Sampling and Sample Designs

8/8/2019 Sampling and Sample Designs

1/69

Research Methods in Research Methods in Economics Economics

ECO 4451ECO 4451

Sampling and Statistical TestingSampling and Statistical Testing

www.solvedassignments.net www.solvedassignments.net


2/69

Sampling TerminologySampling TerminologyPopulationPopulation

The complete set of items of interest The complete set of items of interest

Population elementPopulation element

An individual member of population An individual member of populationCensusCensus

A complete enumeration of all elements in population A complete enumeration of all elements in population

SampleSample A subset of the population selected for investigation A subset of the population selected for investigation



3/69

Terminology (Cont) Terminology (Cont)F rameF rame

P opulation P opulation frame = list of all elements inframe = list of all elements inpopulationpopulationS ample S ample frame = list of elements from whichframe = list of elements from whichsample will be drawnsample will be drawn



4/69

Wh y Sample (not census)? Wh y Sample (not census)?

CostCostSufficiently accurate for most purposes if wellSufficiently accurate for most purposes if well

designed probability sampledesigned probability sampleSometimes decrease in accuracy from attempt toSometimes decrease in accuracy from attempt tomake complete censusmake complete census

Destruction of sample unitsDestruction of sample units



5/69

Wh y Sample (cont)? Wh y Sample (cont)?

B ut sampling introduces error in that it isB ut sampling introduces error in that it is virtually impossible for a sample to perfectly virtually impossible for a sample to perfectly

represent the population from which it wasrepresent the population from which it wasdrawn.drawn. Two categories of errors: Two categories of errors:

NonNon--sampling errorsampling errorSampling errorSampling error



6/69

R epresentative?R epresentative?H ow well does the sample represent theH ow well does the sample represent thepopulation?population?

Population SamplePopulation SampleP arameters P arameters S tatistics S tatistics

( , ) Q W ( , ) X s

Estimation www.solvedassignments.net www.solvedassignments.net


7/69

Wh ats a population? Wh ats a population? Technically the population is the complete set Technically the population is the complete setof elements of interest.of elements of interest.

F or example, in a study of corporate profits, theF or example, in a study of corporate profits, thepopulation is the set of profits of all corporationspopulation is the set of profits of all corporations

We think of univariate, bivariate or We think of univariate, bivariate ormultivariate populations.multivariate populations.If we are interested in whether profits are relatedIf we are interested in whether profits are relatedto CEO compensation, we have a bivariateto CEO compensation, we have a bivariate

population where the elements are the sets of population where the elements are the sets of pairs of profits & compensation.pairs of profits & compensation.



8/69


9/69

A ccuracy is A ccuracy is

The degree to which bias is absent from the The degree to which bias is absent from the

estimator.estimator.

To have Accuracy To have AccuracyOverestimates and Underestimates mustOverestimates and Underestimates mustbalance out in repeated sampling.balance out in repeated sampling.



10/69

P recisionP recision

Is low sampling error.Is low sampling error.Repeated samples would yield similar estimates.Repeated samples would yield similar estimates.

Is measured by the standard error of estimate, aIs measured by the standard error of estimate, atype of standard deviation measurement we willtype of standard deviation measurement we willdiscuss later.discuss later.



11/69

Errors from Investigating a SampleErrors from Investigating a Sample

(rath

er th

an a census)(rath

er th

an a census)Nonsampling (systematic) errorNonsampling (systematic) error

Results from some imperfection in research designResults from some imperfection in research designor mistakes in execution of design.or mistakes in execution of design.Sampling frame errorSampling frame errorNonNon--response biasresponse biasResponse or recording errorResponse or recording error



12/69

Systematic (Nonsampling) ErrorsSystematic (Nonsampling) Errors

Sampling frame errorSampling frame errorSome population elements not represented in sampling frameSome population elements not represented in sampling frame

NonNon--response errorresponse error When results are affected because some elements selected When results are affected because some elements selectedinto sample do not respond or are not measuredinto sample do not respond or are not measured

Response or recording errorResponse or recording errorErrors in making or recording responses or measurementsErrors in making or recording responses or measurements



