slides by john loucks st. edward’s university

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© 2009 Thomson South-Western. All Rights Reserved© 2009 Thomson South-Western. All Rights Reserved

Slides by

JOHNLOUCKSSt. Edward’sUniversity

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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

Selecting a SampleSelecting a Sample

Other Sampling MethodsOther Sampling Methods

pp Sampling Distribution ofSampling Distribution of

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IntroductionIntroduction

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

A A samplesample is a subset of the population. is a subset of the population. A A samplesample is a subset of the population. is a subset of the population.

An An elementelement is the entity on which data are collected. is the entity on which data are collected. An An elementelement is the entity on which data are collected. is the entity on which data are collected.

The reason we select a sample is to collect data toThe reason we select a sample is to collect data to answer a research question about a population.answer a research question about a population. The reason we select a sample is to collect data toThe reason we select a sample is to collect data to answer a research question about a population.answer a research question about a population.

A A frameframe is a list of the elements that the sample will is a list of the elements that the sample will be selected from.be selected from. A A frameframe is a list of the elements that the sample will is a list of the elements that the sample will be selected from.be selected from.

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The sample results provide only The sample results provide only estimatesestimates of the of the values of the population characteristics.values of the population characteristics. The sample results provide only The sample results provide only estimatesestimates of the of the values of the population characteristics.values of the population characteristics.

With With proper sampling methodsproper sampling methods, the sample results, the sample results can provide “good” estimates of the populationcan provide “good” estimates of the population characteristics.characteristics.

With With proper sampling methodsproper sampling methods, the sample results, the sample results can provide “good” estimates of the populationcan provide “good” estimates of the population characteristics.characteristics.

IntroductionIntroduction

The reason is simply that the sample contains only aThe reason is simply that the sample contains only a portion of the population.portion of the population. The reason is simply that the sample contains only aThe reason is simply that the sample contains only a portion of the population.portion of the population.

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Selecting a SampleSelecting a Sample

Sampling from a Finite PopulationSampling from a Finite Population Sampling from a ProcessSampling from a Process

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Sampling from a Finite PopulationSampling from a Finite Population

Finite populationsFinite populations are often defined by lists such as: are often defined by lists such as:

• Organization membership rosterOrganization membership roster

• Credit card account numbersCredit card account numbers

• Inventory product numbersInventory product numbers

A A simple random sample of size simple random sample of size nn from a finite from a finite

population of size population of size NN is a sample selected such is a sample selected such that eachthat each

possible sample of size possible sample of size nn has the same has the same probability ofprobability of

being selected.being selected.

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

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

Replacing each sampled element before selectingReplacing each sampled element before selecting subsequent elements is called subsequent elements is called sampling withsampling with replacementreplacement..


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St. Andrew’s College received 900 St. Andrew’s College received 900 applications forapplications for

admission in the upcoming year from admission in the upcoming year from prospectiveprospective

students. The applicants were numbered, from students. The applicants were numbered, from 1 to1 to

900, as their applications arrived. The Director 900, as their applications arrived. The Director ofof

Admissions would like to select a simple Admissions would like to select a simple randomrandom

sample of 30 applicants.sample of 30 applicants.

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


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Excel’s RAND function generatesExcel’s RAND function generates random numbers between 0 and 1random numbers between 0 and 1

Excel’s RAND function generatesExcel’s RAND function generates random numbers between 0 and 1random numbers between 0 and 1

Step 1:Step 1: Assign a random number to each of the 900 Assign a random number to each of the 900 applicants.applicants.

Step 2:Step 2: Select the 30 applicants corresponding to the Select the 30 applicants corresponding to the 30 smallest random numbers.30 smallest random numbers.

