1 1 slide © 2003 thomson/south-western slides prepared by john s. loucks st. edward’s university

1 1 Slide

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© 2003 Thomson/South-Western© 2003 Thomson/South-Western

Slides Prepared bySlides Prepared byJOHN S. LOUCKSJOHN S. LOUCKS

St. Edward’s UniversitySt. Edward’s University

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Chapter 7Chapter 7Sampling and Sampling DistributionsSampling and Sampling Distributions

Simple Random SamplingSimple Random Sampling Point EstimationPoint Estimation Introduction to Sampling DistributionsIntroduction to Sampling Distributions Sampling Distribution of Sampling Distribution of Sampling Distribution ofSampling Distribution of Sampling MethodsSampling Methods

xxpp nn = 100 = 100

nn = 30 = 30

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Statistical InferenceStatistical Inference

The purpose of The purpose of statistical inferencestatistical inference is to obtain is to obtain information about a population from information about a population from information contained in a sample.information contained in a sample.

A A populationpopulation is the set of all the elements of is the set of all the elements of interest.interest.

A A samplesample is a subset of the population. is a subset of the population. The sample results provide only The sample results provide only estimatesestimates of of

the values of the population characteristics.the values of the population characteristics. A A parameterparameter is a numerical characteristic of a is a numerical characteristic of a

population.population. With With proper sampling methodsproper sampling methods, the sample , the sample

results will provide “good” estimates of the results will provide “good” estimates of the population characteristics.population characteristics.

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Simple Random SamplingSimple Random Sampling

Finite PopulationFinite Population

• A A simple random sample from a finite simple random sample from a finite population population of size of size NN is a sample selected is a sample selected such that each possible sample of size such that each possible sample of size nn has has the same probability of being selected.the same probability of being selected.

• Replacing each sampled element before Replacing each sampled element before selecting subsequent elements is called selecting subsequent elements is called sampling with replacementsampling with replacement..

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Finite PopulationFinite Population

• Sampling without replacementSampling without replacement is the is the procedure used most often.procedure used most often.

• In large sampling projects, computer-In large sampling projects, computer-generated generated random numbersrandom numbers are often used are often used to automate the sample selection process.to automate the sample selection process.

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Infinite PopulationInfinite Population

• A simple random sample from an infinite A simple random sample from an infinite population is a sample selected such that population is a sample selected such that the following conditions are satisfied.the following conditions are satisfied.• Each element selected comes from the Each element selected comes from the

same population.same population.• Each element is selected independently.Each element is selected independently.


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Infinite PopulationInfinite Population

• The population is usually considered infinite The population is usually considered infinite if it involves an ongoing process that makes if it involves an ongoing process that makes listing or counting every element listing or counting every element impossible.impossible.

• The random number selection procedure The random number selection procedure cannot be used for infinite populations.cannot be used for infinite populations.


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Point EstimationPoint Estimation

In In point estimationpoint estimation we use the data from the we use the data from the sample to compute a value of a sample sample to compute a value of a sample statistic that serves as an estimate of a statistic that serves as an estimate of a population parameter.population parameter.

We refer to as the We refer to as the point estimatorpoint estimator of the of the population mean population mean ..

ss is the is the point estimatorpoint estimator of the population of the population standard deviation standard deviation ..

is the is the point estimatorpoint estimator of the population of the population proportion proportion pp..

xx

pp

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Point EstimationPoint Estimation

When the expected value of a point estimator When the expected value of a point estimator is equal to the population parameter, the point is equal to the population parameter, the point estimator is said to be estimator is said to be unbiasedunbiased..

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Sampling ErrorSampling Error

The absolute difference between an unbiased The absolute difference between an unbiased point estimate and the corresponding point estimate and the corresponding population parameter is called the population parameter is called the sampling sampling errorerror..

Sampling error is the result of using a subset of Sampling error is the result of using a subset of the population (the sample), and not the entire the population (the sample), and not the entire population to develop estimates.population to develop estimates.

The sampling errors are:The sampling errors are:

for sample meanfor sample mean

for sample standard for sample standard deviationdeviation

for sample proportionfor sample proportion

| |x | |x

| |s | |s

| |p p| |p p

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Example: St. Andrew’sExample: St. Andrew’s

St. Andrew’s College receives 900 applicationsSt. Andrew’s College receives 900 applications

annually from prospective students. The annually from prospective students. The applicationapplication

forms contain a variety of information including forms contain a variety of information including thethe

individual’s scholastic aptitude test (SAT) score individual’s scholastic aptitude test (SAT) score andand

whether or not the individual desires on-campus whether or not the individual desires on-campus

housing.housing.