13/69

Errors from SampleErrors from Samplerat h er t h an census (Cont)rat h er t h an census (Cont)

Sampling (random) errorSampling (random) errorDifference between sample statistic and populationDifference between sample statistic and populationparameter that results from chance variation in elementsparameter that results from chance variation in elements

selected for inclusion in sample.selected for inclusion in sample. Two determinants of sampling error Two determinants of sampling error

H omogeneity (larger sampling error) vs. heterogeneity H omogeneity (larger sampling error) vs. heterogeneity (smaller sampling error) of population(smaller sampling error) of population

Sample size (larger sample reduces sampling error)Sample size (larger sample reduces sampling error)



14/69

ErrorsErrors

Target Population Target Population

Sampling F rameSampling F rame

Planned SamplePlanned Sample

Actual Sample Actual Sample

S ampling frame error

Nonresponse error

S ampling error



15/69

Define the target population

Select a sampling frame

Conduct fieldwork

Determine if a probability or nonprobabilitysampling method will be chosen

Plan procedurefor selecting sampling units

Determine sample size

Select actual sampling units

S tages in theS electionof a S ample



16/69

Sampling UnitsSampling Units

A single element or group of elements subject to A single element or group of elements subject toselection in sample.selection in sample.

When sampling occurs in one stage, the elements When sampling occurs in one stage, the elementsselected in the sample are the sampling units.selected in the sample are the sampling units.

Example: simple random sample of college students.Example: simple random sample of college students.

In multiIn multi--stage sampling we distinguishstage sampling we distinguishPrimary Sampling Units (PSU)Primary Sampling Units (PSU) first or topfirst or top--levellevelSecondary Sampling UnitsSecondary Sampling Units second levelsecond level

Tertiary Sampling Units Tertiary Sampling Units third.third.



17/69

Sampling Units (Cont)Sampling Units (Cont)

M ultiM ulti--stage sampling stage sampling Primary, secondary, tertiary sampling unitsPrimary, secondary, tertiary sampling units

Example: first select a region (PSU), thenExample: first select a region (PSU), thencolleges within region (SSU), then students atcolleges within region (SSU), then students atthe colleges (TSU).the colleges (TSU).



18/69

Two Major Categories of Sampling Two Major Categories of Sampling

Probability sampling Probability sampling Known, nonzero probability for selecting any Known, nonzero probability for selecting any element from sampling frameelement from sampling frame

This probability may be same or different for This probability may be same or different fordifferent elements.different elements.Sampling error can be estimatedSampling error can be estimated

Nonprobability sampling Nonprobability sampling

Probability of selecting any particular elementProbability of selecting any particular elementof population is unknownof population is unknownSampling error is unknownSampling error is unknown



19/69

Nonprobability SamplingNonprobability Sampling

ConvenienceConvenience Judgment Judgment

QuotaQuotaSnowballSnowball



20/69

P robability SamplingP robability Sampling

Simple random sampleSimple random sampleSystematic sampleSystematic sample

Stratified sampleStratified sampleCluster sampleCluster sampleM ultistage cluster sampleM ultistage cluster sample


21/69

Wh at is t h e Wh at is t h e A

ppropriate Sample Design? A

ppropriate Sample Design?Degree of accuracy & precisionDegree of accuracy & precisionResources available, including time.Resources available, including time.

Advanced knowledge of the population Advanced knowledge of the populationNational versus localNational versus localNeed for statistical analysisNeed for statistical analysis



22/69

Statistical A nalysis of SamplesStatistical A nalysis of Samples

Descriptive statisticsDescriptive statisticsDescribe characteristics of sampleDescribe characteristics of sample

Using sample statistics, like measures of centralUsing sample statistics, like measures of centraltendency and dispersion, to describe a sample of tendency and dispersion, to describe a sample of observations.observations.

Inferential statisticsInferential statisticsM ake an inference about an unknown populationM ake an inference about an unknown populationfrom a samplefrom a sampleEstimation and hypothesis testing.Estimation and hypothesis testing.



