TI-84CalculatorInstructions

Adaptedfrom:ThePracticeofStatistics4ebyStarnes,Yates,andMoore

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HISTOGRAMSONTHECALCULATOR 4

MAKINGCALCULATORBOXPLOTS 7

COMPUTINGNUMERICALSUMMARIESWITHTECHNOLOGY 8

THESTANDARDNORMALCURVE 10

FROMZ-SCORESTOAREAS,ANDVICEVERSA 12

NORMALPROBABILITYPLOTS 14

SCATTERPLOTSONTHECALCULATOR 15

LEAST-SQUARESREGRESSIONLINESONTHECALCULATOR 16

RESIDUALPLOTSANDSONTHECALCULATOR 18

ANALYZINGRANDOMVARIABLESONTHECALCULATOR 20

SIMULATINGWITHRANDNORM 22

BINOMIALCOEFFICIENTSONTHECALCULATOR 23

BINOMIALPROBABILITYONTHECALCULATOR 24

GEOMETRICPROBABILITYONTHECALCULATOR 25

CONFIDENCEINTERVALFORAPOPULATIONPROPORTION 26

INVERSETONTHECALCULATOR 27

ONE-SAMPLETINTERVALSFORΜONTHECALCULATOR 28

ONE-PROPORTIONZTESTONTHECALCULATOR 30

COMPUTINGP-VALUESFROMTDISTRIBUTIONSONTHECALCULATOR 32

ONE-SAMPLETTESTONTHECALCULATOR 33

CONFIDENCEINTERVALFORADIFFERENCEINPROPORTIONS 35

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SIGNIFICANCETESTFORADIFFERENCEINPROPORTIONS 36

TWO-SAMPLETINTERVALSONTHECALCULATOR 37

TWO-SAMPLETTESTSWITHCOMPUTERSOFTWAREANDCALCULATORS 39

FINDINGP-VALUESFORCHI-SQUARETESTSONTHECALCULATOR 41

CHI-SQUAREGOODNESS-OF-FITTESTONTHECALCULATOR 42

CHI-SQUARETESTSFORTWO-WAYTABLESONTHECALCULATOR 43

REGRESSIONINFERENCEONTHECALCULATOR 46

TRANSFORMINGTOACHIEVELINEARITYONTHECALCULATOR 48

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HistogramsontheCalculatorYoucanconstructhistogramsusingyourTI-84.Wewillusethefollowingexampletoillustratetheprocess:Whatpercentofyourhomestate’sresidentswerebornoutsidetheUnitedStates?Thecountryasawholehas12.5%foreign-bornresidents,butthestatesvaryfrom1.2%inWestVirginiato27.2%inCalifornia.Thetablebelowpresentsthedataforall50states.1.EnterthedataforthepercentofstateresidentsbornoutsidetheUnitedStatesinyourStatistics/ListEditor.

• PressSTATandchoose1:Edit…• TypethevaluesintolistL1.

2.SetupahistogramintheStatisticsPlotsmenu.

• Press2ndY=(STATPLOT).• PressENTERor1togointoPlot1

3.UseZoomStattoletthecalculatorchooseclassesandmakeahistogram.

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• PressZOOMandchoose9:ZoomStat.• PressTRACEandtheleftandrightarrowkeystoexaminetheclasses.

4.Adjusttheclassestomatchthoseinthebelowfigure:thengraphthehistogram.

• PressWINDOWandenterthevaluesshown.• PressGRAPH• PressTRACEandtheleftandrightarrowkeystoexaminetheclasses.

5.Seeifyoucanmatchthehistogrambelow:

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MakingCalculatorBoxplotsTheTI-89canplotuptothreeboxplotsinthesameviewingwindow.Let’susethecalculatortomakeside-by-sideboxplotsofthetraveltimetoworkdataforthesamplesfromNorthCarolinaandNewYorkshownbelow:

1.EnterthetraveltimedataforNorthCarolinainL1andforNewYorkinL2.2.Setuptwostatisticsplots:Plot1toshowaboxplotoftheNorthCarolinadataandPlot2toshowaboxplotoftheNewYorkdata.Note:Thecalculatorofferstwotypesofboxplots:a“modified”boxplotthatshowsoutliersandastandardboxplotthatdoesn’t.We’llalwaysusethemodifiedboxplot.3.Usethecalculator’sZoomfeaturetodisplaytheside-by-sideboxplots.ThenTracetoviewthefive-numbersummary.

• PressZOOMandselect9:ZoomStat.• PressTRACE..

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ComputingnumericalsummarieswithtechnologyLet’sfindnumericalsummariesforthetraveltimesofNorthCarolinaandNewYorkworkers.We’llstartbyshowingyouthenecessarycalculatortechniquesandthenlookatoutputfromcomputersoftware.