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The director of admissions would like to know The director of admissions would like to know thethe

following information:following information:

• the average SAT score for the applicants, the average SAT score for the applicants, andand

• the proportion of applicants that want to the proportion of applicants that want to live on campus.live on campus.

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We will now look at three alternatives for We will now look at three alternatives for obtainingobtaining

the desired information.the desired information.

• Conducting a census of the entire 900 Conducting a census of the entire 900 applicantsapplicants

• Selecting a sample of 30 applicants, using a Selecting a sample of 30 applicants, using a random number tablerandom number table

• Selecting a sample of 30 applicants, using Selecting a sample of 30 applicants, using computer-generated random numberscomputer-generated random numbers

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Taking a Census of the 900 ApplicantsTaking a Census of the 900 Applicants

• SAT ScoresSAT Scores• Population MeanPopulation Mean

• Population Standard DeviationPopulation Standard Deviation

ix 990900

ix 990900

ix

2( )80

900

ix

2( )80

900


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Taking a Census of the 900 ApplicantsTaking a Census of the 900 Applicants

• Applicants Wanting On-Campus HousingApplicants Wanting On-Campus Housing• Population ProportionPopulation Proportion

p648

.72900

p648

.72900

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Take a Sample of 30 ApplicantsTake a Sample of 30 Applicants

Using a Random Number TableUsing a Random Number Table

Since the finite population has 900 Since the finite population has 900 elements, we will need 3-digit random elements, we will need 3-digit random numbers to randomly select applicants numbers to randomly select applicants numbered from 1 to 900.numbered from 1 to 900.

We will use the We will use the lastlast three digits of the 5- three digits of the 5-digit random numbers in the digit random numbers in the thirdthird column of column of the textbook’s random number table. the textbook’s random number table.

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Using a Random Number TableUsing a Random Number Table

The numbers we draw will be the numbers The numbers we draw will be the numbers of the applicants we will sample unlessof the applicants we will sample unless

• the random number is greater than 900 orthe random number is greater than 900 or• the random number has already been the random number has already been

used.used.

We will continue to draw random numbers until We will continue to draw random numbers until wewe

have selected 30 applicants for our sample.have selected 30 applicants for our sample.

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Use of Random Numbers for SamplingUse of Random Numbers for Sampling

3-Digit3-Digit ApplicantApplicant

Random NumberRandom Number Included in Included in SampleSample

744744 No. 744 No. 744 436436 No. 436 No. 436 865865 No. 865 No. 865 790790 No. 790 No. 790 835835 No. 835 No. 835 902902 Number exceeds Number exceeds

900900 190190 No. 190 No. 190 436436 Number already Number already

usedused etc.etc. etc. etc.

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Sample DataSample Data

RandomRandom

No.No. NumberNumber ApplicantApplicant SAT ScoreSAT Score On-CampusOn-Campus 11 744 744 Connie Reyman Connie Reyman 1025 1025

YesYes 22 436 436 William Fox William Fox 950 950 Yes Yes 33 865 865 Fabian Avante Fabian Avante 1090 1090 No No 44 790 790 Eric Paxton Eric Paxton 1120 1120

YesYes 55 835 835 Winona Wheeler Winona Wheeler 1015 1015

No No .. . . . . . . . . 3030 685 685 Kevin Cossack Kevin Cossack 965 965 No No


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Using Computer-Generated Random NumbersUsing Computer-Generated Random Numbers

• Excel provides a function for generating Excel provides a function for generating random numbers in its worksheet.random numbers in its worksheet.

• 900 random numbers are generated, one 900 random numbers are generated, one for each applicant in the population.for each applicant in the population.

• Then we choose the 30 applicants Then we choose the 30 applicants corresponding to the 30 smallest random corresponding to the 30 smallest random numbers as our sample.numbers as our sample.

• Each of the 900 applicants have the same Each of the 900 applicants have the same probability of being included.probability of being included.

