1 1 Slide © 2005 Thomson/South-Western Chapter 7, Part B Sampling and Sampling Distributions Other...

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© 2005 Thomson/South-Western© 2005 Thomson/South-Western

Chapter 7, Part BChapter 7, Part BSampling and Sampling DistributionsSampling and Sampling Distributions

Other Sampling MethodsOther Sampling Methods

pp Sampling Distribution ofSampling Distribution of Properties of Point EstimatorsProperties of Point Estimators

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

from the population.from the population.

Population Population with proportionwith proportion

pp = ? = ?

Making Inferences about a Population Making Inferences about a Population ProportionProportion

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

thethesample sample

proportionproportion . .

pp

The value of is usedThe value of is usedto make inferencesto make inferences

about the value of about the value of pp..

pp

Sampling Distribution ofSampling Distribution ofpp

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E p p( ) E p p( )


where:where:pp = the population proportion = the population proportion

The The sampling distribution of sampling distribution of is the probability is the probabilitydistribution of all possible values of the sampledistribution of all possible values of the sampleproportion .proportion .pp

pp

ppExpected Value ofExpected Value of

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pp pn

N nN

( )11

pp pn

N nN

( )11

pp pn

( )1 pp pn

( )1

is referred to as the is referred to as the standard error standard error of theof theproportionproportion..

p p


Finite PopulationFinite Population Infinite PopulationInfinite Population

ppStandard Deviation ofStandard Deviation of

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

pp

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

npnp >> 5 5 nn(1 – (1 – pp) ) >> 5 5andand

Form of the Sampling Distribution ofForm of the Sampling Distribution ofpp

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

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

Form of the Sampling Distribution ofForm of the Sampling Distribution ofpp

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Recall that 72% of theRecall that 72% of the

prospective students applyingprospective students applying

to St. Andrew’s College desireto St. Andrew’s College desire

on-campus housing.on-campus housing.

Example: St. Andrew’s CollegeExample: St. Andrew’s College


What is the probability thatWhat is the probability that

a simple random sample of 30 applicants will providea simple random sample of 30 applicants will provide

an estimate of the population proportion of applicantan estimate of the population proportion of applicant

desiring on-campus housing that is within plus ordesiring on-campus housing that is within plus or

minus .05 of the actual population proportion?minus .05 of the actual population proportion?

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For our example, with For our example, with nn = 30 and = 30 and pp = .72, the normal distribution is an acceptable = .72, the normal distribution is an acceptable approximation because:approximation because:

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

andand

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


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p

.72(1 .72).082

30

p

.72(1 .72).082

30

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

SamplingSamplingDistributionDistribution

of of pp


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Step 1: Step 1: Calculate the Calculate the zz-value at the -value at the upperupper endpoint of endpoint of the interval.the interval.

zz = (.77 = (.77 .72)/.082 = .61 .72)/.082 = .61

PP((zz << .61) = .7291 .61) = .7291

Step 2:Step 2: Find the area under the curve to the left of the Find the area under the curve to the left of the upperupper endpoint. endpoint.


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Cumulative Probabilities forCumulative Probabilities for the Standard Normal the Standard Normal

DistributionDistributionz .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

. . . . . . . . . . .

.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224

.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549

.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852

.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133

.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389

. . . . . . . . . . .

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

. . . . . . . . . . .

.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224

.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549

.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852

.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133

.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389

. . . . . . . . . . .


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.77.77.72.72

Area = .7291Area = .7291

pp


of of pp

.082p .082p


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Step 3: Step 3: Calculate the Calculate the zz-value at the -value at the lowerlower endpoint of endpoint of the interval.the interval.

