Chapter 7 Sampling and Sampling Distributions

Chapter 7Chapter 7Sampling and Sampling DistributionsSampling and Sampling Distributions

xx Sampling Distribution ofSampling Distribution of

Introduction to Sampling DistributionsIntroduction to Sampling Distributions

Point EstimationPoint Estimation

Simple Random SamplingSimple Random Sampling

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

St. Andrew’s College receivesSt. Andrew’s College receives

900 applications annually from900 applications annually from

prospective students. Theprospective students. The

application form contains application form contains

a variety of informationa variety of information

including the individual’sincluding the individual’s

scholastic aptitude test (SAT) score and whether scholastic aptitude test (SAT) score and whether or notor not

the individual desires on-campus housing.the individual desires on-campus housing.


The director of admissionsThe director of admissions

would like to know thewould like to know the

following information:following information:

• the average SAT score forthe average SAT score for

the 900 applicants, andthe 900 applicants, and

• the proportion ofthe proportion of

applicants that want to live on campus.applicants that want to live on campus.


We will now look at threeWe will now look at three

alternatives for obtaining thealternatives for obtaining the

desired information.desired information. Conducting a census of theConducting a census of the entire 900 applicantsentire 900 applicants Selecting a sample of 30Selecting a sample of 30

applicants, using a random number tableapplicants, using a random number table Selecting a sample of 30 applicants, using ExcelSelecting a sample of 30 applicants, using Excel

Conducting a CensusConducting a Census

If the relevant data for the entire 900 applicants If the relevant data for the entire 900 applicants were in the college’s database, the population were in the college’s database, the population parameters of interest could be calculated using parameters of interest could be calculated using the formulas presented in Chapter 3.the formulas presented in Chapter 3.

We will assume for the moment that conducting We will assume for the moment that conducting a census is practical in this example.a census is practical in this example.

990900

ix 990

900ix

2( )80

900ix

2( )80

900ix

Conducting a CensusConducting a Census

648.72

900p

648.72

900p

Population Mean SAT ScorePopulation Mean SAT Score

Population Standard Deviation for SAT ScorePopulation Standard Deviation for SAT Score

Population Proportion Wanting On-Campus HousingPopulation Proportion Wanting On-Campus Housing

as Point Estimator of as Point Estimator of xx

as Point Estimator of as Point Estimator of pppp

29,910997

30 30ix

x 29,910997

30 30ix

x

2( ) 163,99675.2

29 29ix x

s

2( ) 163,99675.2

29 29ix x

s

20 30 .68p 20 30 .68p

Point EstimationPoint Estimation

Note:Note: Different random numbers would haveDifferent random numbers would haveidentified a different sample which would haveidentified a different sample which would haveresulted in different point estimates.resulted in different point estimates.

ss as Point Estimator of as Point Estimator of

PopulationPopulationParameterParameter

PointPointEstimatorEstimator

PointPointEstimateEstimate

ParameterParameterValueValue

= Population mean= Population mean SAT score SAT score

990990 997997

= Population std.= Population std. deviation for deviation for SAT score SAT score

8080 s s = Sample std.= Sample std. deviation fordeviation for SAT score SAT score

75.275.2

pp = Population pro- = Population proportion wantingportion wanting campus housing campus housing

.72.72 .68.68

Summary of Point EstimatesSummary of Point EstimatesObtained from a Simple Random SampleObtained from a Simple Random Sample

= Sample mean= Sample mean SAT score SAT score xx

= Sample pro-= Sample proportion wantingportion wanting campus housing campus housing

pp

Process of Statistical InferenceProcess of Statistical Inference

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

the value of the value of ..

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

the sample meanthe sample mean . .xx

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

from the population.from the population.

Population Population with meanwith mean

= ?= ?

Sampling Distribution of Sampling Distribution of xx

The The sampling distribution of sampling distribution of is the probability is the probability

distribution of all possible values of the sample distribution of all possible values of the sample

mean .mean .

xx

xx


where: where: = the population mean= the population mean

EE( ) = ( ) = xx

xxExpected Value ofExpected Value of


Finite PopulationFinite Population Infinite PopulationInfinite Population

x n

N nN

( )1

x n

N nN

( )1

x n

x n

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

x x

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

• is the finite correction factor.is the finite correction factor.( ) / ( )N n N 1( ) / ( )N n N 1

xxStandard Deviation ofStandard Deviation of

Form of the Sampling Distribution of Form of the Sampling Distribution of xx

If we use a large (If we use a large (nn >> 30) simple random sample, the 30) simple random sample, thecentral limit theoremcentral limit theorem enables us to conclude that the enables us to conclude that thesampling distribution of can be approximated bysampling distribution of can be approximated bya normal distribution.a normal distribution.

xx

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

xx

8014.6

30x

n

80

14.630

xn

( ) 990E x ( ) 990E x xx

Sampling Distribution of Sampling Distribution of for SAT Scoresfor SAT Scoresxx

SamplingSamplingDistributionDistribution

of of xx

With a mean SAT score of 990 and a standard deviation ofWith a mean SAT score of 990 and a standard deviation of

80, what is the probability that a simple random sample80, what is the probability that a simple random sample

of 30 applicants will provide an estimate of theof 30 applicants will provide an estimate of the

population mean SAT score that is within +/population mean SAT score that is within +/10 of10 of

the actual population mean the actual population mean ? ?