One-variablestatisticsonthecalculatorEntertheNorthCarolinadatainL1andtheNewYorkdatainL2.1.FindthesummarystatisticsfortheNorthCarolinatraveltimes.

• PressSTAT,RightArrow(CALCtab);choose1:1-VarStats.• PressENTER.Nowpress2nd1(L1)andENTER.• Pressthedownarrowkeytoseetherestoftheone-variablestatisticsforNorthCarolina.

2.RepeatStep1usinglist2tofindthesummarystatisticsfortheNewYorktraveltimes.OutputfromstatisticalsoftwareWeUsedMinitabstatisticalsoftwaretoproducedescriptivestatisticsfortheNewYorkandNorthCarolinatraveltimedata.Minitaballowsyoutochoosewhichnumericalsummariesareincludedintheoutput.

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ThestandardNormalcurveTheTI-84canbeusedtodrawandfindareasunderastandardNormalcurve.TodrawastandardNormalcurve:First,turnoffallstatisticsplots.1.EntertheformulaforthestandardNormaldensitycurveinY1.

• DefineY1=normalpdf(x,0,1).• PressY=.WiththecursornexttoY1=,press2ndVARS(DISTR)andchoose1:normalpdf(.• PressX,T,θ,n,0,1),tocompletetheformula.

2.Adjustwindowssettingsandgraph.

• PressWINDOW.EnterXmin=-4,Xmax=4,Xscl=1,Ymin=-1,Ymax=.5,Yscl=.1.• PressGRAPHtodisplaythecurve.

TofindareasunderthestandardNormalcurve:Gotothehomescreen.UsetheshadeNormcommandtofindthedesiredarea.TI-84:Press2ndVARS(DISTR),arrowrighttoDRAW,andchoose1:Shade-Norm(.CompletethecommandshadeNorm(lowerbound,upperbound,µ,σ).Note:AftereachtimeyouuseshadeNorm,executethecommandclrDrawtoremovetheshading.(clrDrawcanbefoundintheDRAWmenu).Tofind

• Theareatotheleftofz=2.22,use-100forthelowerbound:shadeNorm(-100,2.22,0,1).

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• Theareatotherightofz=-1.78,use100fortheupperbound:shadeNorm(-1.78,100,0,1).

• Theareabetweenz=-1.25andz=0.81:shadeNorm(-1.25,0.81,0,1).

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Fromz-scorestoareas,andviceversaFindingareas:ThenormalcdfcommandontheTI-84canbeusedtofindareasunderaNormalcurve.ThismethodisquickerthanshadeNormbuthasthedisadvantageofnotprovidingapictureoftheareaitisfinding.Thesyntaxisfamiliar:normalcdf(lowerbound,upperbound,µ,σ).Let’susethefollowingexampletoillustratethisprocess:Onthedrivingrange,TigerWoodspracticeshisswingwithaparticularclubbyhittingmany,manyballs.WhenTigerhitshisdriver,thedistancetheballtravelsfollowsaNormaldistributionwithmean304yardsandstandarddeviation8yards.Recallthatmu=304yardsandsigma=8yards.1.WhatproportionofTiger’sdrivesontherangetravelatleast290yards?

• Press2ndVARS(DISTR)andchoose2:normcdf(.• Completethecommandnormcdf(290,400,304,8)andpressENTER.

Note:Wechose400astheupperboundbecauseit’smany,manystandarddeviationsabovethemean.Theseresultsagreewithourpreviousanswerusingthez-table:0.9599.2.WhatpercentofTiger’sdrivestravelbetween305and325yards?Thescreenshotsbelowindicatethatourearlierresultof0.4440usingthewasalittleoff.Thisdiscrepancywascausedbythefactthatweroundedourz-scorestotwodecimalplacesinordertousethetable.Workbackward:TheTI-84invNormfunctioncalculatesthevaluecorrespondingtoagivenpercentileinaNormaldistribution.Forthiscommand,thesyntaxisinvNorm(percentile, µ, σ).Let’sillustratethisprocesswiththefollowingexample:

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Highlevelsofcholesterolinthebloodincreasetheriskofheartdisease.For14-year-oldboys,thedistributionofbloodcholesterolisapproximatelyNormalwithmeanμ=170milligramsofcholesterolperdeciliterofblood(mg/dl)andstandarddeviationσ=30mg/dl.Recallthatmu=170mg/dlandsigma=30mg/dl.3.Whatisthefirstquartileofthedistributionofbloodcholesterol?

• Press2ndVARS(DISTR)andchoose3:invNorm(.• CompletethecommandinvNorm(.25,170,30)andpressENTER.• ComparethiswiththeresultofinvNorm(.25).