Step 4:Step 4: Find the area under the curve to the left of the Find the area under the curve to the left of the lowerlower endpoint. endpoint.

zz = (.67 = (.67 .72)/.082 = - .61 .72)/.082 = - .61

PP((zz << -.61) = -.61) = PP((zz >> .61) .61)

= .2709= .2709= 1 = 1 . 7291 . 7291

= 1 = 1 PP((zz << .61) .61)


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.67.67 .72.72

Area = .2709Area = .2709

pp


of of pp

.082p .082p


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PP(.67 (.67 << << .77) = .4582 .77) = .4582pp

Step 5: Step 5: Calculate the area under the curve betweenCalculate the area under the curve between the lower and upper endpoints of the interval.the lower and upper endpoints of the interval.

PP(-.61 (-.61 << zz << .61) = .61) = PP((zz << .61) .61) PP((zz << -.61) -.61)

= .7291 = .7291 .2709 .2709= .4582= .4582

The probability that the sample proportion of applicantsThe probability that the sample proportion of applicantswanting on-campus housing will be within +/-.05 of thewanting on-campus housing will be within +/-.05 of theactual population proportion :actual population proportion :


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.77.77.67.67 .72.72

Area = .4582Area = .4582

pp


of of pp

.082p .082p


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Properties of Point EstimatorsProperties of Point Estimators

Before using a sample statistic as a point Before using a sample statistic as a point estimator, statisticians check to see whether estimator, statisticians check to see whether the sample statistic has the following the sample statistic has the following properties associated with good point properties associated with good point estimators.estimators.

ConsistencyConsistency

EfficiencyEfficiency

UnbiasedUnbiased

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If the expected value of the sample If the expected value of the sample statistic is equal to the population parameter statistic is equal to the population parameter being estimated, the sample statistic is said to being estimated, the sample statistic is said to be an be an unbiased estimatorunbiased estimator of the population of the population parameter.parameter.

UnbiasedUnbiased

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Given the choice of two unbiased Given the choice of two unbiased estimators of the same population parameter, estimators of the same population parameter, we would prefer to use the point estimator we would prefer to use the point estimator with the smaller standard deviation, since it with the smaller standard deviation, since it tends to provide estimates closer to the tends to provide estimates closer to the population parameter.population parameter.

The point estimator with the smaller The point estimator with the smaller standard deviation is said to have greater standard deviation is said to have greater relative efficiencyrelative efficiency than the other. than the other.

EfficiencyEfficiency

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A point estimator is A point estimator is consistentconsistent if the if the values of the point estimator tend to become values of the point estimator tend to become closer to the population parameter as the closer to the population parameter as the sample size becomes larger.sample size becomes larger.

ConsistencyConsistency

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Other Sampling MethodsOther Sampling Methods

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

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

Stratified Random SamplingStratified Random Sampling

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

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

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

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Stratified Random SamplingStratified Random Sampling

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

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

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

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

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

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

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Cluster SamplingCluster Sampling

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

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

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

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

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Cluster SamplingCluster Sampling

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

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

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

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

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

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

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Systematic SamplingSystematic Sampling

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

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

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

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

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Systematic SamplingSystematic Sampling

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

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

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

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

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

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Convenience SamplingConvenience Sampling

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

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

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

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

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

Convenience SamplingConvenience Sampling

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Judgment SamplingJudgment Sampling

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

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

It is a It is a nonprobability sampling techniquenonprobability sampling technique.. It is a It is a nonprobability sampling techniquenonprobability sampling technique..

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

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

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Judgment SamplingJudgment Sampling

AdvantageAdvantage: It is a relatively easy way of selecting a: It is a relatively easy way of selecting a sample.sample. AdvantageAdvantage: It is a relatively easy way of selecting a: It is a relatively easy way of selecting a sample.sample.

DisadvantageDisadvantage: The quality of the sample results: The quality of the sample results depends on the judgment of the person selecting thedepends on the judgment of the person selecting the sample.sample.

DisadvantageDisadvantage: The quality of the sample results: The quality of the sample results depends on the judgment of the person selecting thedepends on the judgment of the person selecting the sample.sample.

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End of Chapter 7, Part BEnd of Chapter 7, Part B

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