Sampling Methods and Sampling Distributions Learning Objectives l Explain Types of Samples l...

Sampling Methods and

Sampling Distributions

Learning Objectives Explain Types of Samples

Describe the Properties of Estimators

Explain Sampling Distribution

Describe the Relationship betweenPopulations & Sampling Distributions

State the Central Limit Theorem

Solve Probability Problems Involving Sampling Distributions

Sampling

Methods

Types of SamplesType ofSample

ProbabilityNon

Probability

SimpleRandom

StratifiedSystematic Cluster

ChunkJudge-ment

Quota

Simple Random Sample

1.Each Population Element Has an Equal Chance of Being Selected

2.Selecting 1 Subject Does Not Affect Selecting Others

3.May Use Random Number Table, Lottery, ‘Fish Bowl’

Random Number Table

Column

00000 00001 11111 11111Row 12345 67890 12345 67890

01 49280 88924 35779 00283

02 61870 41657 07468 08612

03 43898 65923 25078 86129

Types of Samples

Type ofSample

ProbabilityNon

Probability

SimpleRandom

StratifiedSystematic Cluster

ChunkJudge-ment

Quota

Systematic Sample

1. Items of population arranged in some way- alphabetically, by date received

2.Every kth Element Is Selected After a Random Start within the First k Elements

3.Used in Telephone Surveys

© 1984-1994 T/Maker Co.

Types of SamplesType ofSample

ProbabilityNon

Probability

SimpleRandom

StratifiedSystematic Cluster

ChunkJudge-ment

Quota

Stratified Sample

1. Divide Population into Subgroups Mutually Exclusive

Collectively Exhaustive

At Least 1 Common Characteristic of Interest

All Students

Commuters Residents

Sample2. Select Simple Random Samples from Subgroups

Types of SamplesType ofSample

ProbabilityNon

Probability

SimpleRandom

StratifiedSystematic Cluster

ChunkJudge-ment

Quota

Cluster Sample

Divide Population into Clusters If Managers

are Elements, then Companies are Clusters

Select Clusters Randomly

Survey All or a Random Sample of Elements in Cluster

Companies (Clusters)

Sample

Types of Samples

Type ofSample

ProbabilityNon

Probability

SimpleRandom

StratifiedSystematic Cluster

ChunkJudge-ment

Quota

Nonprobability Samples

1.JudgmentUse Experience to Select Sample

e.g., Test Markets

2.QuotaSimilar to Stratified Sampling Except

No Random Sampling

3.Chunk (Convenience)Use Elements Most Available

Errors Due to Sampling

Sampling Error - occurs because sample is taken instead of census Errors are due to chance Equally likely to be too high or too low Improve by increasing sample size

Nonsampling Error - Bias A directional error Can not be reduced by increasing sample

size

Sampling Distributions

Statistical Methods

StatisticalMethods

DescriptiveStatistics

InferentialStatistics

Inferential Statistics

InvolvesEstimationHypothesis

Testing

PurposeMake Decisions

about Population Characteristics

Population?

Inference Process

Population

Sample

Sample Statistic (X, P )

Estimates & Tests

1. Random Variables Used to Estimate a Population Parameter

-Sample Mean, Sample Proportion, Sample Median

2. Sample Mean is an Estimator of Population Mean

If = 3 then 3 Is the Estimate of

3. Theoretical Basis Is Sampling Distribution

Estimators

X

Properties of Mean Unbiasedness

Mean of Sampling Distribution Equals Population Mean

Efficiency Sample Mean Comes Closer to Population Mean Than

Any Other Unbiased Estimator

Consistency As Sample Size Increases, Variation of Sample Mean

from Population Mean Decreases

Unbiasedness

x= x

Unbiased Biased

X

P(X)

CA

x C

Efficiency

x

Sampling Distribution of Median

Sampling Distribution

of Mean

X

P(X)

A

B

Consistency

x

Smaller Sample

Size

Larger Sample

Size

X

P(X)

A

B

Sampling Distribution

Theoretical Probability Distribution Random Variable is Sample Statistic

Sample Mean, Sample Proportion, etc.

Results from Drawing All Possible Samples of a Fixed Size

List of All Possible [X, P(X) ] PairsSampling Distribution of Mean

DevelopingSampling Distributions

Suppose There’s a Population ...

Population Size, N = 4

Random Variable, X, Is # Errors in Work

Values of X: 1, 2, 3, 4

Uniform Distribution

1 2.5 -1.5 2.252 2.5 -0.5 0.253 2.5 0.5 0.254 2.5 1.5 2.25

10 5

X (X -) (X - )2

10

42 5.

