test on variables in statistics

8/8/2019 Test on Variables in Statistics

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Test On VariablesTest On Variables

Group 5Group 5

AsimAsim KumarKumar VermaVermaDeeptimanDeeptiman GuhaGuhaHimanshuHimanshu AroraAroraKaavishKaavish KidwaiKidwaiNirmalNirmal ModhModh

In Surveys, the foolish ask questions, wise cannot answersIn Surveys, the foolish ask questions, wise cannot answers..--Oscar WildeOscar Wilde


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Two Types ofVariablesTwo Types ofVariables

Quantitative:Quantitative: Numericvaluethatitmakessenseto do arithmeticNumericvaluethatitmakessenseto do arithmetic

operations (+,operations (+, --, x, /), x, /) Findinganarithmetic average makessenseFindinganarithmetic average makessense

EX:EX:

HeightHeight

WeightWeight AgeAge

IncomeIncome

Test ScoresTest Scores


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Types ofVariablesTypes ofVariables

CategoricalCategorical

Records which ofseveralgroups or categoriesRecords which ofseveralgroups or categoriesto whichanindividualbelongsto whichanindividualbelongs

Ifthereare only 2 possiblecategories,theIfthereare only 2 possiblecategories,thevariableiscalledbinaryvariableiscalledbinary

Ex:Ex: GenderGender

RaceRace

Marital statusMarital status


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Stepsin Hypothesis TestingStepsin Hypothesis Testing

1.1. Hypothesis formulationHypothesis formulation

2.2. SpecifythesignificancelevelSpecifythesignificancelevel

3.3. CollectthedataandcalculatetheteststatisticCollectthedataandcalculatetheteststatistic4.4. FetchthetabulatedvalueFetchthetabulatedvalue

5.5. MakethestatisticaldecisionMakethestatisticaldecision


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Statistical Test

Parametric Non Parametric

When ngreater than 30

When nless than

30


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HypothesistestingHypothesistesting

Goal:Makestatement(s) regarding unknown populationGoal:Makestatement(s) regarding unknown populationparameter valuesbased onsampledataparameter valuesbased onsampledata

Elements ofahypothesistest:Elements ofahypothesistest: NullhypothesisNullhypothesis -- Statementregardingthevalue(s) ofStatementregardingthevalue(s) of

unknown parameter(s). Typically willimplyno associationunknown parameter(s). Typically willimplyno associationbetween explanatoryand responsevariablesin our applicationsbetween explanatoryand responsevariablesin our applications

Alternative hypothesisAlternative hypothesis-- Statementcontradictoryto thenullStatementcontradictoryto thenullhypothesis (willalwayscontainaninequality)hypothesis (willalwayscontainaninequality)

TeststatisticTeststatistic-- Quantitybased onsampledataandnullQuantitybased onsampledataandnull

hypothesis usedto testbetweennullandalternativehypothesis usedto testbetweennullandalternativehypothesishypothesis RejectionregionRejectionregion-- Values oftheteststatistic for whichweValues oftheteststatistic for whichwe

rejectthenullin favor ofthealternativehypothesisrejectthenullin favor ofthealternativehypothesis


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TestOnVariablesTestOnVariables

Inthesedecisionsaremade on quantitativemeasuresInthesedecisionsaremade on quantitativemeasures

T testT test

StatisticalStatistical hypothesishypothesis testtest inin whichwhich thethetesttest statisticstatisticfollowsfollows aa student'sstudent's t t dist ributiondistribution ifif thethe nullnullhypothesishypothesis isis truetruenn > 3030


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1.1. StateStateHH00 HH00:: QuQu3.03.0

2.2. StateStateHH11 HH11 ::QQ

3.3. ChooseChoose EE EE = .05= .05

4.4. ChooseChoose nn n= 100n= 100

5.5. Choose TestChoose Test:: Z testZ test

Hypothesis Testing: Steps

Test the Assumption that the true meangrade point average of juniors is at least 3.


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6. Set Up Critical Value(s)6. Set Up Critical Value(s) Ze= 1.645Ze= 1.645

7. Collect Data7. Collect Data 100 students sampled100 students sampled

8. Compute Test Statistic8. Compute Test Statistic Computed Test Stat.=Computed Test Stat.= --22

9. Make Statistical Decision9. Make Statistical Decision Since Z mod greater than ZeSince Z mod greater than ZeReject NullHypothesisReject NullHypothesis

10. Express Decision10. Express Decision The true mean grade point isThe true mean grade point isless than 3.0less than 3.0

Hypothesis Testing: Steps

Test the Assumption that grade point average ofjuniors is at least 3.

(continued)


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Z0

E

Reject H0

Z0

Reject H0

E

H0: Qu

H1: Q < 0H

0: Qe0

H1: Q > 0

Must BeSignificantlyBelowQ=0

Small values dont contradictH0

Dont Reject H0!

RejectionRegion


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Hypothesis TestHypothesis Test HypothesisHypothesis testingtesting isis usedused whenwhen wewe wantwant toto

knowknow whetherwhether thethe samplesample datadata supportsupport somesomepreconceived preconceived theorytheory wewe holdhold aboutabout thethe

parametersparameters ofof thethe populationpopulation modelmodel.. IfIf wewe cancan findfind aa numericalnumerical valuevalue ofof thethe

relevantrelevant probability,probability, thenthen wewe mightmight bebe ableable totoassessassess whetherwhether thethe beliefbelief isis supportedsupported oror notnot::

aa highhigh probability probability ofof obtainingobtaining thethe datadatasupportssupports thethe belief,belief, aa lowlow oneone doesdoes notnot..


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ttestttest

tt teststests isis simplysimply comparingcomparing twotwo meansmeans toto seesee ifif theythey

areare significantlysignificantly differentdifferent fromfrom eacheach otherother..

TheThe moremore technicaltechnical definitiondefinition oror descriptiondescription ofof aa tt testtestisis anyany statisticalstatistical testtest thatthat usesuses thethe t,t, oror Student'sStudent's t,t,

familyfamily ofof distributionsdistributions..

MostMost EconomistsEconomists thinkthink ofof godgod asas workingworking greatgreat multiplemultiple

regressionsregressions inin thethe skysky EdgarEdgar FiedlerFiedler


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StudentstStudentst--testtestIfIf randomrandom samplessamples ofof sizesize lessless thanthan 3030 areare takentaken fromfrom aa normalnormal distributiondistribution andand thethe samplessamples

usedused toto estimateestimate thethe variance,variance, thenthen thethe statisticstatistic

--

s/ns/n

isis notnot normallynormally distributeddistributed.. TheThe probabilitiesprobabilities inin thethe tailstails ofof thisthis distributiondistribution areare greatergreaterthanthan forfor thethe standardstandard normalnormal distributiondistribution

ThisThis isis reasonablereasonable sincesince

--

z=z=

s/ns/n

containscontains onlyonly oneone randomrandom variablevariable ,, whilewhile--

s/ns/n

containscontains twotwo randomrandom variablesvariables yy andand ss.. AsAs nn increasesincreases thisthis newnew distributiondistribution approachesapproachesthethe standardstandard normalnormal distributiondistribution..


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Properties of t TestProperties of t Test

StudentsStudents tt distributionsdistributions areare

11.. UnimodalUnimodal;;

22.. AsymptoticAsymptotic toto thethe horizontalhorizontal axisaxis;;33.. SymmetricalSymmetrical aboutabout zero,zero, E(t)E(t);;

44.. DependentDependent onon v,v, thethe degreesdegrees ofof freedomfreedom (for(for thethestatisticstatistic underunder discussion,discussion, v=nv=n--11));;

55.. MoreMore variablevariable thanthan thethe standardstandard normalnormaldistribution,distribution, V(t)=v/(vV(t)=v/(v--22)) forfor nn >> 22;;

66.. ApproximatelyApproximately standardstandard normalnormal ifif vv isis largelarge..


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Excel CommandExcel Command

ToTo testtest whetherwhether thethe populationpopulation meanmean couldcould bebe ,,useuse ::

MeanTestMeanTest usesuses thethe originaloriginal datadata andand thethehypotheticalhypothetical valuevalue ofof thethe populationpopulation meanmean..

StudentTPValueStudentTPValue usesuses onlyonly thethe valuevalue ofof thethe testteststatisticstatistic tt == mm--//SmSm andand degreesdegrees ofof freedomfreedom nn -- 11..

TheThe twotwo commandscommands useuse thethe StudentStudent tt distributiondistributiontoto performperform thethe testingtesting..


