1 1 Slide © 2004 Thomson/South-Western Slides Prepared by JOHN S. LOUCKS St. Edward’s University Slides Prepared by JOHN S. LOUCKS St. Edward’s University.

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© 2004 Thomson/South-Western© 2004 Thomson/South-Western

Slides Prepared bySlides Prepared by

JOHN S. LOUCKSJOHN S. LOUCKSSt. Edward’s UniversitySt. Edward’s University

Slides Prepared bySlides Prepared by

JOHN S. LOUCKSJOHN S. LOUCKSSt. Edward’s UniversitySt. Edward’s University

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Chapter 7Chapter 7Sampling and Sampling DistributionsSampling and Sampling Distributions

Sampling MethodsSampling Methods

ppSampling Distribution ofSampling Distribution ofxxSampling Distribution ofSampling Distribution of

Introduction to Sampling DistributionsIntroduction to Sampling Distributions

Point EstimationPoint Estimation

Simple Random SamplingSimple Random Sampling

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

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

Statistical InferenceStatistical Inference

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.

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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.

A A parameterparameter is a numerical characteristic of a is a numerical characteristic of a population.population. A A parameterparameter is a numerical characteristic of a is a numerical characteristic of a population.population.

With With proper sampling methodsproper sampling methods, the sample 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.

Statistical InferenceStatistical Inference

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Simple Random Sampling:Simple Random Sampling:Finite PopulationFinite 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 from a finitefinite

population of size population of size NN is a sample selected is a sample selected suchsuch

that each possible sample of size that each possible sample of size nn has has the samethe same

probability of being selected.probability of being selected.

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Simple Random Sampling:Simple Random Sampling:Finite PopulationFinite Population

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

Simple Random Sampling:Simple Random Sampling:Infinite PopulationInfinite Population

A A simple random sample from an infinite populationsimple random sample from an infinite population is a sample selected such that the following conditionsis a sample selected such that the following conditions are satisfied.are satisfied.

• Each element selected comes from the sameEach element selected comes from the same population.population.

• Each element is selected independently.Each element is selected independently.

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Simple Random Sampling:Simple Random Sampling:Infinite PopulationInfinite Population

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

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.

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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.

Point EstimationPoint Estimation

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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Sampling ErrorSampling Error

Statistical methods can be used to make probabilityStatistical methods can be used to make probability statements about the size of the sampling error.statements about the size of the sampling error.

Sampling error is the result of using a subset of theSampling error is the result of using a subset of the population (the sample), and not the entirepopulation (the sample), and not the entire population.population.

The absolute value of the difference between anThe absolute value of the difference between an unbiased point estimate and the correspondingunbiased point estimate and the corresponding population parameter is called the population parameter is called the sampling errorsampling error..

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

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Sampling ErrorSampling Error

The sampling errors are:The sampling errors are:

| |p p| |p p for sample proportionfor sample proportion

| |s | |s for sample standard deviationfor sample standard deviation

| |x | |x for sample meanfor sample mean

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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.

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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.

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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 Selecting a sample of 30 applicants, using

ExcelExcel

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Conducting a CensusConducting a Census

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

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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

990900

ix 990

900ix

2( )80

900ix

2( )80

900ix

Conducting a CensusConducting a Census

648.72

900p

648.72

900p

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Simple Random SamplingSimple Random Sampling

The applicants were numbered, from 1 to 900, asThe applicants were numbered, from 1 to 900, as their applications arrived.their applications arrived.

She decides a sample of 30 applicants will be used.She decides a sample of 30 applicants will be used.

Furthermore, the Director of Admissions must obtainFurthermore, the Director of Admissions must obtain estimates of the population parameters of interest forestimates of the population parameters of interest for a meeting taking place in a few hours.a meeting taking place in a few hours.

Now suppose that the necessary information onNow suppose that the necessary information on the current year’s applicants was not yet enteredthe current year’s applicants was not yet entered in the college’s database.in the college’s database.

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Taking a Sample of 30 ApplicantsTaking a Sample of 30 Applicants

Simple Random Sampling:Simple Random Sampling:Using a Random Number TableUsing a Random Number Table

• We will use the We will use the lastlast three digits of the 5-digit three digits of the 5-digit random numbers in the random numbers in the thirdthird column of the column of the textbook’s random number table, and continuetextbook’s random number table, and continue into the fourth column as needed.into the fourth column as needed.

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

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• (We will go through all of column 3 and part (We will go through all of column 3 and part ofof

column 4 of the random number table, column 4 of the random number table, encounteringencountering

in the process five numbers greater than in the process five numbers greater than 900 and900 and

one duplicate, 835.)one duplicate, 835.)

• We will continue to draw random numbers untilWe will continue to draw random numbers until we have selected 30 applicants for our sample.we have selected 30 applicants for our sample.