23/69

Descriptive StatisticsDescriptive StatisticsM easures of central tendency M easures of central tendency

M

ean, median, modeM

ean, median, modeM easures of dispersionM easures of dispersion Variance (or standard deviation), range Variance (or standard deviation), range

M easures of frequency M easures of frequency Counts, proportionsCounts, proportions

Often presented in a table.Often presented in a table.Possibly separate by different groups or subPossibly separate by different groups or sub- -samples, particularly if samples, particularly if your paper involves a comparison between groups.your paper involves a comparison between groups.

Usually some brief discussion of the descriptive statistics isUsually some brief discussion of the descriptive statistics isappropriate.appropriate.

Give the reader some idea about the type of units in the sample.Give the reader some idea about the type of units in the sample.Give the reader a feel for the scale of the data.Give the reader a feel for the scale of the data.Give information about the amount of variation.Give information about the amount of variation.

Inspection of descriptive statistics often reveals the source of Inspection of descriptive statistics often reveals the source of problems you may be having with statistical procedures.problems you may be having with statistical procedures.


24/69

Frequency (number of

people making deposits Amount in each range)

less than $3,000 499$3,000 - $4,999 530$5,000 - $9,999 562$10,000 - $14,999 718$15,000 or more 811

3,120

F requency Distribution of F requency Distribution of

DepositsDeposits



25/69

Amount Percentless than $3,000 16$3,000 - $4,999 17$5,000 - $9,999 18$10,000 - $14,999 23

$15,000 or more 26100

P ercentage Distribution of P ercentage Distribution of A

mounts of Deposits A

mounts of Deposits



26/69

Amount Probabilityless than $3,000 .16$3,000 - $4,999 .17$5,000 - $9,999 .18$10,000 - $14,999 .23

$15,000 or more .261.00

P robability Distribution of P robability Distribution of A

mounts of Deposits A

mounts of Deposits



27/69

Measures of Central TendencyMeasures of Central Tendency

MeanMean -- arithmetic averagearithmetic average, Population; , sample, Population; , sample

MedianMedian -- midpoint of the distributionmidpoint of the distributionModeMode -- the value that occurs most oftenthe value that occurs most often

X



28/69

P opulation MeanP opulation Mean

( ) E X Q !

Av erage v alue in population.



29/69

n X X i7!

Sample MeanSample Mean

W here n denotes the total number of elements in sample.



30/69

Number of Salesperson Sales callsMike 4Patty 3Billie 2

Bob 5John 3Frank 3Chuck 1Samantha 5

26

Daily Sales Calls by SalespersonsDaily Sales Calls by Salespersons

S ample mean=3.25 , median=3 , mode=3.

Range=4 , I-q Range=1 , Variance=1.93 , S td.De v .=1.39



31/69

Measures of DispersionMeasures of Dispersion

or Spreador SpreadRangeRangeM ean absolute deviationM ean absolute deviation

Variance VarianceStandard deviationStandard deviation



32/69

Product A Product B

196 150198 160199 176199 181200 192200 200200 201201 202201 213

201 224202 240202 261

Sales for P roducts A and B, Bot h A verage 200Sales for P roducts A and B, Bot h A verage 200

B ut sales of product B hav e greater v ariability. www.solvedassignments.net www.solvedassignments.net


33/69

150 160 170 180 190 200 210

5

4

3

2

1

Low Dispersion

Value of Variable

L ow Dispersion VsL ow Dispersion Vs

High

DispersionHigh

Dispersion


34/69

150 160 170 180 190 200 210

5

4

3

2

1

High dispersion

Value of Variable

L ow Dispersion VsL ow Dispersion Vs

Hig h DispersionHig h Dispersion


35/69

Deviation ScoresDeviation Scores

The differences between each observation value The differences between each observation valueand the mean:and the mean:

)( X X d ii !



36/69

A verage A verage DeviationDeviation

n

X X i )(



37/69

Mean Squared DeviationMean Squared Deviation

n

X X i2

)(



38/69

Variance = Mean SquaredDeviation Variance = Mean SquaredDeviation

2

2

S Sample

Populat ion

W



39/69

Sample VarianceSample Variance

22 )1

i X X S n

7 !



