6 - 1 2000 Prentice-Hall, Inc. Statistics for Business and Economics Sampling Distributions Chapter...

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Statistics for Business Statistics for Business and Economicsand Economics

Sampling DistributionsSampling DistributionsChapter 6Chapter 6

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Learning ObjectivesLearning Objectives

1.1. Describe the Properties of EstimatorsDescribe the Properties of Estimators

2.2. Explain Sampling DistributionExplain Sampling Distribution

3.3. Describe the Relationship between Describe the Relationship between Populations & Sampling DistributionsPopulations & Sampling Distributions

4.4. State the Central Limit TheoremState the Central Limit Theorem

5.5. Solve Probability Problems Involving Solve Probability Problems Involving Sampling DistributionsSampling Distributions

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Inferential StatisticsInferential Statistics

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Statistical MethodsStatistical Methods

StatisticalMethods

DescriptiveStatistics

InferentialStatistics

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Inferential StatisticsInferential Statistics

1.1. Involves:Involves: EstimationEstimation Hypothesis Hypothesis

TestingTesting

2.2. PurposePurpose Make Decisions Make Decisions

about Population about Population CharacteristicsCharacteristics

Population?Population?

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Inference ProcessInference Process

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Inference ProcessInference Process

PopulationPopulation

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Inference ProcessInference Process

PopulationPopulation

SampleSample

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Inference ProcessInference Process

PopulationPopulation

SampleSample

Sample Sample statistic statistic

((XX))

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Inference ProcessInference Process

PopulationPopulation

SampleSample

Sample Sample statistic statistic

((XX))

Estimates Estimates & tests& tests

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1.1. Random Variables Used to Estimate a Random Variables Used to Estimate a Population ParameterPopulation Parameter Sample Mean, Sample Proportion, Sample Sample Mean, Sample Proportion, Sample

MedianMedian

2.2. Example: Sample MeanExample: Sample MeanXX Is an Estimator of Is an Estimator of Population Mean Population Mean IfIfXX = 3 then = 3 then 33 Is the Is the EstimateEstimate of of

3.3. Theoretical Basis Is Sampling DistributionTheoretical Basis Is Sampling Distribution

EstimatorsEstimators

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Sampling DistributionsSampling Distributions

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1.1. TheoreticalTheoretical Probability Distribution Probability Distribution

2.2. Random Variable is Random Variable is Sample StatisticSample Statistic Sample Mean, Sample Proportion etc.Sample Mean, Sample Proportion etc.

3.3. Results from Drawing Results from Drawing AllAll Possible Possible Samples of a Samples of a FixedFixed Size Size

4.4. List of All Possible [List of All Possible [X, P(X, P(X) ] PairsX) ] Pairs Sampling Distribution of MeanSampling Distribution of Mean

Sampling Sampling DistributionDistribution

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DevelopingDevelopingSampling Sampling

DistributionsDistributionsSuppose There’s a Suppose There’s a Population ... Population ...

Population Size, Population Size, NN = 4 = 4

Random Variable, Random Variable, xx, , Is # Errors in WorkIs # Errors in Work

Values of Values of xx: 1, 2, 3, 4: 1, 2, 3, 4

Uniform DistributionUniform Distribution

© 1984-1994 T/Maker Co.

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Population Population CharacteristicsCharacteristics

.0

.1

.2

.3

1 2 3 4

Population DistributionPopulation DistributionSummary MeasuresSummary Measures

12.1

5.2

1

2

1

N

X

N

X

N

ii

N

ii

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All Possible Samples All Possible Samples

of Size of Size nn = 2 = 2

1st 2nd ObservationObs 1 2 3 41 1,1 1,2 1,3 1,4

2 2,1 2,2 2,3 2,4

3 3,1 3,2 3,3 3,4

4 4,1 4,2 4,3 4,4

16 Samples16 Samples

Sample with replacementSample with replacement

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All Possible Samples All Possible Samples

of Size of Size nn = 2 = 2

1st 2nd ObservationObs 1 2 3 41 1,1 1,2 1,3 1,4

2 2,1 2,2 2,3 2,4

3 3,1 3,2 3,3 3,4

4 4,1 4,2 4,3 4,4

1st 2nd ObservationObs 1 2 3 41 1.0 1.5 2.0 2.5

2 1.5 2.0 2.5 3.0

3 2.0 2.5 3.0 3.5

4 2.5 3.0 3.5 4.0

16 Samples16 Samples 16 Sample Means16 Sample Means

Sample with replacementSample with replacement

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Sampling Sampling DistributionDistribution

of All Sample of All Sample MeansMeans

.0

.1

.2

.3

1.0 1.5 2.0 2.5 3.0 3.5 4.0X

P(X)

1st 2nd ObservationObs 1 2 3 41 1.0 1.5 2.0 2.5

2 1.5 2.0 2.5 3.0

3 2.0 2.5 3.0 3.5

4 2.5 3.0 3.5 4.0

16 Sample Means16 Sample Means Sampling Sampling DistributionDistribution

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Summary Measures Summary Measures ofof

All Sample MeansAll Sample Means

x

ii

NX

N

1 10 15 4 0

162 5. . . .

