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MINITABisprobablyoneofthemostuserfriendlypackagesavailableforintroductorystatisticscoursesonboththeundergraduatelevelandgraduatelevelaswellasforprofessionaluses.ItisavailableinApple,PowerP.C.nativeandI.B.M.compatibleformats.Dataisenteredincolumnsandmanipulatedinaspreadsheetenvironment.MINITABcommandsusuallypointtothelocationthatdataisstoredandaregenerallyverybriefand'  descriptiveoftheirfunction. Intheclassroom,MINITABshouldbeusedasmorethanjustatoolforcrunchingdataandstatisticalanalysis.Usedproperly,MINITABcanprovideavaluableopportunitytodevelopstudentscriticalthinkingskills.SimplyaskingstudentstoutilizeMINITABbydoingmechanical,routinestatisticalproceduresisnotpedagogicallyappropriate.Instead,studentsshouldbeguidedtouseMINITABtodiscoverortointuitivelyjustifystatisticalprinciples.Furthermore,ifstudentsareaskedgoodthoughtprovokingquestions,MINITABcanbeatooltoengagestudentsinhigherlevelsofthinkingsuchasanalysis,synthesisandevaluation. ToillustratehowMINITABcanbeusedinexploratoryexercises,considerthefollowingexampleinvolvingtheCentralLimitTheorem.Supposeacomputercompanyrecordsthenumberoftelephonecallsreceivedeachhourandthencalculatesthatanaverageofsixphonecallsisreceivedeachhour.StudentsinanelementarystatisticscourseshouldknowthatthenumberoftelephonecallsreceivedcouldbemodeledbyaPoissonDistributionwithalambdaparameterofsix.(SeeFigure1)  C 4$ ` \T ` CXFigure1:PDFforthePoissondistributionwith=6.    Nowsupposethatthiscomputercompanyhasthemanagerperformthefollowingexperiment:Thenumberoftelephonecallsthecompanyreceivesisrecordedforeachhourforatotalof100hours.Hethencalculatestheaveragenumberofcallsperhourforthe100hours.Ifthemanagerrepeatsthisexperimentntimes,thenhehasasetofnindependentsampleaverages. ThefollowingMINITABmacromaybeusedtomodelthisexperimentforvariousvaluesofn.ThismacroisanessentialpartofanexploratoryexercisetohelpstudentsunderstandtheCentralLimitTheorem(CLT).Forthesakeofillustration,thisexperimentwillonlyberepeatedtwice,first29timesandthen98times.IfstudentsareaskedtousethisMINITABmacro,theywillseenotonlyasetofnindependentsampleaveragesgeneratedbutalsoahistogramoftheresultingdistributionofsampleaverages. FollowingistheMINITABmacrotogeneratethehistogramsinFigures2and3:  notek1isonemorethanthenumberofdesiredmeans  notek1isthecolumnstostorethemeans  notek3isthenumberofsamplemeans  letk2=k11  random100c1ck2;  poisson6.  letk3=1  execute'mean'k2  ghistogramck1.Xwherethemacromeanis  letck1(k3)=mean(ck3)  letk3=k3+1XInthisexploratoryexerciseontheCLT,studentsrerunthemacroforincreasingvaluesofnandareaskedthefollowingquestions(SeeFigures2and3): 1.Howdoesincreasingnaffectthesymmetryofthe histogramsofmeans? 2.Asnincreasesthehistogramsoftheresultingmeans approachesthehistogramofwhattypeofdistribution? 3.Theoriginaldistributionwaspositivelyskewed.Areyou surprisedbyyouranswerstothequestionsabove?Whyorwhy not? C 4$ ` \L `C  X Figure2:Histogramofthe29meansgeneratedby29independentsamples,eachofsize100. C 4$ ` \W} `W}CX Figure3:Histogramofthe98meansgeneratedby98independentsamples,eachofsize100.    ThepreviousexampleisanillustrationoftheCentralLimitTheoremforadiscretedistribution,namelythePoissondistribution.ItwouldbebeneficialtoalsoguidestudentsthroughanexampleoftheCentralLimitTheoremforacontinuousdistribution.Inordertoaccomplishthisletsnowconsiderstudyinganewexperiment,thevariabilityofameasurementinthequalitycontrolofamanufacturingprocess.SupposeindependentsamplevariancesaretakenfortheprocessandfurtherthatthesesamplevariancesaredistributedChisquarewithsixdegreesoffreedom.