1 Chapter 7 Sampling and Sampling Distributions Simple Random Sampling Point Estimation Introduction...

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Chapter 7Sampling and Sampling Distributions

Simple Random SamplingPoint EstimationIntroduction to Sampling DistributionsSampling Distribution of Sampling Distribution ofSampling Methods

xxpp nn = 100 = 100

nn = 30 = 30

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Statistical Inference

The purpose of statistical inference is to obtain information about a population from information contained in a sample.A population is the set of all the elements of interest.A ________ is a subset of the population.The sample results provide only ________ of the values of the population characteristics.A parameter is a numerical characteristic of a population.With proper sampling methods, the sample results will provide “good” estimates of the population characteristics.

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Simple Random Sampling

Finite Population– A simple random sample from a finite

population of size N is a sample selected such that each possible sample of size n has the same probability of being selected.

– Replacing each sampled element before selecting subsequent elements is called sampling with ___________.

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Finite Population– Sampling without ___________ is the

procedure used most often.– In large sampling projects, computer-

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

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Infinite Population– A simple random sample from an infinite

population is a sample selected such that the following conditions are satisfied.• Each element selected comes from the

same population.• Each element is selected independently.


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Infinite Population– The population is usually considered infinite

if it involves an ongoing process that makes listing or counting every element impossible.

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


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Point Estimation

In point estimation we use the data from the sample to compute a value of a sample statistic that serves as an estimate of a population parameter.We refer to as the _____________ of the population mean .s is the point estimator of the population standard deviation . is the point estimator of the population proportion p.

xx

pp

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Point Estimation

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

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Sampling Error

The absolute difference between an unbiased point estimate and the corresponding population parameter is called the sampling error.Sampling error is the result of using a subset of the population (the sample), and not the entire population to develop estimates.The sampling errors are:

for sample mean for sample standard

deviationfor sample proportion

| |x | |x

| |s | |s

| |p p| |p p

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Example: St. Andrew’s

St. Andrew’s College receives 900 applicationsannually from prospective students. The

applicationforms contain a variety of information including

theindividual’s scholastic aptitude test (SAT) score

andwhether or not the individual desires on-campus housing.

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The director of admissions would like to know the

following information:– the average SAT score for the applicants,

and– the proportion of applicants that want to

live on campus.

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We will now look at three alternatives for obtaining

the desired information.– Conducting a census of the entire 900

applicants– Selecting a sample of 30 applicants, using a

random number table– Selecting a sample of 30 applicants, using

computer-generated random numbers

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Taking a Census of the 900 Applicants– SAT Scores

• Population Mean

• Population Standard Deviation

ix 990900

ix 990900

ix

2( )80

900

ix

2( )80

900


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Taking a Census of the 900 Applicants– Applicants Wanting On-Campus Housing

• Population Proportion

p648

.72900

p648

.72900

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Take a Sample of 30 ApplicantsUsing a Random Number Table

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

We will use the last three digits of the 5-digit random numbers in the third column of the textbook’s random number table (________________).

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Take a Sample of 30 ApplicantsUsing a Random Number Table

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

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

We will continue to draw random numbers until we

have selected _______ applicants for our sample.

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Use of Random Numbers for Sampling

3-Digit Applicant Random Number Included in Sample

744 No. 744 436 No. 436 865 No. 865 790 No. 790 835 No. 835 902 Number exceeds

900 190 No. 190 436 Number already

used etc. etc.


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Sample Data

RandomNo. Number Applicant SAT Score On-Campus 1 744 Connie Reyman 1025

Yes 2 436 William Fox 950 Yes 3 865 Fabian Avante 1090 No 4 790 Eric Paxton 1120

Yes 5 835 Winona Wheeler 1015

No . . . . . 30 685 Kevin Cossack 965 No


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Take a Sample of 30 ApplicantsUsing Computer-Generated Random Numbers– Excel provides a function for generating

random numbers in its worksheet.– 900 random numbers are generated, one for

each applicant in the population.– Then we choose the ____ applicants

corresponding to the 30 smallest random numbers as our sample.

– Each of the 900 applicants have the same probability of being included.

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Using Excel to Selecta Simple Random Sample

Formula 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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n Value WorksheetValue WorksheetA B C D

1Applicant Number

SAT Score

On-Campus Housing

Random Number

2 1 1008 Yes 0.413273 2 1025 No 0.795144 3 952 Yes 0.662375 4 1090 Yes 0.002346 5 1127 Yes 0.712057 6 1015 No 0.180378 7 965 Yes 0.716079 8 1161 No 0.90512


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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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Point Estimates– as Point Estimator of

– s as Point Estimator of

– as Point Estimator of p

xx

pp

ixx29,910

99730 30

ixx29,910

99730 30

ix x

s2( ) 163,996

75.229 29

ix x

s2( ) 163,996

75.229 29

p 20 30 .68 p 20 30 .68


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Point Estimates

Note: Different random numbers would have identified a different sample which would have resulted in different point estimates.

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Sampling Distribution of

Process of Statistical Inference

Population Population with meanwith mean

= ?= ?

Population Population with meanwith mean

= ?= ?

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

from the population.from the population.

xx

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

the sample meanthe sample mean . .

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

the sample meanthe sample mean . .xx

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

the value of the value of ..

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

the value of the value of ..

xx

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The sampling distribution of is the probability distribution of all possible values of the sample mean .Expected Value of

E( ) =

where: = the population mean

What does this imply?– The sample mean (from a simple random

sample) is an unbiased estimator of population mean.


xx

xxxx

xx

xx

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: Standard Deviation of: Standard Deviation of

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

Finite PopulationFinite Population Infinite Infinite Population Population

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

• is the finite correction is the finite correction factor.factor.

x n

N nN

( )1

x n

N nN

( )1

x n

x n

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

xx

xx

x x

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If we use a large (n > 30) simple random sample, the central limit theorem enables us to conclude that the sampling distribution of can be approximated by a normal probability distribution.