79.016

5.20.45.25.15.20.1 222

1

2

N

XN

ixi

x

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ComparisonComparison

.0

.1

.2

.3

1 1.5 2 2.5 3 3.5 4

X

P(X)

.0

.1

.2

.3

1 2 3 4

P(X)

PopulationPopulation Sampling DistributionSampling Distribution

x 2 5.

x 0 79. 112.

2 5.

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Standard Error of Standard Error of MeanMean

1.1. Standard Deviation of All Possible Standard Deviation of All Possible Sample Means,Sample Means,XX Measures Scatter in All Sample Means,Measures Scatter in All Sample Means,XX

2.2. Less Than Pop. Standard DeviationLess Than Pop. Standard Deviation

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Standard Error of Standard Error of MeanMean

1.1. Standard Deviation of All Possible Standard Deviation of All Possible Sample Means,Sample Means,XX Measures Scatter in All Sample Means,Measures Scatter in All Sample Means,XX

2.2. Less Than Pop. Standard DeviationLess Than Pop. Standard Deviation

3.3. Formula (Sampling With Replacement)Formula (Sampling With Replacement)

x n

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Properties of Sampling Properties of Sampling Distribution of MeanDistribution of Mean

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Properties of Properties of Sampling Sampling

Distribution of MeanDistribution of Mean1.1. UnbiasednessUnbiasedness

Mean of Sampling Distribution Equals Population Mean of Sampling Distribution Equals Population MeanMean

2.2. EfficiencyEfficiency Sample Mean Comes Closer to Population Mean Sample Mean Comes Closer to Population Mean

Than Any Other Unbiased EstimatorThan Any Other Unbiased Estimator

3.3. ConsistencyConsistency As Sample Size Increases, Variation of Sample Mean As Sample Size Increases, Variation of Sample Mean

from Population Mean Decreasesfrom Population Mean Decreases

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UnbiasednessUnbiasedness

X

P(X)

CA

UnbiasedUnbiased BiasedBiased

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EfficiencyEfficiency

X

P(X)

A

B

Sampling Sampling distribution distribution of medianof median

Sampling Sampling distribution distribution

of meanof mean

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ConsistencyConsistency

X

P(X)

A

B

Smaller Smaller sample sample

sizesize

Larger Larger sample sample

sizesize

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Sampling from Sampling from Normal PopulationsNormal Populations

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= 50

= 10

X

Sampling from Sampling from Normal PopulationsNormal Populations

Central TendencyCentral Tendency

DispersionDispersion

Sampling Sampling withwith replacementreplacement

Population DistributionPopulation Distribution

Sampling DistributionSampling Distribution

x n

x

X = 50- X

n =16n =16XX = 2.5 = 2.5

n = 4n = 4XX = 5 = 5

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Standardizing Standardizing Sampling Distribution Sampling Distribution

of Meanof Mean

XX

X

Sampling Distribution

= 0

= 1

Z

Z X X

n

x

x

Standardized Normal Distribution

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Thinking ChallengeThinking Challenge

You’re an operations You’re an operations analyst for AT&T. Long-analyst for AT&T. Long-distance telephone calls distance telephone calls are normally distribution are normally distribution with with = 8 = 8 min. & min. & = 2 = 2 min. min. If you select random If you select random samples of samples of 25 25 calls, what calls, what percentage of the percentage of the samplesample means means would be between would be between 7.87.8 & & 8.2 8.2 minutes?minutes? © 1984-1994 T/Maker Co.

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Sampling Sampling Distribution Distribution

Solution*Solution*

8

X = .4

7.8 8.2 X

Sampling Distribution

Z Xn

Z Xn

7 8 82 25

50

8 2 82 25

50

. .

. .

0

= 1

-.50 Z.50

.3830.3830

.1915.1915.1915.1915

Standardized Normal Distribution

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Sampling from Sampling from Non-Normal Non-Normal PopulationsPopulations

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= 50

= 10

X

Sampling from Sampling from Non-Normal Non-Normal PopulationsPopulations

Central TendencyCentral Tendency

DispersionDispersion

Sampling Sampling withwith replacementreplacement

Population DistributionPopulation Distribution

Sampling DistributionSampling Distribution

x n

x

X = 50- X

n =30n =30XX = 1.8 = 1.8

n = 4n = 4XX = 5 = 5

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Central Limit TheoremCentral Limit Theorem

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Central Limit Central Limit TheoremTheorem

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X

Central Limit Central Limit TheoremTheorem

As As sample sample size gets size gets large large enough enough (n (n 30) ...30) ...

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X

Central Limit Central Limit TheoremTheorem

As As sample sample size gets size gets large large enough enough (n (n 30) ...30) ...

sampling sampling distribution distribution becomes becomes almost almost normal.normal.

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X

Central Limit Central Limit TheoremTheorem

As As sample sample size gets size gets large large enough enough (n (n 30) ...30) ...

sampling sampling distribution distribution becomes becomes almost almost normal.normal.

x n

x

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ConclusionConclusion

1.1. Described the Properties of EstimatorsDescribed the Properties of Estimators

2.2. Explained Sampling DistributionExplained Sampling Distribution

3.3. Described the Relationship between Described the Relationship between Populations & Sampling DistributionsPopulations & Sampling Distributions

4.4. Stated the Central Limit TheoremStated the Central Limit Theorem

5.5. Solved Probability Problems Involving Solved Probability Problems Involving Sampling DistributionsSampling Distributions

End of Chapter

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