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Using Excel to SelectUsing Excel to Selecta Simple Random Samplea Simple Random Sample

Formula WorksheetFormula WorksheetA B C D

1Applicant Number

SAT Score

On-Campus Housing

Random Number

2 1 1008 Yes =RAND()3 2 1025 No =RAND()4 3 952 Yes =RAND()5 4 1090 Yes =RAND()6 5 1127 Yes =RAND()7 6 1015 No =RAND()8 7 965 Yes =RAND()9 8 1161 No =RAND()

Note: Rows 10-901 are not shown.Note: Rows 10-901 are not shown.

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Value WorksheetValue WorksheetA B C D

1Applicant Number

SAT Score

On-Campus Housing

Random Number

2 1 1008 Yes 0.413273 2 1025 No 0.795144 3 952 Yes 0.662375 4 1090 Yes 0.002346 5 1127 Yes 0.712057 6 1015 No 0.180378 7 965 Yes 0.716079 8 1161 No 0.90512


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Value Worksheet (Sorted)Value Worksheet (Sorted)A B C D

1Applicant Number

SAT Score

On-Campus Housing

Random Number

2 12 1107 No 0.000273 773 1043 Yes 0.001924 408 991 Yes 0.003035 58 1008 No 0.004816 116 1127 Yes 0.005387 185 982 Yes 0.005838 510 1163 Yes 0.006499 394 1008 No 0.00667


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Point EstimatesPoint Estimates

• as Point Estimator of as Point Estimator of

• ss as Point Estimator of as Point Estimator of

• as Point Estimator of as Point Estimator of pp

xx

pp

ixx29,910

99730 30

ixx29,910

99730 30

ix x

s2( ) 163,996

75.229 29

ix x

s2( ) 163,996

75.229 29

p 20 30 .68 p 20 30 .68


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Point EstimatesPoint Estimates

Note:Note: Different random numbers would have Different random numbers would have

identified a different sample which would have identified a different sample which would have resulted in different point estimates.resulted in different point estimates.

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Sampling Distribution ofSampling Distribution of

Process of Statistical InferenceProcess of Statistical Inference

Population Population with meanwith mean

= ?= ?

Population Population with meanwith mean

= ?= ?

A simple random sampleA simple random sampleof of nn elements is selected elements is selected

from the population.from the population.

xx

The sample data The sample data provide a value forprovide a value for

the sample meanthe sample mean . .

The sample data The sample data provide a value forprovide a value for

the sample meanthe sample mean . .xx

The value of is used toThe value of is used tomake inferences aboutmake inferences about

the value of the value of ..

The value of is used toThe value of is used tomake inferences aboutmake inferences about

the value of the value of ..

xx

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The The sampling distribution of sampling distribution of is the is the probability distribution of all possible values of probability distribution of all possible values of the sample the sample

mean .mean . Expected Value ofExpected Value of

EE( ) = ( ) =

where: where:

= the population mean = the population mean

Sampling Distribution of Sampling Distribution of xx

xx

xxxx

xx

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Standard Deviation ofStandard Deviation of

Finite PopulationFinite Population Infinite Infinite Population Population

• A finite population is treated as being A finite population is treated as being infinite if infinite if nn//NN << .05. .05.

• is the finite correction is the finite correction factor.factor.

• is referred to as the is referred to as the standard error of the standard error of the meanmean..

xx

x n

N nN

( )1

x n

N nN

( )1

x n

x n

( ) / ( )N n N 1( ) / ( )N n N 1

x x

xxSampling Distribution of Sampling Distribution of

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If we use a large (If we use a large (nn >> 30) simple random 30) simple random sample, the sample, the central limit theoremcentral limit theorem enables us enables us to conclude that the sampling distribution of to conclude that the sampling distribution of can be approximated by a normal probability can be approximated by a normal probability distribution.distribution.

When the simple random sample is small (When the simple random sample is small (nn < < 30), the sampling distribution of can be 30), the sampling distribution of can be considered normal only if we assume the considered normal only if we assume the population has a normal probability population has a normal probability distribution.distribution.

xx

xx

Sampling Distribution of Sampling Distribution of xx

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Sampling Distribution of for the SAT ScoresSampling Distribution of for the SAT Scoresxx


xn

8014.6

30

xn

8014.6

30

E x( ) 990E x( ) 990

xx

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Sampling Distribution of for the SAT ScoresSampling Distribution of for the SAT Scores

What is the probability that a simple What is the probability that a simple random sample of 30 applicants will provide random sample of 30 applicants will provide an estimate of the population mean SAT score an estimate of the population mean SAT score that is within plus or minus 10 of the actual that is within plus or minus 10 of the actual population mean population mean ? ?