5

4112.

Population Mean and Standard Deviation

(# of errors)

Population Characteristics

Population DistributionSummary Measures

x

ii

N

x

i xi

N

X

N

X

N

1

2

1

2 5

112

.

.( )

.0

.1

.2

.3

1 2 3 4

Inference Process

Population

Sample

Sample Statistic (X, Ps )

Estimates & Tests

All Possible Samples of Size n = 2

16 Samples 16 Sample Means

Sample With Replacement

1st 2nd ObservationObs 1 2 3 4

1 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 4

1 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

Sampling Distribution of All Sample Means

1st 2nd ObservationObs 1 2 3 4

1 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 Means Sampling Distribution

X f p(X)

1.0 1 1/16

1.5 2 2/16

2.0 3 3/16

2.5 4 4/16

3.0 3 3/16

3.5 2 2/16

4.0 1 1/16

Sampling Distribution of All Sample Means

1st 2nd ObservationObs 1 2 3 4

1 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 Means Sampling Distribution

.0

.1

.2

.3

1.0 1.5 2.0 2.5 3.0 3.5 4.0X

P(X)

1.0 2.5 -1.5 2.251.5 2.5 -1.0 1.001.5 2.5 -1.0 1.002.0 2.5 -0.5 0.252.0 2.5 -0.5 0.252.0 2.5 -0.5 0.252.5 2.5 0.0 0.002.5 2.5 0.0 0.002.5 2.5 0.0 0.002.5 2.5 0.0 0.003.0 2.5 0.5 0.253.0 2.5 0.5 0.253.0 2.5 0.5 0.253.5 2.5 1.0 1.003.5 2.5 1.0 1.004.0 2.5 1.5 2.25

40 10.00

X x x)2(X-x)(X-

x N 10 15 4 0

162.5. . .i

NX

i14016

Summary Measures of All Possible Sample Means

x

ii

N

X

N

1

10 15 4 0

162 5

. . ..

x

i xi

N

X

N

( )2

1

79. ( ) ( ) ( )2 2 2

10 2 5 15 2 5 4 0 2 5

16

. . . . . .

1016

Comparison of Population & Sampling Distribution

Population Sampling Distribution

.0

.1

.2

.3

1 2 3 4

P(X)

.0

.1

.2

.3

1 1.5 2 2.5 3 3.5 4

X

P(X)

x 2 5.

x .79

x 2 5.

x 112.

Standard Error of Mean(Standard Deviation of the Sampling Distribution of Means)

Standard Deviation of All Possible Sample Means,X Measures Scatter in All Sample Means,X

Less Than Population Standard Deviation Formula

(Sampling With Replacement)

x

i xi

N

X

N

( )2

1

n

x = x

x n

Sampling is done with replacementor

Population is infiniteor

n/N < .05

Sampling Distribution of the Sample Means Summary

Sampling from Normal Populations

Central Tendency

Dispersion

Sampling With Replacement

Population Distribution

Sampling Distributionn =16n =4

x x

xx

n

X = 50

X= 10

X

X = 50- X

x= 2.5x = 5

Z = 0

z= 1

Z

Standardizing Sampling Distribution of Mean

Sampling Distribution

Standardized Normal Distribution

ZX x

x

n

X x

x

X

X

X

Thinking Challenge

You’re an operations analyst for AT&T. Long-distance telephone calls are normally distribution with x = 8 min. & x = 2 min. If you select random samples of 25 calls, what percentage of the sample means would be between 7.8 & 8.2 minutes? © 1984-1994 T/Maker Co.

Sampling Distribution Solution*

ZX

nx

x

8 2 8

2 2550

..

nx 2 25Z

X x

7 8 850

..

Sampling Distribution

8

X = .4

7.8 8.2 X

Standardized Normal Distribution

0

Z = 1

Z

.1915.1915

-.50 .50

.3830

Sampling from Normal Populations

Central Tendency

Dispersion

Sampling With Replacement

Population Distribution

Sampling Distributionn =16

x= 2.5n =4

x = 5

x x

xx

n

X = 50

X= 10

X

X = 50- X

Sampling from Non-Normal Populations

Central Tendency

Dispersion

Sampling With Replacement

Population Distribution

Sampling Distributionn =30x =1.8

n = 4x= 5

x x

xx

n

X = 50

X= 10

X

X = 50- X

Central Limit TheoremFor a population with a mean u and a standard deviation, the sampling distribution of the means of all possible samples of size n generated from the population will be approximately normally distributed assuming that the sample size is sufficiently large.