40/69

Variance Variance

The variance is given in squared units The variance is given in squared units The standard deviation is the square root of The standard deviation is the square root of

variance, and so is in original units. variance, and so is in original units.



41/69

P opulation Standard DeviationP opulation Standard Deviation

2W W !



42/69

Sample Standard DeviationSample Standard Deviation

2S S ! www.solvedassignments.net www.solvedassignments.net


43/69

Sample Standard DeviationSample Standard Deviation

1

2 7!

n

X X iS www.solvedassignments.net www.solvedassignments.net


44/69


45/69

DistributionsDistributions

Population distributionPopulation distribution frequency frequency distribution of elements in populationdistribution of elements in populationSample distributionSample distribution -- frequency distributionfrequency distributionof elements in sampleof elements in sampleSampling distributionSampling distribution theoreticaltheoreticaldistribution of a sample statistic indistribution of a sample statistic in repeated repeated sampling.sampling.

Key concept in inferential statistics.Key concept in inferential statistics.Example: sampling distribution of sample meanExample: sampling distribution of sample meanis normal.is normal.



46/69

Q x

P opulation DistributionP opulation Distribution



47/69

C' XS

Sample DistributionSample Distribution



48/69

X S X X Q

Sampling DistributionSampling Distribution



49/69

T h e Normal Distribution T h e Normal Distribution

Describes the probability distribution expectedDescribes the probability distribution expectedof many random occurrences.of many random occurrences.B

ell shaped curveB

ell shaped curve Almost all of its values are within plus or minus Almost all of its values are within plus or minus3 standard deviations3 standard deviationsI.Q. is an exampleI.Q. is an example



50/69

2.14%

13.59% 34.13% 34.13% 13.59%

2.14%

Normal DistributionNormal Distribution



51/69

85 115100 14570

Normal Curve: IQ ExampleNormal Curve: IQ Example


52/69

Standardized Normal DistributionStandardized Normal DistributionSymmetrical about its meanSymmetrical about its meanM ean identifies highest pointM ean identifies highest pointInfinite number of casesInfinite number of cases - - a continuous distributiona continuous distribution

Area under curve has a probability density = 1.0 Area under curve has a probability density = 1.0M ean of zero, standard deviation of 1M ean of zero, standard deviation of 1



53/69

Standard Normal CurveStandard Normal Curve

The curve is bell The curve is bell--shaped or symmetricalshaped or symmetrical About 68% of the elements will fall within 1 About 68% of the elements will fall within 1standard deviation of the meanstandard deviation of the mean

About 95% of the elements will fall within About 95% of the elements will fall withinapproximately 2 (i.e., 1.96) standard deviationsapproximately 2 (i.e., 1.96) standard deviationsof the meanof the mean

Almost all (>99%) of the elements will fall Almost all (>99%) of the elements will fall within 3 standard deviations of the mean within 3 standard deviations of the mean



54/69

01-1-2

2 z

A Standardized Normal Curve A Standardized Normal Curve


55/69

T h e Standardized Normal is t h e T h e Standardized Normal is t h e

Distribution of ZDistribution of Z

z +z


56/69

W !

x z

P opulation Standardized ScoresP opulation Standardized Scores



57/69

W Q

!x

z

Standardized ValuesStandardized Values

Used to compare an individual value to theUsed to compare an individual value to thepopulation mean in units of the standardpopulation mean in units of the standarddeviationdeviation



58/69

L inear Transformation of A ny NormalL inear Transformation of A ny Normal Variable Into a Standardized Normal Variable Variable Into a Standardized Normal Variable

-2 -1 0 1 2

Sometimes the

distribution is stretchedSometimes the

distribution is shrunk

QQ X

Q!

x z


59/69

Central L imit T h eoremCentral L imit T h eorem

The CLT says that if the sample size The CLT says that if the sample size n n is large,is large,On averageOn average across repeated samples, the mean of sampleacross repeated samples, the mean of samplemeans equals the population mean.means equals the population mean.

The variance of the sample means across different samples The variance of the sample means across different samples

equals the population variance divided by equals the population variance divided by n n .. The distribution of sample means across different sample The distribution of sample means across different sampleis normal.is normal.