In other words, what is the probability that will beIn other words, what is the probability that will be

between 980 and 1000?between 980 and 1000?

xx


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 = (1000 = (1000 990)/14.6= .68 990)/14.6= .68

.2517.2517

Step 2:Step 2: Find the area under the curve between the mean Find the area under the curve between the mean and the and the upperupper endpoint. endpoint.



Probabilities forProbabilities for the Standard Normal the Standard Normal

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

. . . . . . . . . . .

.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224

.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549

.7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852

.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133

.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389

. . . . . . . . . . .

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

. . . . . . . . . . .

.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224

.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549

.7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852

.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133

.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389

. . . . . . . . . . .


xx990990


of of xx14.6x 14.6x

10001000

Area = .2517Area = .2517

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 Find the area under the curve between the mean between the mean and the and the lowerlower endpoint. endpoint.

zz = (980 = (980 990)/14.6= - .68 990)/14.6= - .68

= .2517= .2517



xx990990


of of xx14.6x 14.6x

980980

Area = .2517Area = .2517


xx980980 990990

Area = .2517Area = .2517


of of xx14.6x 14.6x

10001000

Area = .2517Area = .2517


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(-.68 (-.68 << zz << .68) = .68) =

= .2517 = .2517 .2517 .2517= .5034= .5034

The probability that the sample mean SAT The probability that the sample mean SAT score willscore willbe between 980 and 1000 is:be between 980 and 1000 is:

PP(980 (980 << << 1000) = .5034 1000) = .5034xx

xx10001000980980 990990


Area = .5034Area = .5034


of of xx14.6x 14.6x

Relationship Between the Sample SizeRelationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of xx

Suppose we select a simple random sample of 100Suppose we select a simple random sample of 100 applicants instead of the 30 originally considered.applicants instead of the 30 originally considered.

EE( ) = ( ) = regardless of the sample size. In regardless of the sample size. In ourour example,example, E E( ) remains at 990.( ) remains at 990.

xxxx

Whenever the sample size is increased, the standardWhenever the sample size is increased, the standard error of the mean is decreased. With the increaseerror of the mean is decreased. With the increase in the sample size to in the sample size to nn = 100, the standard error of the = 100, the standard error of the mean is decreased to:mean is decreased to:

xx

808.0

100x

n

80

8.0100

xn


( ) 990E x ( ) 990E x xx

14.6x 14.6x With With nn = 30, = 30,

8x 8x With With nn = 100, = 100,

Recall that when Recall that when nn = 30, = 30, PP(980 (980 << << 1000) = .5034. 1000) = .5034.xx


We follow the same steps to solve for We follow the same steps to solve for PP(980 (980 << << 1000) 1000) when when nn = 100 as we showed earlier when = 100 as we showed earlier when nn = 30. = 30.

xx

Now, with Now, with nn = 100, = 100, PP(980 (980 << << 1000) = .7888. 1000) = .7888.xx

Because the sampling distribution with Because the sampling distribution with nn = 100 has a = 100 has a smaller standard error, the values of have lesssmaller standard error, the values of have less variability and tend to be closer to the populationvariability and tend to be closer to the population mean than the values of with mean than the values of with nn = 30. = 30.

xx

xx


xx10001000980980 990990

Area = .7888Area = .7888


of of xx8x 8x

Chapter 7 Chapter 7 Sampling and Sampling DistributionsSampling and Sampling Distributions

Other Sampling MethodsOther Sampling Methods

pp Sampling Distribution ofSampling Distribution of

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

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

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

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

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?

p

.72(1 .72).082

30

p

.72(1 .72).082

30

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


of of pp


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

.2291.2291

Step 2:Step 2: Find the area under the curve Find the area under the curve between the mean between the mean and and upperupper endpoint. endpoint.


Probabilities forProbabilities for the Standard Normal the Standard Normal

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

. . . . . . . . . . .

.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224

.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549

.7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852

.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133

.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389

. . . . . . . . . . .

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

. . . . . . . . . . .

.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224

.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549

.7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852

.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133

.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389

. . . . . . . . . . .


.77.77.72.72

Area = .2291Area = .2291

pp


of of pp

.082p .082p


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 Find the area under the curve between the mean between the mean and the and the lowerlower endpoint. endpoint.

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

.2291.2291


.67.67 .72.72

Area = .2291Area = .2291

pp


of of pp

.082p .082p


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) =

= .2291 = .2291 .2291 .2291= .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 :


.77.77.67.67 .72.72

Area = .4582Area = .4582

pp


of of pp

.082p .082p


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 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).).

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.

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.

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.

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.

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

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..

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

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.

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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Chapter 7 Sampling and Sampling Distributions