Sampling and Sampling Distributions Simple Random Sampling Point Estimation Sampling Distribution.

Sampling and Sampling Distributions

•Simple Random Sampling

•Point Estimation

•Sampling Distribution

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.

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

Simple Random Sampling:Finite Population

• Finite populations are often defined by lists such as:– Organization membership roster

– Credit card account numbers

– Inventory product numbers

A simple random sample of A simple random sample of size size nn from a finite from a finite population of size population of size NN is a is a sample selected such that sample selected such that each possible sample of each possible sample of size size nn has the same has the same probability of being probability of being selected.selected.

A Good Sample is a Random Sample

Simple Random Sampling:Finite 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..

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

Simple Random Sampling:Infinite 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.

Simple Random Sampling:Infinite Population

In the case of infinite In the case of infinite populations, it is impossible to populations, it is impossible to obtain a list of all elements in the obtain a list of all elements in the population. The random number population. The random number selection procedure selection procedure cannotcannot be be used for infinite populations.used for infinite populations.

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

Sampling 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..

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

Example: St. Andrew’s

St. Andrew’s College receives

900 applications annually from

prospective students. The

application form contains

a variety of information

including the individual’s

scholastic aptitude test (SAT) score and whether or not

the individual desires on-campus housing.

Example: St. Andrew’s

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 Excel

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.

• Population Mean SAT Score

• Population Standard Deviation for SAT Score

• Population Proportion Wanting On-Campus Housing

990900

ix 990

900ix

2( )80

900ix

2( )80

900ix

Conducting a Census

648.72

900p

648.72

900p

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.

• Taking a Sample of 30 Applicants

Simple Random Sampling:Using a Random Number Table

• We will use the We will use the lastlast three digits of the 5- three digits of the 5-digitdigit random numbers in the random numbers in the thirdthird column of column of thethe textbook’s random number table, and textbook’s random number table, and continuecontinue into the fourth column as needed. into the fourth column as needed. (See Table 7.1, p. 275)(See Table 7.1, p. 275)

• 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.



• (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.

Use 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

See 3rd column of Table 7.1, p. 275

• Sample 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

.. . . . . . . . .


Using Excel to Selecta 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.


• 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.

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

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


Put Random Numbers in Ascending OrderPut Random Numbers in Ascending Order

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

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


• 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


Point Estimates

xx

• as Point Estimator of

• s as Point Estimator of

• as Point Estimator of ppp

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.

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 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 populationfrom the population..

Population Population with meanwith mean

= ?= ?

Sampling Distribution of Sampling Distribution of xx

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

Sampling Distribution of

x x

x

The sampling distribution of The sampling distribution of is the probability distribution is the probability distribution of all possible values of the of all possible values of the sample meansample mean

x

)(xE


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


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

Central Limit Theorem

In selecting simple random samples of size n, the sampling distribution of the mean can be approximated by a normal probability distribution as the sample size becomes large.

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((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.



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

. . . . . . . . . . .


xx990990


of of xx14.6x 14.6x

10001000

Area = .7517Area = .7517

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)



xx980980 990990

Area = .2483Area = .2483


of of xx14.6x 14.6x


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

xx10001000980980 990990


Area = .5034Area = .5034


of of xx14.6x 14.6x

Sampling and Sampling Distributions Simple Random Sampling Point Estimation Sampling Distribution.

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