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

Chapter 7, Part AChapter 7, Part ASampling and Sampling DistributionsSampling and Sampling Distributions

x�� Sampling Distribution ofSampling Distribution of

�� Introduction to Sampling DistributionsIntroduction to Sampling Distributions

�� Point EstimationPoint Estimation

�� Simple Random SamplingSimple Random Sampling


The purpose of The purpose of statistical inferencestatistical inference is to obtainis to obtaininformation about a population from informationinformation about a population from informationcontained 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 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 theof thevalues of the population characteristics.values of the population characteristics.

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

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

Statistical InferenceStatistical Inference


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 finitefrom a finite

population of size population of size NN is a sample selected suchis a sample selected such

that each possible sample of size that each possible sample of size nn has the samehas the 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, computerIn large sampling projects, computer--generatedgeneratedrandom numbersrandom numbers are often used to automate theare often used to automate thesample selection process.sample selection process.

�� Sampling without replacementSampling without replacement is the procedureis the procedureused most often.used most often.

�� Replacing each sampled element before selectingReplacing each sampled element before selectingsubsequent elements is called subsequent elements is called sampling withsampling withreplacementreplacement..


■■ Infinite populations are often defined by an Infinite 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 process would consist of items generated as though the process would operate indefinitely.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 beused 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 toobtain a list of all elements in the population.obtain a list of all elements in the population.


ss is the is the point estimatorpoint estimator of the population standardof the population standarddeviation deviation σσ..

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 servesas 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 populationof the populationmean mean µµ..

x

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

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Sampling ErrorSampling Error

�� Statistical methods can be used to make probabilityStatistical methods can be used to make probabilitystatements 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 thepopulation (the sample), and not the entirepopulation (the sample), and not the entirepopulation.population.

�� The absolute value of the difference between anThe absolute value of the difference between anunbiased point estimate and the correspondingunbiased point estimate and the correspondingpopulation 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 equalto the population parameter, the point estimator is saidto the population parameter, the point estimator is saidto be to be unbiasedunbiased..


Sampling ErrorSampling Error

■■ The sampling errors are:The sampling errors are:

| |p p− for sample proportionfor sample proportion

| |s σ− for sample standard deviationfor sample standard deviation

| |x µ− 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 or notscholastic aptitude test (SAT) score and whether or not

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

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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 ExcelSelecting a sample of 30 applicants, using Excel


Conducting a CensusConducting a Census

■■ If the relevant data for the entire 900 applicants were If the relevant data for the entire 900 applicants were

in the college’s database, the population parameters of in the college’s database, the population parameters of

interest could be calculated using the formulas interest could be calculated using the formulas

presented in Chapter 3.presented in Chapter 3.

■■ We will assume for the moment that conducting a We will assume for the moment that conducting a

census is practical in this example.census is practical in this example.

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990900

ixµ = =∑

2( )80

900

ix µσ

−= =∑

Conducting a CensusConducting a Census

648.72

900p = =

■■ Population Mean SAT ScorePopulation Mean SAT Score

■■ Population Standard Deviation for SAT ScorePopulation Standard Deviation for SAT Score

■■ Population Proportion Wanting OnPopulation Proportion Wanting On--Campus HousingCampus Housing


Simple Random SamplingSimple Random Sampling

�� The applicants were numbered, from 1 to 900, asThe applicants were numbered, from 1 to 900, astheir 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 obtainestimates of the population parameters of interest forestimates of the population parameters of interest fora meeting taking place in a few hours.a meeting taking place in a few hours.

�� Now suppose that the necessary data on theNow suppose that the necessary data on thecurrent year’s applicants were not yet entered in thecurrent year’s applicants were not yet entered in thecollege’s database.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 5three digits of the 5--digitdigitrandom numbers in the random numbers in the thirdthird column of thecolumn of thetextbook’s random number table, and continuetextbook’s random number table, and continueinto the fourth column as needed.into the fourth column as needed.

•• Because the finite population has 900 elements, weBecause the finite population has 900 elements, wewill need 3will need 3--digit random numbers to randomlydigit random numbers to randomlyselect applicants numbered from 1 to 900.select applicants numbered from 1 to 900.




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

column 4 of the random number table, encounteringcolumn 4 of the random number table, encountering

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

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

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

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

•• the random number is greater than 900 orthe random number is greater than 900 or

•• the random number has already been used.the random number has already been used.

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■■ Use of Random Numbers for SamplingUse of Random Numbers for Sampling


744744436436865865790790835835902902190190836836

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

33--DigitDigitRandom 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


■■ Sample DataSample Data


11 744 744 Conrad HarrisConrad Harris 10251025 YesYes22 436436 Enrique RomeroEnrique Romero 950950 YesYes

33 865865 Fabian AvanteFabian Avante 10901090 NoNo

44 790790 Lucila CruzLucila Cruz 11201120 YesYes

55 835835 Chan ChiangChan Chiang 930930 NoNo.. .. .. .. ..