In other words, what is the probability In other words, what is the probability that will be between 980 and 1000?that will be between 980 and 1000?

xx


xx

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xx

Sampling distribution of

Sampling distribution of xx

10001000980980 990990

Area = ?Area = ?

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Sampling Distribution of for the SAT ScoresSampling Distribution of for the SAT Scores

Using the standard normal probability table Using the standard normal probability table with with

zz = 10/14.6= .68, we have area = (.2518)(2) = = 10/14.6= .68, we have area = (.2518)(2) = .5036 .5036

The probability is .5036 that the sample mean The probability is .5036 that the sample mean will be within +/-10 of the actual population will be within +/-10 of the actual population mean.mean.


xx

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xx


Sampling distribution of xx

10001000980980 990990

Area = 2(.2518) = .5036Area = 2(.2518) = .5036

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The The sampling distribution of sampling distribution of is the is the probability distribution of all possible values of probability distribution of all possible values of the sample proportion .the sample proportion .

Expected Value ofExpected Value of

where:where:

pp = the population proportion = the population proportion

Sampling Distribution of Sampling Distribution of pp

pp

pp

pp

E p p( ) E p p( )

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Sampling Distribution of Sampling Distribution of pp

pp

pp pn

N nN

( )11

pp pn

N nN

( )11

pp pn

( )1 pp pn

( )1

p p

Standard Deviation ofStandard Deviation of

Finite PopulationFinite Population Infinite Population Infinite Population

• is referred to as the is referred to as the standard error of the standard error of the proportionproportion..

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The sampling distribution of can be The sampling distribution of can be approximated by a normal probability approximated by a normal probability distribution whenever the sample size is large.distribution whenever the sample size is large.

The sample size is considered large whenever The sample size is considered large whenever these conditions are satisfied:these conditions are satisfied:

npnp >> 5 5

andand

nn(1 – (1 – pp) ) >> 5 5

pp

pp

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For values of For values of pp near .50, sample sizes as small near .50, sample sizes as small as 10 permit a normal approximation.as 10 permit a normal approximation.

With very small (approaching 0) or large With very small (approaching 0) or large (approaching 1) values of (approaching 1) values of pp, much larger , much larger samples are needed.samples are needed.

pp

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Sampling Distribution of for In-State Sampling Distribution of for In-State ResidentsResidents

The normal probability distribution is an The normal probability distribution is an acceptable approximation because:acceptable approximation because:

npnp = 30(.72) = 21.6 = 30(.72) = 21.6 >> 5 5

and and

nn(1 - (1 - pp) = 30(.28) = 8.4 ) = 30(.28) = 8.4 >> 5. 5.

pp

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pp


p

.72(1 .72).082

30

p

.72(1 .72).082

30

( ) .72E p ( ) .72E p

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What is the probability that a simple What is the probability that a simple random sample of 30 applicants will provide random sample of 30 applicants will provide an estimate of the population proportion of an estimate of the population proportion of applicants desiring on-campus housing that is applicants desiring on-campus housing that is within plus or minus .05 of the actual within plus or minus .05 of the actual population proportion?population proportion?

In other words, what is the probability In other words, what is the probability that that

will be between .67 and .77?will be between .67 and .77?

pp


pp

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pp



0.770.770.670.67 0.720.72

Area = ?Area = ?

pp

pp

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For For zz = .05/.082 = .61, the area = (.2291)(2) = = .05/.082 = .61, the area = (.2291)(2) = .4582..4582.

The probability is .4582 that the sample The probability is .4582 that the sample proportion will be within +/-.05 of the actual proportion will be within +/-.05 of the actual population proportion.population proportion.


pp

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pp



0.770.770.670.67 0.720.72

Area = 2(.2291) = .4582Area = 2(.2291) = .4582pp

pp

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Sampling MethodsSampling Methods

Stratified Random SamplingStratified Random Sampling Cluster SamplingCluster Sampling Systematic SamplingSystematic Sampling Convenience SamplingConvenience Sampling Judgment SamplingJudgment Sampling

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Stratified Random SamplingStratified Random Sampling

The population is first divided into groups of The population is first divided into groups of elements called elements called stratastrata..

Each element in the population belongs to one Each element in the population belongs to one and only one stratum.and only one stratum.