60/69

P opulation P arameters andP opulation P arameters andSample StatisticsSample Statistics

D i s t r i b u t i o n M e a n S t a n d a r dD e v i a t i o n

P o p u l a t i o n Q W S a m p l e X S

S a m p l i n g X

Q X S



61/69


62/69

Overview Overview

Tests concerning population means. Tests concerning population means.One sample test.One sample test.

Two independent samples test. Two independent samples test.K > 2 independent samples.K > 2 independent samples.M atched samples.M atched samples.

Tests concerning population proportions. Tests concerning population proportions.One sample.One sample.M ore general tests.M ore general tests.



63/69

Examples of Difference betweenExamples of Difference betweenMeans TestsMeans Tests

Consider Does the Death Penalty Deter M urder?Consider Does the Death Penalty Deter M urder?by Tammra H unt.by Tammra H unt.Compare murder rates (# per 100,000) with andCompare murder rates (# per 100,000) with and

without death penalty. without death penalty.CrossCross--section of states: is murder rate higher on averagesection of states: is murder rate higher on averagein states without executions?in states without executions?

Time series of states: did the murder rate fall in states Time series of states: did the murder rate fall in statesimplementing the death penalty when allowed by implementing the death penalty when allowed by Supreme Court?Supreme Court?Panel data allows an approach based on differencesPanel data allows an approach based on differences- -inin--differences .differences .



64/69

BetweenBetween- -State Differences: 2003State Differences: 2003Consider two populations, A (with death penalty) andConsider two populations, A (with death penalty) andB (without death penalty).B (without death penalty).

0

,

: . (S

o0

.): . (S o 0 .)

iA A iA iB B iB

A B A B

A A B A B

y e y e

H H

Q Q

Q Q H Q Q Q Q H Q Q

! !

! ! !!

Note the alternati v e is one-sided , because theresearch hypothesis is that the death penalty detersmurder.



65/69

Testing t h e null h ypot h esis Testing t h e null h ypot h esis

The test can be conducted by computing the The test can be conducted by computing thett--statistic (note one sample size is less thanstatistic (note one sample size is less than30) manually.30) manually.

Or it can be conducted automatically using Or it can be conducted automatically using statistical software, or Excel.statistical software, or Excel.In Excel, select Tools,In Excel, select Tools,

Data Analysis,Data Analysis,tt--Test: Two Sample Test Assuming Equal Variances Test: Two Sample Test Assuming Equal Variances



66/69

BetweenBetween- -State Differences: 2003State Differences: 2003

t-Tes t: T o-Sa ple Assu ing Equal Variancesea th Penal ty No Dea th Penal tyurder Rat e urder Rat e

ea n 5 328947368 2 883333333Va ria nce 6 279409673 2 614242424Observ at ions 38 12

Pooled Va ria nce 5 439475512ypo thesized Mea n Di erence 0d 48t S tat 3 166699287P(T


67/69


68/69

Testing t h e wit h in Testing t h e wit h in--state differencestate difference

The test can be conducted manually. The test can be conducted manually.Compute the AfterCompute the After B efore difference for each stateB efore difference for each state

with the death penalty. with the death penalty.Get the sample mean and variance of these differences.Get the sample mean and variance of these differences.

Test the null hypothesis that the difference is zero against Test the null hypothesis that the difference is zero againstthe alternative that it is negative.the alternative that it is negative.

Or it can be conducted automatically using statisticalOr it can be conducted automatically using statisticalsoftware, or Excel.software, or Excel.

In Excel, select Tools,In Excel, select Tools,Data Analysis,Data Analysis,

tt--Test: Paired Two Sample for M eans Test: Paired Two Sample for M eans



69/69

W ith in W ith in--State DifferencesState Differences

t-Test: P ired T o S mple for e ns

Before 4-year Mean After 4-year MeanM e n 8.913157895 8.768421053V ri nce 14.21759246 15.29465149Observ tions 38 38Pe rson orrel tion 0.934731402Hypothesized M e n ifference 0df 37t St t 0.639816975P(T

Sampling and Sample Designs

Documents

Transcript of Sampling and Sample Designs