Sampling from a Finite Population Using Sampling from a Finite Population Using ExcelExcel


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Excel Formula WorksheetExcel Formula Worksheet

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


A B

1Applicant Number

2 1 =RAND()3 2 =RAND()4 3 =RAND()5 4 =RAND()6 5 =RAND()7 6 =RAND()8 7 =RAND()9 8 =RAND()

Random Number

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Excel Value WorksheetExcel Value Worksheet



A B

1Applicant Number

2 13 24 35 46 57 68 79 8

Random Number0.610210.837620.589350.199340.866580.605790.809600.33224

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Put Random Numbers in Ascending OrderPut Random Numbers in Ascending Order

Step 4Step 4 Choose Choose Sort Smallest to LargestSort Smallest to LargestStep 3Step 3 In the In the EditingEditing group, click group, click Sort & FilterSort & FilterStep 2Step 2 Click the Click the HHome tab on the Ribbonome tab on the Ribbon

Step 1Step 1 Select any cell in the range B2:B901Select any cell in the range B2:B901


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Excel Value Worksheet Excel Value Worksheet (Sorted)(Sorted)



A B

1Applicant Number

23456789

Random Number0.000270.001920.003030.004810.005380.005830.006490.00667

1277340858116185510394

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Populations are often defined by an Populations are often defined by an ongoing ongoing processprocess whereby the elements of the population whereby the elements of the population consist of items generated as though the consist of items generated as though the process would operate indefinitely.process would operate indefinitely.

Sampling from a ProcessSampling from a Process

Some examples of on-going processes, with infiniteSome examples of on-going processes, with infinite populations, are:populations, are:

• parts being manufactured on a production lineparts being manufactured on a production line• transactions occurring at a banktransactions occurring at a bank• telephone calls arriving at a technical help desktelephone calls arriving at a technical help desk• customers entering a storecustomers entering a store

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The sampled population is such that a frame cannotThe sampled population is such that a frame cannot be constructed.be constructed.

In the case of infinite populations, it is impossible toIn the case of infinite populations, it is impossible to obtain a list of all elements in the population.obtain a list of all elements in the population.

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

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A A random sample from an infinite populationrandom sample from an infinite population is a is a sample selected such that the following conditionssample selected such that the following conditions are satisfied.are satisfied.

• Each of the sampled elements is independent.Each of the sampled elements is independent.

• Each of the sampled elements follows the sameEach of the sampled elements follows the same

probability distribution as the elements in theprobability distribution as the elements in the

population.population.

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

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

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


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

xx

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

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Recall that St. Andrew’s College received 900Recall that St. Andrew’s College received 900applications from prospective students. The applications from prospective students. The application form contains a variety of application form contains a variety of

informationinformationincluding the individual’s scholastic aptitude including the individual’s scholastic aptitude

test test (SAT) score and whether or not the individual (SAT) score and whether or not the individual

desiresdesireson-campus housing.on-campus housing.



At a meeting in a few hours, the Director ofAt a meeting in a few hours, the Director ofAdmissions would like to announce the average Admissions would like to announce the average

SATSATscore and the proportion of applicants that score and the proportion of applicants that

want towant tolive on campus, for the population of 900 live on campus, for the population of 900

applicants.applicants.

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However, the necessary data on the However, the necessary data on the applicants haveapplicants have

not yet been entered in the college’s not yet been entered in the college’s computerizedcomputerized

database. So, the Director decides to estimate database. So, the Director decides to estimate

thethe

values of the population parameters of interest values of the population parameters of interest

basedbased

on sample statistics. The sample of 30 on sample statistics. The sample of 30

applicantsapplicants

selected earlier with Excel’s RAND() function selected earlier with Excel’s RAND() function

will bewill be

used.used.

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Excel Value Worksheet Excel Value Worksheet (Sorted)(Sorted)


Point Estimation Using ExcelPoint Estimation Using Excel

A B

1Applicant Number

23456789

Random Number0.000270.001920.003030.004810.005380.005830.006490.00667

1277340858116185510394

C D

SAT Score

On-Campus Housing

1107 No1043 Yes991 Yes1008 No1127 Yes982 Yes1163 Yes1008 No

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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


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

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990900

ix 990

900ix

2( )80

900ix

2( )80

900ix

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 Population Proportion Wanting On-Campus HousingHousing

Once all the data for the 900 applicants were Once all the data for the 900 applicants were enteredentered

in the college’s database, the values of the in the college’s database, the values of the populationpopulation

parameters of interest were calculated.parameters of interest were calculated.


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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

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Practical AdvicePractical Advice

The The target populationtarget population is the population we want to is the population we want to make inferences about.make inferences about. The The target populationtarget population is the population we want to is the population we want to make inferences about.make inferences about.

