chapter 7 sampling and sampling distributions sampling distribution of sampling distribution of...

Chapter 7 Chapter 7 Sampling and Sampling DistributionsSampling and Sampling Distributions

xx Sampling Distribution ofSampling Distribution of

Introduction to Sampling DistributionsIntroduction to Sampling Distributions

Point EstimationPoint Estimation

Simple Random SamplingSimple Random Sampling

Sampling Distribution of Sampling Distribution of pp

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

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

Statistical InferenceStatistical Inference

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

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

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.

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 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.

Simple Random Sampling:Simple Random Sampling:Finite PopulationFinite Population

Computer-generated Computer-generated random numbersrandom numbers are often used are often used to automate the sample selection process.to automate the sample selection process.

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

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.

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 ..


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

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.

The difference between an unbiased point estimate and The difference between an unbiased point estimate and the corresponding population parameter is called thethe corresponding 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..

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.

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

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.

Conducting a CensusConducting a Census

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

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

990900

ix 990

900ix

2( )80

900ix

2( )80

900ix

Conducting a CensusConducting a Census

648.72

900p

648.72

900p

Population Mean SAT ScorePopulation Mean SAT Score

Population Standard Deviation for SAT ScorePopulation Standard Deviation for SAT Score

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

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 data on theNow suppose that the necessary data on the current year’s applicants were not yet entered in thecurrent year’s applicants were not yet entered in the college’s database.college’s database.

Use of Random Numbers for SamplingUse of Random Numbers for Sampling

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

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

Sample DataSample Data

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

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

.. . . . . . . . .

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

PopulationPopulationParameterParameter

PointPointEstimatorEstimator

PointPointEstimateEstimate

ParameterParameterValueValue

= Population mean= Population mean SAT score SAT score

990990 997997

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

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

75.275.2

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

.72.72 .68.68

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

= Sample mean= Sample mean SAT score SAT score xx

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

pp

Process of Statistical InferenceProcess of Statistical Inference

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

the value of the value of ..

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

the sample meanthe sample mean . .xx

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

from the population.from the population.

Population Population with meanwith mean

= ?= ?

Sampling Distribution of Sampling Distribution of xx

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

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

mean .mean .

xx

xx


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

EE( ) = ( ) = xx

xxExpected Value ofExpected Value of


Finite PopulationFinite Population Infinite PopulationInfinite Population

x n

N nN

( )1

x n

N nN

( )1

x n

x n

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

x x

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

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

xxStandard Deviation ofStandard Deviation of

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

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

xx

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

xx

8014.6

30x

n

80

14.630

xn

( ) 990E x ( ) 990E x xx

Sampling Distribution of Sampling Distribution of for SAT Scoresfor SAT Scoresxx

SamplingSamplingDistributionDistribution

of of xx

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


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(µ ≤ (µ ≤ zz << .68) = .2517 .68) = .2517

Step 2:Step 2: Find the area under the curve between Find the area under the curve between µµ and Z and Z



xx990990


of of xx14.6x 14.6x

10001000

Area = .2517Area = .2517

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

Step 4:Step 4: Find the area under the curve between µ and Find the area under the curve between µ and lowerlower endpoint. endpoint.

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


PP(µ ≥ (µ ≥ zz ≥ - .68) = .2517 ≥ - .68) = .2517


xx990990


of of xx14.6x 14.6x

10001000

Area = .2517Area = .2517

989800

.2517.2517


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

PP(-.68 (-.68 << zz << .68) = .68) = .2517 + .2517 = .5034 .2517 + .2517 = .5034

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

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

xx10001000980980 990990


Area = .5034Area = .5034


of of xx14.6x 14.6x

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

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

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

xxxx

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

xx

808.0

100x

n

80

8.0100

xn


( ) 990E x ( ) 990E x xx

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

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

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


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

xx

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

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

xx

xx


xx10001000980980 990990

Area = .7888Area = .7888


of of xx8x 8x

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

from the population.from the population.

Population Population with proportionwith proportion

pp = ? = ?

Making Inferences about a Population Making Inferences about a Population ProportionProportion

The sample data The sample data provide a value for provide a value for

thethesample sample

proportionproportion . .

pp

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

about the value of about the value of pp..

pp

Sampling Distribution ofSampling Distribution ofpp

E p p( ) E p p( )


where:where:pp = the population proportion = the population proportion

The The sampling distribution of sampling distribution of is the probability is the probabilitydistribution of all possible values of the sampledistribution of all possible values of the sampleproportion .proportion .pp

pp

ppExpected Value ofExpected Value of

pp pn

N nN

( )11

pp pn

N nN

( )11

pp pn

( )1 pp pn

( )1

is referred to as the is referred to as the standard error standard error of theof theproportionproportion..

p p


Finite PopulationFinite Population Infinite PopulationInfinite Population

ppStandard Deviation ofStandard Deviation of

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:

npnp >> 5 5 nn(1 – (1 – pp) ) >> 5 5andand

Form of the Sampling Distribution ofForm of the Sampling Distribution ofpp

Recall that 72% of theRecall that 72% of the

prospective students applyingprospective students applying

to St. Andrew’s College desireto St. Andrew’s College desire

on-campus housing.on-campus housing.

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


What is the probability thatWhat is the probability that

a simple random sample of 30 applicants will providea simple random sample of 30 applicants will provide

an estimate of the population proportion of applicantan estimate of the population proportion of applicant

desiring on-campus housing that is within plus ordesiring on-campus housing that is within plus or

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

p

.72(1 .72).082

30

p

.72(1 .72).082

30

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


of of pp


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

zz = (.77 = (.77 .72)/.082 = .61 .72)/.082 = .61

PP(µ ≤ (µ ≤ zz << .61) = .2291 .61) = .2291

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


.77.77.72.72

Area = .2291Area = .2291

pp


of of pp

.082p .082p


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

Step 4:Step 4: Find the area under the curve between µ and the Find the area under the curve between µ and the lowerlower endpoint. endpoint.

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

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

= .2291= .2291


.77.77.72.72

Area = .2291Area = .2291

pp


of of pp

.082p .082p


..22912291

.67.67

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

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

PP(-.61 (-.61 << zz << .61) = .61) = .2291 + .2291.2291 + .2291

= .4582= .4582

The probability that the sample proportion of applicantsThe probability that the sample proportion of applicantswanting on-campus housing will be within +/-.05 of thewanting on-campus housing will be within +/-.05 of theactual population proportion :actual population proportion :


.77.77.67.67 .72.72

Area = .4582Area = .4582

pp


of of pp

.082p .082p
