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11SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern

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

�� Other Sampling MethodsOther Sampling Methods

p�� Sampling Distribution ofSampling Distribution of

�� Properties of Point EstimatorsProperties of Point Estimators


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

from the population.from the population.

Population Population with proportionwith proportion

pp = ?= ?

■■ Making Inferences about a Population ProportionMaking Inferences about a Population Proportion

The sample data The sample data provide a value for theprovide a value for thesample proportionsample proportion ..p

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

about the value of about the value of pp..

p

Sampling Distribution ofSampling Distribution of p

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E p p( ) =


where:where:

pp = the population proportion= the population proportion

The The sampling distribution of sampling distribution of is the probabilityis the probabilitydistribution of all possible values of the sampledistribution of all possible values of the sampleproportion .proportion .p

p

pExpected Value ofExpected Value of


σ pp p

nN nN

= − −−

( )11

σ pp p

n= −( )1

is referred to as the is referred to as the standard error of thestandard error of theproportionproportion..σ p


Finite PopulationFinite Population Infinite PopulationInfinite Population

pStandard Deviation ofStandard Deviation of

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

p

The sample size is considered large whenever theseThe sample size is considered large whenever theseconditions are satisfied:conditions are satisfied:

npnp >> 55 nn(1 (1 –– pp) ) >> 55andand

Form of the Sampling Distribution ofForm of the Sampling Distribution of p


For values of For values of pp near .50, sample sizes as small as 10near .50, sample sizes as small 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 samples , much larger samples are needed.are needed.

Form of the Sampling Distribution ofForm of the Sampling Distribution of p

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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

onon--campus housing.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 ondesiring on--campus housing that is within plus orcampus housing that is within plus or

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


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

nn(1 (1 -- pp) = 30(.28) = 8.4 ) = 30(.28) = 8.4 >> 55

andand

npnp = 30(.72) = 21.6 = 30(.72) = 21.6 >> 55


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σ −= =p

.72(1 .72).082

30

( ) .72E p =p

SamplingSamplingDistributionDistribution

of of p



Step 1: Step 1: Calculate the Calculate the zz--value at the value at the upperupper endpoint ofendpoint ofthe 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 theFind the area under the curve to the left of theupperupper endpoint.endpoint.


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Cumulative Probabilities forCumulative Probabilities forthe Standard Normal Distributionthe Standard Normal Distribution

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

. . . . . . . . . . .



.77.77.72.72

Area = .7291Area = .7291

p


of of p

.082pσ =


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

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

zz = (.67 = (.67 -- .72)/.082 = .72)/.082 = -- .61.61

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

= .2709= .2709

= 1 = 1 -- . 7291. 7291

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



.67.67 .72.72

Area = .2709Area = .2709

p


of of p

.082pσ =


8


PP(.67 (.67 << << .77) = .4582.77) = .4582p

Step 5: Step 5: Calculate the area under the curve betweenCalculate the area under the curve betweenthe 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 onwanting on--campus housing will be within +/campus housing will be within +/--.05 of the.05 of theactual population proportion :actual population proportion :



.77.77.67.67 .72.72

Area = .4582Area = .4582

p


of of p

.082pσ =


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Properties of Point EstimatorsProperties of Point Estimators

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

ConsistencyConsistency

EfficiencyEfficiency

UnbiasedUnbiased



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

UnbiasedUnbiased

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

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

EfficiencyEfficiency



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


The population is first divided into groups ofThe population is first divided into groups ofelements 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 andonly one stratum.only one stratum.

Best results are obtained when the elements withinBest results are obtained when the elements withineach 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.

Formulas are available for combining the stratumFormulas are available for combining the stratumsample results into one population parametersample results into one population parameterestimate.estimate.

AdvantageAdvantage: If strata are homogeneous, this method: If strata are homogeneous, this methodis as “precise” as simple random sampling but withis as “precise” as simple random sampling but witha 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 bedepartment, location, age, industry type, and so on.department, location, age, industry type, and so on.


Cluster SamplingCluster Sampling

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

Ideally, each cluster is a representative smallIdeally, each cluster is a representative small--scalescaleversion 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.

All elements within each sampled (chosen) clusterAll elements within each sampled (chosen) clusterform 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 becost effective (i.e. many sample observations can becost effective (i.e. many sample observations can beobtained in a short time).obtained in a short time).

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

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


Systematic SamplingSystematic Sampling

If a sample size of If a sample size of nn is desired from a populationis desired from a populationcontaining containing NN elements, we might sample oneelements, we might sample oneelement 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 elementselementsfrom the population list.from the population list.

We then select every We then select every nn//NNth element that follows inth element that follows inthe 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 randomsample, especially if the list of the populationsample, especially if the list of the populationelements is a random ordering.elements is a random ordering.

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

ExampleExample: Selecting every 100: Selecting every 100thth listing in a telephonelisting in a telephonebook after the first randomly selected listingbook after the first randomly selected listing


Convenience SamplingConvenience Sampling

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

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

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 arerelatively easy.relatively easy.

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

Convenience SamplingConvenience Sampling


Judgment SamplingJudgment Sampling

The person most knowledgeable on the subject of theThe person most knowledgeable on the subject of thestudy selects elements of the population that he orstudy selects elements of the population that he orshe 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..

ExampleExample: A reporter might sample three or four: A reporter might sample three or foursenators, judging them as reflecting the generalsenators, judging them as reflecting the generalopinion 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 asample.sample.

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


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

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SBE10_07b [Read-Only] [Compatibility Mode]