic
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
ProportionsProportions
1.1. Involve qualitative variablesInvolve qualitative variables
2.2. Fraction or % of population in a categoryFraction or % of population in a category
3.3. If two qualitative outcomes, binomial If two qualitative outcomes, binomial distributiondistribution Possess or don’t possess characteristicPossess or don’t possess characteristic
4.4. Sample proportion (Sample proportion (pp))
pxn
number of successes
sample sizep
xn
number of successes
sample size
^̂
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Sampling Sampling Distribution Distribution of Proportionof Proportion
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Sampling Sampling Distribution Distribution of Proportionof Proportion
Sampling DistributionSampling Distribution
.0.0
.1.1
.2.2
.3.3
.0.0 .2.2 .4.4 .6.6 .8.8 1.01.0
PP^̂
P(PP(P^̂ ))
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
1.1. Approximated by Approximated by normal distributionnormal distribution
excludes 0 or nexcludes 0 or n
Sampling Sampling Distribution Distribution of Proportionof Proportion
Sampling DistributionSampling Distribution
.0.0
.1.1
.2.2
.3.3
.0.0 .2.2 .4.4 .6.6 .8.8 1.01.0
PP^̂
P(PP(P^̂ ))np np p 3 1b g np np p 3 1b g
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
1.1. Approximated by Approximated by normal distributionnormal distribution
excludes 0 or nexcludes 0 or n
2.2. MeanMean
Sampling Sampling Distribution Distribution of Proportionof Proportion
Pp Pp
Sampling DistributionSampling Distribution
.0.0
.1.1
.2.2
.3.3
.0.0 .2.2 .4.4 .6.6 .8.8 1.01.0
PP^̂
P(PP(P^̂ ))np np p 3 1b g np np p 3 1b g
Pp Pp
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
ppaa
1.1. Approximated by Approximated by normal distributionnormal distribution
excludes 0 or nexcludes 0 or n
2.2. MeanMean
3.3. Standard errorStandard error
Sampling Sampling Distribution Distribution of Proportionof Proportion
Pp Pp
Sampling DistributionSampling Distribution
pp̂̂pp
nn
11 ff
.0.0
.1.1
.2.2
.3.3
.0.0 .2.2 .4.4 .6.6 .8.8 1.01.0
PP^̂
P(PP(P^̂ ))np np p 3 1b g np np p 3 1b g
Pp Pp
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Confidence Interval Confidence Interval EstimatesEstimates
OnePopulation
ZDistribution
tDistribution
LargeLargeSampleSample
ZDistribution
Mean ProportionSmallSmall
SampleSample
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Confidence Interval Confidence Interval Proportion Proportion
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Confidence Interval Confidence Interval Proportion Proportion
1.1. AssumptionsAssumptions Two categorical outcomesTwo categorical outcomes Population follows binomial distributionPopulation follows binomial distribution Normal approximation can be usedNormal approximation can be used
does not include 0 or 1does not include 0 or 1np np p 3 1b g np np p 3 1b g
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Confidence Interval Confidence Interval Proportion Proportion
1.1. AssumptionsAssumptions Two categorical outcomesTwo categorical outcomes Population follows binomial distributionPopulation follows binomial distribution Normal approximation can be usedNormal approximation can be used
does not include 0 or 1does not include 0 or 1
2.2. Confidence interval estimateConfidence interval estimate
( )
( )p z
p pn
p p zp p
n
2 2
1 1 ( )
( )p z
p pn
p p zp p
n
2 2
1 1
np np p 3 1b g np np p 3 1b g
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Estimation Example Estimation Example ProportionProportion
A random sample of 400 graduates A random sample of 400 graduates showed 32 went to grad school. Set up a showed 32 went to grad school. Set up a 95% confidence interval estimate for 95% confidence interval estimate for pp..
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Estimation Example Estimation Example ProportionProportion
A random sample of 400 graduates A random sample of 400 graduates showed 32 went to grad school. Set up a showed 32 went to grad school. Set up a 95% confidence interval estimate for 95% confidence interval estimate for pp..
( )
( )
. .. ( . )
. .. ( . )
. .
/ /p Zp p
np p Z
p pn
p
p
2 21 1
08 19608 1 08
40008 196
08 1 08400
053 107
( )
( )
. .. ( . )
. .. ( . )
. .
/ /p Zp p
np p Z
p pn
p
p
2 21 1
08 19608 1 08
40008 196
08 1 08400
053 107
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Thinking 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?
AloneAlone GroupGroup Class Class
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Confidence Interval Confidence Interval Solution*Solution*
( )
( )
. .. (. )
. .. (. )
. .
/ /p zp p
np p z
p pn
p
p
2 21 1
175 1645175 825
200175 1645
175 825200
1308 2192
( )
( )
. .. (. )
. .. (. )
. .
/ /p zp p
np p z
p pn
p
p
2 21 1
175 1645175 825
200175 1645
175 825200
1308 2192
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Finding Sample SizesFinding Sample Sizes
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Finding Sample Finding Sample Sizes Sizes
for Estimating for Estimating
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Finding Sample Finding Sample Sizes Sizes
for Estimating for Estimating
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Finding Sample Finding Sample Sizes Sizes
for Estimating for Estimating
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Finding Sample Finding Sample Sizes Sizes
for Estimating for Estimating
(1)
(2)
ZX Error
Error Z Zn
nZ
Error
x x
x
( )3
2 2
2
(1)
(2)
ZX Error
Error Z Zn
nZ
Error
x x
x
( )3
2 2
2
Error is also called bound, Error is also called bound, BB
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Finding Sample Finding Sample Sizes Sizes
for Estimating for Estimating I don’t want to sample too much or too little!
(1)
(2)
ZX Error
Error Z Zn
nZ
Error
x x
x
( )3
2 2
2
(1)
(2)
ZX Error
Error Z Zn
nZ
Error
x x
x
( )3
2 2
2
Error is also called bound, Error is also called bound, BB
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Sample Size Sample Size ExampleExample
What sample size is needed to be What sample size is needed to be 90%90% confident of being correct within confident of being correct within 5 5? A ? A pilot study suggested that the standard pilot study suggested that the standard deviation is deviation is 4545..
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Sample Size Sample Size ExampleExample
What sample size is needed to be What sample size is needed to be 90%90% confident of being correct within confident of being correct within 5 5? A ? A pilot study suggested that the standard pilot study suggested that the standard deviation is deviation is 4545..
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
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Thinking 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 sample confident that the 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 sample sizesample size do you use? do you use?
AloneAlone GroupGroup Class Class
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
Sample Size Sample Size Solution*Solution*
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
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
ConclusionConclusion
1.1. Stated what is estimatedStated what is estimated
2.2. Distinguished point & interval estimatesDistinguished point & interval estimates
3.3. Explained interval estimatesExplained interval estimates
4.4. Computed confidence interval estimates Computed confidence interval estimates for population mean & proportionfor population mean & proportion
5.5. Computed sample sizeComputed sample size
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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.
This Class...This Class...
1.1. What was the most important thing you What was the most important thing you learned in class today?learned in class today?
2.2. What do you still have questions about?What do you still have questions about?
3.3. How can today’s class be improved?How can today’s class be improved?
Please take a moment to answer the following questions in writing: