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
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