TECHNOLOGYTIP:Fornormpdf,shadeNorm,normcdf,andinvNorm,thedefaultvaluesareμ=0andσ=1.Thefirstcommandshowsthatthefirstquartile(25thquartile)ofthecholesteroldistributionfor14year-oldmalesis149.8mg/dl.(Ouranswerusingthez-tableis149.9mg/dl.)ThesecondcommandshowsthatinthestandardNormaldistribution,25%oftheobservationsfallbelowz=-0.067449.(Weusedz=-0.67forourpreviouscalculations,whichexplainsthesmalldiscrepancy.)

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NormalprobabilityplotsTheTI-84canconstructanormalprobabilityplot.Wewillusethefollowingexampletoillustratethisprocess:Herearetheunemploymentratesinthe50statesinNovember2009.Thedataisarrangedfromlowest(NorthDakota’s4.1%)tohighest(Michigan’s14.7%)

TomakeaNormalprobabilityplotforasetofquantitativedata:

• EnterthedatavaluesinL1.• DefinePlot1asshown.

• UsezoomStattoseethefinishedgraph.Interpretation:TheNormalprobabilityplotisquitelinear,soitisreasonabletobelievethatthedatafollowaNormaldistribution.

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ScatterplotsonthecalculatorMakingscatterplotswithtechnologyismucheasierthanconstructingthembyhand.We’llusethefollowingexampletoshowhowtoconstructascatterplotonaTI-84:Ninth-gradestudentsattheWebbSchoolsgoonabackpackingtripeachfall.Studentsaredividedintohikinggroupsofsize8byselectingnamesfromahat.Beforeleaving,studentsandtheirbackpacksareweighed.Herearedatafromonehikinggroupinarecentyear:

• Enterthedatavaluesintoyourlists.Clearlists:L1andL2.PutthebodyweightsinL1andthebackpackweightsinL2.

• Defineascatterplotinthestatisticsplotmenu.Specifythesettingsshown.

• UseZoomStattoobtainagraph.ThecalculatorwillsetthewindowdimensionsautomaticallybylookingatthevaluesinL1andL2.

Noticethattherearenoscalesontheaxesandthattheaxesarenotlabeled.Ifyoucopyascatterplotfromyourcalculatorontoyourpaper,makesurethatyouscaleandlabeltheaxes.YoucanuseTRACEtohelpyougetstarted(likewedid).

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Least-squaresregressionlinesonthecalculatorLet’susethefatgainandNEAdatatoshowhowtofindtheequationoftheleast-squaresregressionlineontheTI-84.Hereisthedata:

1.EntertheNEAchangedataintoL1andthefatgaindataintoL2.Thenmakeascatterplot.Referto“Scatterplotsonthecalculator.”2.Todeterminetheleast-squaresregressionline:

• PressSTAT;chooseCALCandthen8:LinReg(a+bx).FinishthecommandtoreadLinReg(a+bx)L1,L2,Y1andpressENTER.(Y1isfoundunderVARS/Y-VARS/1:Function.)

3.Graphtheregressionline.TurnoffallotherequationsintheY=screenandpressGRAPHtoaddtheleast-squareslinetothescatterplot.4.Savetheselistsforlateruse.Onthehomescreen,L1→NEA:L2→FAT.

Althoughthecalculatorwillreportthevaluesforaandbtoninedecimalplaces,weusuallyroundofftofewerdecimalplaces.Youwouldwritetheequationas .

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Note:TheTI-84commandtellsthecalculatortocomputetheequationoftheleast-squaresregressionlineusingL1astheexplanatoryvariableandL2astheresponsevariableandthentostoretheresultinslotY1.Thismethodisusefulifyouwanttographtheregressionlineoruseitsequationtomakepredictions.Ifyou’reinterestedinonlytheequationoftheline,LinReg(a+bx)L1,L2willdo.Ifr2andrdonotappearontheTI-84screen,dothisone-timeseriesofkeystrokes:Press2nd0(CATALOG),scrolldowntoDiagnosticOn,andpressENTER.PressENTERagaintoexecutethecommand.Thescreenshouldsay“Done.”Thenpress2ndENTER(ENTRY)torecalltheregressioncommandandENTERagaintocalculatetheleast-squaresline.Ther2andrvaluesshouldnowappear.

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ResidualplotsandsonthecalculatorWewanttocalculateresidualsandmakearesidualplotontheTI-84usingthefollowingexample:

Youshouldhavealreadymadeascatter-plot,calculatedtheequationoftheleast-squaresregressionline,andgraphedthelineonyourplot.Earlier,wefoundthat .1.DefineL3asthepredictedvaluesfromtheregressionequation.