Whenever a sample is used to make inferences aboutWhenever a sample is used to make inferences about a population, we should make sure that the targeteda population, we should make sure that the targeted population and the sampled population are in closepopulation and the sampled population are in close agreement.agreement.

Whenever a sample is used to make inferences aboutWhenever a sample is used to make inferences about a population, we should make sure that the targeteda population, we should make sure that the targeted population and the sampled population are in closepopulation and the sampled population are in close agreement.agreement.

The The sampled populationsampled population is the population from is the population from which the sample is actually taken.which the sample is actually taken. The The sampled populationsampled population is the population from is the population from which the sample is actually taken.which the sample is actually taken.

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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

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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

xx• Expected Value ofExpected Value of

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We will use the following notation to define theWe will use the following notation to define the

standard deviation of the sampling distribution of standard deviation of the sampling distribution of . .

xx

= the standard deviation of = the standard deviation of xx xx

= the standard deviation of the population = the standard deviation of the population

nn = the sample size = the sample size

NN = the population size = the population size

xx• Standard Deviation ofStandard Deviation of

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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 populationis the finite population correction factor.correction factor.

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

xx• Standard Deviation ofStandard Deviation of

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When the population has a normal distribution, theWhen the population has a normal distribution, thesampling distribution of is normally distributedsampling distribution of is normally distributedfor any sample size.for any sample size.

x

In cases where the population is highly skewed orIn cases where the population is highly skewed oroutliers are present, samples of size 50 may beoutliers are present, samples of size 50 may beneeded.needed.

In most applications, the sampling distribution of In most applications, the sampling distribution of can be approximated by a normal distributioncan be approximated by a normal distributionwhenever the sample is size 30 or more.whenever the sample is size 30 or more.

x


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8014.6

30x

n

80

14.630

xn

( ) 990E x ( ) 990E x xx

SamplingSamplingDistributionDistribution

of of for SATfor SATScoresScores

xx



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What is the probability that a simple What is the probability that a simple randomrandom

sample of 30 applicants will provide an sample of 30 applicants will provide an estimate ofestimate of

the population mean SAT score that is within the population mean SAT score that is within +/+/1010

of the actual population mean of the actual population mean ? ?



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

be between 980 and 1000?be between 980 and 1000?

xx

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

PP((zz << .68) = .7517 .68) = .7517

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

. . . . . . . . . . .



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xx990990

14.6x 14.6x

10001000

Area = .7517Area = .7517





xx

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

PP((zz << -.68) = .2483 -.68) = .2483



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Sampling Distribution of Sampling Distribution of for SAT Scoresfor SAT Scoresxx

xx980980 990990

Area = .2483Area = .2483

14.6x 14.6x




xx

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

= .7517 = .7517 .2483 .2483= .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


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xx10001000980980 990990


Area = .5034Area = .5034

14.6x 14.6x




xx

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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 = 100, the standard error of the mean is decreased from 14.6 to:the mean is decreased from 14.6 to:

xx

808.0

100x

n

80

8.0100

xn


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( ) 990E x ( ) 990E x xx

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

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


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• 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) when 1000) 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


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xx10001000980980 990990

Area = .7888Area = .7888

8x 8x




xx

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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

pp• Expected 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 standard error oferror of the proportionthe proportion..

p p


Finite PopulationFinite Population Infinite PopulationInfinite Population

pp• Standard Deviation ofStandard Deviation of

• is the finite populationis the finite population correction factor.correction factor.

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

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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.

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:

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.

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.

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 the prospective students Recall that 72% of the prospective students applyingapplying

to St. Andrew’s College desire on-campus to St. Andrew’s College desire on-campus housing.housing.



What is the probability that a simple random sampleWhat 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 proportion of applicant desiring on-campuspopulation proportion of applicant desiring on-campus

housing that is within plus or minus .05 of the actualhousing that is within plus or minus .05 of the actual

population proportion?population proportion?

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For our example, with For our example, with nn = 30 and = 30 and pp = .72, the = .72, thenormal distribution is an acceptable approximationnormal distribution is an acceptable approximationbecause: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


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

DistributionDistribution


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) = .2709 -.61) = .2709



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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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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 7End of Chapter 7

slides by john loucks st. edward’s university

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