3030 498498 Emily MorseEmily Morse 10101010 NoNo

No.No.RandomRandomNumberNumber ApplicantApplicant

SATSATScoreScore

Live OnLive On--CampusCampus

.. .. .. .. ..

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

•• For example, Excel’s functionFor example, Excel’s function= RANDBETWEEN(1,900)= RANDBETWEEN(1,900)

can be used to generate random numbers betweencan be used to generate random numbers between1 and 900.1 and 900.

•• Computers can be used to generate randomComputers can be used to generate randomnumbers for selecting random samples.numbers for selecting random samples.

Simple Random Sampling:Simple Random Sampling:Using a ComputerUsing a Computer


■■ as Point Estimator of as Point Estimator of µµx

■■ as Point Estimator of as Point Estimator of ppp

29,910997

30 30

ixx = = =∑

2( ) 163,99675.2

29 29

ix xs

−= = =∑

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 σσ

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PopulationPopulationParameterParameter

PointPointEstimatorEstimator

PointPointEstimateEstimate

ParameterParameterValueValue

µµ = Population mean= Population meanSAT score SAT score

990990 997997

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

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

75.275.2

pp = Population pro= Population pro--portion wantingportion wantingcampus 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 meanSAT score SAT score

x

= Sample pro= Sample pro--portion wantingportion wantingcampus housing campus housing

p


■■ 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 µµ..

x The sample data The sample data provide a value forprovide a value forthe sample meanthe sample mean ..x

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

from the population.from the population.

Population Population with meanwith mean

µµ = ?= ?

Sampling Distribution of Sampling Distribution of x

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

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

mean .mean .

x

x


where: where:

µµ = the population mean= the population mean

EE( ) = ( ) = µµx

xExpected Value ofExpected Value of



Finite PopulationFinite Population Infinite PopulationInfinite Population

σ σx n

N n

N= −

−( )

1σ σ

x n=

•• is referred to as the is referred to as the standard error of thestandard error of themeanmean..σ x

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

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

xStandard Deviation ofStandard Deviation of

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Form of the Sampling Distribution of Form of the Sampling Distribution of x

If we use a large (If we use a large (nn >> 30) simple random sample, the30) simple random sample, thecentral limit theoremcentral limit theorem enables us to conclude that theenables us to conclude that thesampling distribution of can be approximated bysampling distribution of can be approximated bya normal distribution.a normal distribution.

x

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.

x


8014.6

30x

n

σσ = = =

( ) 990E x =x

Sampling Distribution of Sampling Distribution of for SAT Scoresfor SAT Scoresx

SamplingSamplingDistributionDistribution

of of x

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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 µµ ? ?

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?

x



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

zz = (1000 = (1000 -- 990)/14.6= .68990)/14.6= .68

PP((zz << .68) = .7517.68) = .7517

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


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Cumulative Probabilities forCumulative Probabilities forthe Standard Normal Distributionthe Standard Normal Distribution

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

. . . . . . . . . . .

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

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

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

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

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

. . . . . . . . . . .



x990990


of of x14.6xσ =

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 ofendpoint ofthe interval.the interval.

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

zz = (980 = (980 -- 990)/14.6= 990)/14.6= -- .68.68

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

= .2483= .2483

= 1 = 1 -- . 7517. 7517

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




x980980 990990

Area = .2483Area = .2483


of of x14.6xσ =

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

PP((--.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 score willThe probability that the sample mean SAT score willbe between 980 and 1000 is:be between 980 and 1000 is:

PP(980 (980 << << 1000) = .50341000) = .5034x


x10001000980980 990990


Area = .5034Area = .5034


of of x14.6xσ =

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Relationship Between the Sample SizeRelationship Between the Sample Sizeand the Sampling Distribution of and the Sampling Distribution of x

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

�� EE( ) = ( ) = µµ regardless of the sample size. In ourregardless of the sample size. In ourexample,example, EE( ) remains at 990.( ) remains at 990.

xx

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

xσ

808.0

100x

n

σσ = = =



( ) 990E x =x

14.6xσ =With With nn = 30,= 30,

8xσ =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.x


�� 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.

x

�� Now, with Now, with nn = 100, = 100, PP(980 (980 << << 1000) = .7888.1000) = .7888.x

�� Because the sampling distribution with Because the sampling distribution with nn = 100 has a= 100 has asmaller standard error, the values of have lesssmaller standard error, the values of have lessvariability and tend to be closer to the populationvariability and tend to be closer to the populationmean than the values of with mean than the values of with nn = 30.= 30.

x

x



x10001000980980 990990

Area = .7888Area = .7888


of of x8xσ =

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End of Chapter 7, Part AEnd of Chapter 7, Part A

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