Best results are obtained when the elements Best results are obtained when the elements within each stratum are as much alike as within each stratum are as much alike as possible (i.e. possible (i.e. homogeneous grouphomogeneous group).).

A simple random sample is taken from each A simple random sample is taken from each stratum.stratum.

Formulas are available for combining the Formulas are available for combining the stratum sample results into one population stratum sample results into one population parameter estimate.parameter estimate.

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Stratified Random SamplingStratified Random Sampling

AdvantageAdvantage: If strata are homogeneous, this : If strata are homogeneous, this method is as “precise” as simple random method is as “precise” as simple random sampling but with a smaller total sample size.sampling but with a smaller total sample size.

ExampleExample: The basis for forming the strata : The basis for forming the strata might be department, location, age, industry might be department, location, age, industry type, etc.type, etc.

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Cluster SamplingCluster Sampling

The population is first divided into separate The population is first divided into separate groups of elements called groups of elements called clustersclusters..

Ideally, each cluster is a representative small-Ideally, each cluster is a representative small-scale version of the population (i.e. scale version of the population (i.e. heterogeneous group).heterogeneous group).

A simple random sample of the clusters is then A simple random sample of the clusters is then taken.taken.

All elements within each sampled (chosen) All elements within each sampled (chosen) cluster form the sample.cluster form the sample.

… … continuedcontinued

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Cluster SamplingCluster Sampling

AdvantageAdvantage: The close proximity of elements : The close proximity of elements can be cost effective (I.e. many sample can be cost effective (I.e. many sample observations can be obtained in a short time).observations can be obtained in a short time).

DisadvantageDisadvantage: This method generally requires : This method generally requires a larger total sample size than simple or a larger total sample size than simple or stratified random sampling.stratified random sampling.

ExampleExample: A primary application is area : A primary application is area sampling, where clusters are city blocks or sampling, where clusters are city blocks or other well-defined areas.other well-defined areas.

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Systematic SamplingSystematic Sampling

If a sample size of If a sample size of nn is desired from a is desired from a population containing population containing NN elements, we might elements, we might sample one element for every sample one element for every nn//NN elements in elements in the population.the population.

We randomly select one of the first We randomly select one of the first nn//NN elements from the population list.elements from the population list.

We then select every We then select every nn//NNth element that th element that follows in the population list.follows in the population list.

This method has the properties of a simple This method has the properties of a simple random sample, especially if the list of the random sample, especially if the list of the population elements is a random ordering.population elements is a random ordering.

… … continuedcontinued

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Systematic SamplingSystematic Sampling

AdvantageAdvantage: The sample usually will be easier : The sample usually will be easier to identify than it would be if simple random to identify than it would be if simple random sampling were used.sampling were used.

ExampleExample: Selecting every 100: Selecting every 100thth listing in a listing in a telephone book after the first randomly telephone book after the first randomly selected listing.selected listing.

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Convenience SamplingConvenience Sampling

It is a It is a nonprobability sampling techniquenonprobability sampling technique. Items . Items are included in the sample without known are included in the sample without known probabilities of being selected.probabilities of being selected.

The sample is identified primarily by The sample is identified primarily by convenienceconvenience..

AdvantageAdvantage: Sample selection and data : Sample selection and data collection are relatively easy.collection are relatively easy.

DisadvantageDisadvantage: It is impossible to determine how : It is impossible to determine how representative of the population the sample is. representative of the population the sample is.

ExampleExample: A professor conducting research : A professor conducting research might use student volunteers to constitute a might use student volunteers to constitute a sample. sample.

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Judgment SamplingJudgment Sampling

The person most knowledgeable on the subject The person most knowledgeable on the subject of the study selects elements of the population of the study selects elements of the population that he or she feels are most representative of that he or she feels are most representative of the population.the population.

It is a It is a nonprobability sampling techniquenonprobability sampling technique.. AdvantageAdvantage: It is a relatively easy way of : It is a relatively easy way of

selecting a sample.selecting a sample. DisadvantageDisadvantage: The quality of the sample results : The quality of the sample results

depends on the judgment of the person depends on the judgment of the person selecting the sample.selecting the sample.

ExampleExample: A reporter might sample three or four : A reporter might sample three or four senators, judging them as reflecting the general senators, judging them as reflecting the general opinion of the senate.opinion of the senate.

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End of Chapter 7End of Chapter 7

1 1 slide © 2003 thomson/south-western slides prepared by john s. loucks st. edward’s university

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