(SeeFigure4)  C 4$ ` \ ` C Figure4:PDFfortheChisquaredistributionwith6degreesoffreedom.    Onehundredindependentsamplevariancesofthisprocessweretakenandthisexperimentisrepeated29timesandthen98times.Again,studentscanuseMINITABtofindthedistributionofthemeanforthe29repetitionsandforthe98repetitions.StudentswouldbeabletoseeforthemselveshowthehistogramsoftheresultingmeansapproachthehistogramofanormaldistributionandthusillustratetheCentralLimitTheorem.(SeeFigures5and6)  ӀC 4$ ` \1H `C Figure5:Histogramofthe29meansgeneratedby29independentsamples,eachofsize100.  ̀Note,Figure5isanexamplewherethefollowing ruleofthumbfeaturedinmostelementarystatisticstextbooksfails:Ifthesamplesizeisaroundthirtythenthedistributionofthesamplemeanisapproximatelynormal.C 4$ ` \  ` C  Figure6:Histogramofthe98meansgeneratedby98independentsamples,eachofsize100.      UsingMINITABonlytododemonstrationexercisesliketheonesjustdiscussedisnotsufficienttofullydevelopstudentscriticalthinkingskills.MINITABshouldbeusedtocompare,contrast,andevaluatedatasetsbyuncoveringpatternsandrelationships.MINITABgraphiccommandscanhelpuncoverrelationshipsbothwithinandbetweendatasets.Forexample,letsnowconsideradatasetoftheheightininchesof30students.MINITABcommandscanbeusedtoansweraquestionsuchas:Doesthedatacomefromanormaldistribution?(SeeFigure7)ThenscoreMINITABcommandallowsstudentstosupporttheassumptionofnormallydistributeddata.Studentsmaybeaskedthefollowingquestions,afterhavinglearnedaboutorderstatisticsandnormalprobabilityplots:  1.Whatisimpliedbyastraightlineinthenormalprobabilityplot?2.Whatfurtherstatisticalproceduresmaybeusedifthedatais trulynormal?̀C 4$ ` \c `C Figure7:Normalprobabilityplotforheightdata.    WecanalsouseMINITABtohelpcreateexercisesinwhichgoodinterpretivequestionsareasked.Forexample,consideragaintheheightininchesof30students.Ifallthedatawereconvertedtofeet,whatwouldbetherelationshipbetweenthestandarddeviationofthedataininchesandthestandarddeviationofthedatainfeet?Whatprincipleexplainsthisrelationship?(SeeFigure8)TheoutputfromtheMINITABdescribecommandmakesthesequestionseasytoanswer.Forexample,height=66.600,themeanofheight,andheight/12=5.5500,ԀXthemeanofthetransformeddata.Studentscanreadilydeducethatheight/12=height/12.Ingeneral,studentswillderiveforthemselvesgeneralrulesofhowtransformationsaffectbasicdescriptivestatistics.  ̀   NMEANMEDIANTRMEANSTDEVSEMEAN Height3066.60067.00066.5003.1470.575̀ MINMAXQ1Q3 Height62.00074.00064.00069.000̀ NMEANMEDIANTRMEANSTDEVSEMEAN Height/12305.55005.58335.54170.26220.0479̀ MINMAXQ1Q3 Height/125.16676.16675.33335.7500  X Figure8:OutputofMINITABdescribecommandforheightdataininchesandthentransformedintofeet.  X ItwasdemonstratedherethatMINITABcanenhancestudentscriticalthinkingskillsthroughexploratoryexercisesandthroughexercisesthatforcestudentstocompareandcontrastsetsofdata.Theessenceismerelyaskinggoodquestionsofstudentsinordertoforcethemtodrawvalidconclusions.Moreimportantly,MINITABisaveryflexiblesoftwarepackagethataccommodatesdifferentteachingstylesaswellasdifferentlearningstylesandcanbeusedtoillustrateconcretelyabstracttheoreticalconcepts,suchastheCentralLimitTheorem. References: MINITAB,StatisticalSoftware,StandardVersion,Release8forMacintosh,MinitabInc.,1991.