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ComparisionComparision ofstandardnormaldistributionandatofstandardnormaldistributionandatdistributiondistribution


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StandardErrorStandardError

TestTest determinesdetermines whetherwhether differencesdifferences inin twotwo

samplessamples isis significantsignificant enoughenough toto suggestsuggest aadifferencedifference inin respectiverespective populationspopulations..

HenceHence averageaverage expectedexpected differencedifference betweenbetween thethemeansmeans ofof twotwo samplessamples isis requiredrequired.. ThisThis averageaveragedifferencedifference isis knownknown asas thethe StandardStandard ErrorError ofof thethedifferencedifference betweenbetween twotwo meansmeans..


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Paired SamplesPaired Samples

AA dependentdependent samplessamples tt testtest isis alsoalso usedused toto comparecompare twotwomeansmeans onon aa singlesingle dependentdependent variablevariable..

UnlikeUnlike thethe independentindependent samplessamples test,test, however,however, aa dependentdependent

samplessamples tt testtest isis usedused toto comparecompare thethe meansmeans ofof aa singlesinglesamplesample oror ofof twotwo matchedmatched oror pairedpaired samplessamples..

ForFor example,example, ifif aa groupgroup ofof studentsstudents tooktook aa mathmath testtest ininMarchMarch andand thatthat samesame groupgroup ofof studentsstudents tooktook thethe samesame mathmathtesttest twotwo monthsmonths laterlater inin May,May, wewe couldcould comparecompare theirtheir

averageaverage scoresscores onon thethe twotwo testtest datesdates usingusing aa dependentdependentsamplessamples tttesttest..


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ExampleExample

SupposeSuppose wewe wantwant toto knowknow whetherwhether employeesemployees atat ourourwidgetwidget--makingmaking factoryfactory areare moremore productiveproductive afterafter theythey returnreturnfromfrom aa 22--weekweek vacationvacation..

WeWe randomlyrandomly selectselect 3030 ofof mymy employeesemployees andand calculatecalculate thetheaverageaverage numbernumber ofof widgetswidgets mademade byby eacheach employeeemployee duringduringthethe weekweek beforebefore theythey gogo onon vacationvacation..

WeWe findfind that,that, onon average,average, thethe employeesemployees mademade 250250 widgetswidgetseacheach duringduring thethe weekweek.. DuringDuring thethe weekweek afterafter theythey returnreturnfromfrom vacation,vacation, II keepkeep tracktrack ofof howhow manymany widgetswidgets isis mademade

by by thethe samesame samplesample ofof 3030 employeesemployees andand findfind that,that, ononaverage,average, theythey mademade 300300 widgetswidgets eacheach duringduring thethe weekweek afterafterreturningreturning fromfrom theirtheir vacationsvacations..


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ContdContd

T=observedT=observed diffdiff.. betweenbetween prevacationprevacation andand postvacationpostvacationmeans/standardmeans/standard errorerror ofof thethe differencedifference betweenbetween thethe meansmeans

OrOr

T=xT=x--y/y/SdSd

TheThe formulaformula forfor calculatingcalculating thethe standardstandard errorerror ofof thethedifferencedifference between between thethe meansmeans forfor dependentdependent samplessamples isis

slightlyslightly differentdifferent thanthan thethe oneone forfor independentindependent samples,samples, butbutthethe principles principles involvedinvolved (i(i..ee..,, whatwhat thethe standardstandard errorerrorrepresents)represents) areare thethe samesame..


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CaseCase

Effectiveness ofNewly ImplementedEffectiveness ofNewly ImplementedGreen BeltProject ina Call Center ofGreen BeltProject ina Call Center of

Insurance CompanyInsurance Company


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BibliographyBibliography ParadigmParadigm : IMT G Journal: IMT G Journal

Appliedstatisticsinbusinessand economicsbyAppliedstatisticsinbusinessand economicsbyDoaneDoane & Seward& Seward

www.w

ikipedia.

comwww.w

ikipedia.

com Case: An analysis of the Indian Textile IndustryCase: An analysis of the Indian Textile Industry

in Quota Free Regime byin Quota Free Regime by ManishaManisha Sharma &Sharma &AnuAnu PrashaantPrashaant

It is not nice to be wedded to anythingIt is not nice to be wedded to anything not even anot even atheorytheory Samuel ButlerSamuel Butler

test on variables in statistics

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