• The numbers we draw will be the numbers The numbers we draw will be the numbers of the of the applicants we will sample unlessapplicants we will sample unless

• the random number is greater than 900 the random number is greater than 900 oror• the random number has already been the random number has already been used.used.

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Use of Random Numbers for SamplingUse of Random Numbers for Sampling


744744436436865865790790835835902902190190836836

. . . and so on. . . and so on

3-Digit3-DigitRandom NumberRandom Number

ApplicantApplicantIncluded in SampleIncluded in Sample

No. 436No. 436No. 865No. 865No. 790No. 790No. 835No. 835

Number exceeds 900Number exceeds 900No. 190No. 190No. 836No. 836

No. 744No. 744

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Sample DataSample Data


11 744 744 Conrad Harris Conrad Harris 10251025 Yes Yes22 436436 Enrique Romero Enrique Romero 950 950 Yes Yes33 865865 Fabian Avante Fabian Avante 10901090 No No

44 790790 Lucila Cruz Lucila Cruz 11201120 Yes Yes55 835835 Chan Chiang Chan Chiang 930 930 No No.. . . . . . . . .

3030 498 498 Emily Morse Emily Morse 1010 1010 No No

No.No.RandomRandomNumberNumber ApplicantApplicant

SATSAT ScoreScore

Live On-Live On-CampusCampus

.. . . . . . . . .

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

• Each of the 900 applicants has the same Each of the 900 applicants has the same probabilityprobability of being included.of being included.

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

• 900 random numbers are generated, one for each900 random numbers are generated, one for each applicant in the population.applicant in the population.

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

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Formula WorksheetFormula WorksheetA B C D

1Applicant Number

SAT Score

On-Campus Housing

Random Number

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

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

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Value WorksheetValue WorksheetA B C D

1Applicant Number

SAT Score

On-Campus Housing

Random Number

2 1 1008 Yes 0.848913 2 1025 No 0.301814 3 952 Yes 0.357095 4 1090 Yes 0.994336 5 1127 Yes 0.330727 6 1015 No 0.545488 7 965 Yes 0.467589 8 1161 No 0.33167


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Put Random Numbers in Ascending OrderPut Random Numbers in Ascending Order


Step 4Step 4 When the When the SortSort dialog box appears: dialog box appears:

Choose Choose Random Numbers Random Numbers in in thethe

Sort by Sort by text boxtext box

Choose Choose AscendingAscending

Click Click OKOK

Step 3Step 3 Choose the Choose the SortSort option optionStep 2Step 2 Select the Select the DataData menu menuStep 1Step 1 Select cells A2:A901Select cells A2:A901

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

1Applicant Number

SAT Score

On-Campus Housing

Random Number

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


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as Point Estimator of as Point Estimator of

ss as Point Estimator of as Point Estimator of

as Point Estimator of as Point Estimator of pp

xx

pp

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 EstimatesPoint Estimates

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.

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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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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

xxExpected Value ofExpected Value 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 correction factor.is the finite correction factor.( ) / ( )N n N 1( ) / ( )N n N 1

xxStandard Deviation ofStandard Deviation of

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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 probability distribution.a normal probability 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 probability distribution.normal probability distribution.

xx

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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

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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 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


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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

. . . . . . . . . . .

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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xx990990


of of xx14.6x 14.6x

10001000

Area = .7517Area = .7517

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

= .2483= .2483= 1 = 1 . 7517 . 7517

= 1 = 1 PP((zz << .68) .68)


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xx980980 990990

Area = .2483Area = .2483


of of xx14.6x 14.6x

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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


of of xx14.6x 14.6x

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

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xx10001000980980 990990

Area = .7888Area = .7888


of of xx8x 8x

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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 probability distribution whenever theby a normal probability distribution whenever the sample size is large.sample size 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


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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.


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

Sampling Distribution of Sampling Distribution of for the Proportionfor the Proportion of Applicants Wanting On-Campus Housingof Applicants Wanting On-Campus Housing

pp

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


pp


of of pp

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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 the populationof 30 applicants will provide an estimate of the population

proportion of applicants desiring on-campus housing thatproportion of applicants desiring on-campus housing that

is within plus or minus .05 of the actual populationis within plus or minus .05 of the actual population

proportion?proportion?

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

between .67 and .77?between .67 and .77?

pp


pp

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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

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

. . . . . . . . . . .


pp

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.77.77.72.72

Area = .7291Area = .7291

pp


pp


of of pp

.082p .082p

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pp

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

pp


of of pp

.082p .082p

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


pp

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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pp

.77.77.67.67 .72.72

Area = .4582Area = .4582

pp


of of pp

.082p .082p

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Sampling MethodsSampling Methods

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

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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

1 1 Slide © 2004 Thomson/South-Western Slides Prepared by JOHN S. LOUCKS St. Edward’s University Slides Prepared by JOHN S. LOUCKS St. Edward’s University.

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