When the simple random sample is small (n < 30), the sampling distribution of can be considered normal only if we assume the population has a normal probability distribution.

xx

xx

Sampling Distribution of xx

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Sampling Distribution of for the SAT Scores


xn

8014.6

30

xn

8014.6

30

E x( ) 990E x( ) 990

xx

xx

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Sampling Distribution of for the SAT ScoresWhat is the probability that a simple

random sample of 30 applicants will provide an estimate of the population mean SAT score that is within plus or minus 10 of the actual population mean ?

In other words, what is the probability that will be between _____ and ______?

xx


xx

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xx

Sampling distribution of

Sampling distribution of xx

10001000980980 990990

Area = ?Area = ?

xx

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Using the standard normal probability table with z = 10/14.6= .68, we have area = (.2517)(2) =

The probability is _______ that the sample mean will be within +/-10 of the actual population mean.


xx

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xx


Sampling distribution of xx

10001000980980 990990

Area = 2(.2517) = .5034Area = 2(.2517) = .5034

xx

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The sampling distribution of is the probability distribution of all possible values of the sample proportion .

Expected Value of

where:p = the population proportion


pp

pp

pp

E p p( ) E p p( )

pp

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Standard Deviation of

Finite Population Infinite Population

– is referred to as the standard error of the proportion.


pp pn

N nN

( )11

pp pn

N nN

( )11

pp pn

( )1 pp pn

( )1

p p

pp

pp

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The sampling distribution of can be approximated by a normal probability distribution whenever the sample size is large.The sample size is considered large whenever these conditions are satisfied:

np > 5 and n(1 – p) > 5


pp

pp

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n For values of For values of pp near .50, sample sizes as small near .50, sample sizes as small as 10 permit a normal approximation.as 10 permit a normal approximation.

n With very small (approaching 0) or large With very small (approaching 0) or large (approaching 1) values of (approaching 1) values of pp, much larger , much larger samples are needed.samples are needed.

ppSampling Distribution of

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Sampling Distribution of for Applicants Desiring On-campus Housing

The normal probability distribution is an acceptable approximation because:

np = 30(.72) = 21.6 > 5 and

n(1 - p) = 30(.28) = 8.4 > 5.

pp

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p

.72(1 .72).082

30

p

.72(1 .72).082

30

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

pp

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What is the probability that a simple random sample of 30 applicants will provide an estimate of the population proportion of applicants desiring on-campus housing that is within plus or minus .05 of the actual population proportion?

In other words, what is the probability that will be between ______ and ______?

pp


pp

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0.770.770.670.67 0.720.72

Area = ?Area = ?

pp

pp

pp

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For z = .05/.082 = .61, the area = (.2291)(2) = ______.The probability is _______ that the sample proportion will be within +/-.05 of the actual population proportion.


pp

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Sampling Distribution of for In-State Residents



0.770.770.670.67 0.720.72

Area = 2(.2291) = .4582Area = 2(.2291) = .4582pp

pp

pp

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Sampling Methods

Stratified Random SamplingCluster SamplingSystematic SamplingConvenience SamplingJudgment Sampling

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Stratified Random Sampling

The population is first divided into groups of elements called strata.Each element in the population belongs to one and only one stratum.Best results are obtained when the elements within each stratum are as much alike as possible (i.e. homogeneous group).A simple random sample is taken from each stratum.Example: The basis for forming the strata might be department, location, age, industry type, etc.

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Stratified Random Sampling

Advantage: If strata are homogeneous, this method is as “precise” as simple random sampling but with a smaller total sample size.Formulas are available for combining the stratum sample results into one population parameter estimate.

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Cluster Sampling

The population is first divided into separate groups of elements called clusters.Ideally, each cluster is a representative small-scale version of the population (i.e. heterogeneous group).A simple random sample of the clusters is then taken.All elements within each sampled (chosen) cluster form the sample.Example: A primary application is area sampling, where clusters are city blocks or other well-defined areas.

… continued

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Cluster Sampling

Advantage: The close proximity of elements can be cost effective (I.e. many sample observations can be obtained in a short time).Disadvantage: This method generally requires a larger total sample size than simple or stratified random sampling.

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Systematic Sampling

If a sample size of n is desired from a population containing N elements, we might sample one element for every N/n elements in the population.We randomly select one of the first N/n elements from the population list.We then select every N/nth element that follows in the population list.This method has the properties of a simple random sample, especially if the list of the population elements is a random ordering.

… … continuedcontinued

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Systematic Sampling

Advantage: The sample usually will be easier to identify than it would be if simple random sampling were used.Example: Selecting every 100th listing in a telephone book after the first randomly selected listing.

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Convenience Sampling

It is a nonprobability sampling technique. Items are included in the sample without known probabilities of being selected.The sample is identified primarily by convenience.Advantage: Sample selection and data collection are relatively easy.Disadvantage: It is impossible to determine how representative of the population the sample is. Example: A professor conducting research might use student volunteers to constitute a sample.

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Judgment Sampling

The person most knowledgeable on the subject of the study selects elements of the population that he or she feels are most representative of the population.It is a nonprobability sampling technique.Advantage: It is a relatively easy way of selecting a sample.Disadvantage: The quality of the sample results depends on the judgment of the person selecting the sample.Example: A reporter might sample three or four senators, judging them as reflecting the general opinion of the senate.

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End of Chapter 7

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1 Chapter 7 Sampling and Sampling Distributions Simple Random Sampling Point Estimation Introduction...

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