X

Central Limit Theorem

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

sampling distribution becomes almost normal.

Central Limit Theorem

The sampling distribution of means is a normal distribution if population is normally distributed

Even if population is not normally distributed, the sampling distribution of means is approximated by a normal distribution for large n (n>30)

X

Central Limit Theorem

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

sampling distribution becomes almost normal.

Proportions

Categorical Variable (e.g., Gender) % Population Having a Characteristic If Two Outcomes, Binomial Distribution

Possess - Don’t Possess Characteristic

Sample Proportion Formula:

PX

n

number of successes

sample size

Approximated by Normal Distribution n·p 5 n·(1 - p) 5

Mean

Standard Error

Sampling Distribution of Proportion

Sampling Distribution

.0

.1

.2

.3

.0 .2 .4 .6 .8 1.0

P

P(Ps)

P

p

nwhere p = Population

Proportion

P

p p

1( )

Z = 0

z= 1

Z

Standardizing Sampling Distribution of Proportion

Sampling Distribution

Standardized Normal Distribution

P

ZP P p

p p

n

P

P

( )1

P P

Thinking Challenge

You’re manager of a bank. 40% of depositors have multiple accounts. You select a random sample of 200 customers. What is the probability that the sample proportion of depositors with multiple accounts would be between 40% & 43% ?

© 1984-1994 T/Maker Co.

.87

Solution* P(.40 P .43)

Sampling Distribution

.3078

ZZ= 0

Z = 1

Standardized Normal Distribution

n·p 5n·(1 - p) 5

PP = .40

P = .0346

.43

ZP p

p p

n

( )

. .

. ( . ).

1

43 40

40 1 40

200

87

Sampling from Finite Populations

Modify Standard Error if Sample Size (n) Is Large Relative to Population Size (N) n > .05·N (or n/N > .05)

Use Finite Population Correction (fpc) Factorfor Standard Errors if n/N > .05

xx

n

P

p p

n

1( )N n

N

1

N n

N

1)

( )(

x = x

x n

xx

n

N n

N

1

Sampling is done with replacementor

Population is infiniteor

n/N < .05

Sampling is without replaacementand

Population is finiteand

n/N > .05

Sampling Distribution of the Sample Means Summary

Thinking Challenge

You’re manager of a bank. 40% of all 1000 depositors have multiple accounts. You select a random sample of 200 customers. What is the probability that the sample proportion of depositors with multiple accounts would be between 40% & 43% ?

© 1984-1994 T/Maker Co.

ZZ= 0

Z = 1

.97

Solution* P(.40 P .43)

Sampling Distribution

.3340

Standardized Distribution

ZP p

p p

n

N n

N

( )

. .

. ( . ).

1

1

43 40

40 1 40

200

1000 200

1000 1

97

PP = .40

P = .0310

.43

Selecting a Sample Size

Selecting a Sample Size

The Degree of Cofidence Selected

The Maximum Allowable Error The Population Standard

Deviation

Sample Size for Means

E is the allowable error z is the z score associated with degree of confidence is the population standard deviation

nz

E

z

E

2 2

2

2 n

z

E

z

E

2 2

2

2

The marketing manager would like to estimate the population mean annual usage of home heating oil to within 50 gallons of the true value and desires to be 95% confident of correctly estimating the true mean. Based on a previous study taken last year,the marketing manager feels that the standard deviation can be estimated as 325 gallons. What is the sample size need to obtain these results?

n

z

E

2 2

2

2

2

2196 325 384 105 625

2500162 31

. ( )

(50)

( . )( , ).

nhomes need to be sampled

Confidence = 95%E = 50 = 325

z = 1.96

Sample Size for Proportions

E is the maximum allowable errorz is the z value associated with the degree of confidencep is the estimated proportion

n

p p z

E

1 2

2

A political pollister would like to estimate the proportion of voters who will vote for the Democratic candidate in a presidential campaign. The pollster would like 95% confidence that her prediction is correct to within .04 of the true proportion. What sample size is needed?

Confidence = 95%E = .04p = unknown use p = .5

np P z

E

( ) . ( . )( . )

(. ).

1 5 1 5 196

04600 25

2

2

2

2

n = 601 voters

Conclusion Examined Sampling Methods

Described the Properties of Estimators

Explained Sampling Distribution

Described the Relationship between Populations & Sampling Distributions

Stated the Central Limit Theorem

Solved Probability Problems Involving Sampling Distributions

End of Chapter

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