• WithL3highlighted,enterthecommand3.505-0.00344*L1andpressENTER2.DefineL4astheobservedy-valueminusthepredictedy-value.

• WithL4highlighted,enterthecommandL2-L3andpressENTERtoshowtheresiduals.3.TurnoffPlot1andtheregressionequation.SpecifyPlot2withL1asthexvariableandL4astheyvariable.UseZoomStattoseetheresidualplot.Thexaxisintheresidualplotservesasareferenceline:pointsabovethislinecorrespondtopositiveresidualsandpointsbelowthelinecorrespondtonegativeresiduals.WeusedTRACEtoseetheresidualfortheindividualwithanNEAchangeof−94calories.

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4.Finally,wewanttocomputethestandarddeviationsoftheresiduals.Calculateone-variablestatisticsontheresidualslist(L4).Themeanoftheresidualsis0(uptoroundofferror).ThesumofthesquaredresidualsisΣx2=7.663.Tofinds,usetheformula:

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AnalyzingrandomvariablesonthecalculatorLet’sexplorewhatthecalculatorcandousingtherandomvariableX=Apgarscoreofarandomlyselectednewborn.

1.StartbyenteringthevaluesoftherandomvariableinL1andthecorrespondingprobabilitiesinL2.Usethefollowingtable:

2.Tographahistogramoftheprobabilitydistribution:

• SetupastatisticsplotwithXlist:L1andFreq:L2.• Adjustyourwindowsettingsasfollows:Xmin=−1,Xmax=11,Xscl=1,Ymin=−0.1,Ymax=

0.5,Yscl=0.1.

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3.Tocalculatethemeanandstandarddeviationoftherandomvariable,useone-variablestatisticswiththevaluesinL1andtheprobabilities(relativefrequencies)inL2.TI-84:Executethecommand1-VarStatsL1,L2.

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SimulatingwithrandNormTherandNormcommandontheTI-84allowsyoutosimulateobservationsfromaNormaldistributionwithaspecifiedmeanandstandarddeviation.Thisprocesswillbeillustratedusingthefollowingexample:

ThediameterCofarandomlyselectedlargedrinkcupatafast-foodrestaurantfollowsaNormaldistributionwithameanof3.96inchesandastandarddeviationof0.01inches.ThediameterLofarandomlyselectedlargelidatthisrestaurantfollowsaNormaldistributionwithmean3.98inchesandstandarddeviation0.02inches.

Foralidtofitonacup,thevalueofLhastobebiggerthanthevalueofC,butnotbymorethan0.06inches.YoucanfindrandNormunderMATH/PRB.Forinstance,randNorm(3.98,.02)willrandomlyselectavaluefromtheNormaldistributionwithmean3.98andstandarddeviation0.02.Thissimulateschoosingalargecuplidatrandomfromthefast-foodrestaurantofthepriorexampleandmeasuringitsdiameter(ininches).Tosimulatechoosingalargedrinkcupandmeasuringitswidth,youcanuserandNorm(3.96,.01).Toestimatetheprobabilitythatarandomlyselectedlidwillfitonarandomlychosencup:1.Simulatechoosing100largecuplidsatrandom,andstoretheirwidthsinL1:randNorm(3.98,.02,100)−>L12.Simulatechoosing100largedrinkcupsatrandom,andstoretheirdiametersinL2:randNorm(3.96,.01,100)−>L23.Computethedifferencebetweenthelidandcupdiametersforthese100pairsofvalues:L1-L2−>L34.CountthenumberofvaluesinL3thatarebetween0and0.06.(YoumaywanttosortthevaluesinL3first!)Thisnumberdividedby100isyourestimateoftheprobability.

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BinomialcoefficientsonthecalculatorTocalculateabinomialcoefficientlike !! ontheTI-84,proceedasfollows:Type5,pressMATH,arrowovertoPRB,choose3:nCr,andpressENTER.Thentype2andpressENTERagaintoexecutethecommand5nCr2.

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BinomialprobabilityonthecalculatorTherearetwohandycommandsontheTI-84forfindingbinomialprobabilities:

binompdf(n,p,k)computesP(X=k)

binomcdf(n,p,k)computesP(X≤k)

Thesetwocommandscanbefoundinthedistributionsmenu(2nd/VARS)ontheTI-84.Thiswillbeillustratedusingthefollowingexample:Eachchildofaparticularpairofparentshasprobability0.25ofhavingtypeOblood.Geneticssaysthatchildrenreceivegenesfromeachoftheirparentsindependently.Iftheseparentshave5children,thecountXofchildrenwithtypeObloodisabinomialrandomvariablewithn=5trialsandprobabilityp=0.25ofasuccessoneachtrial.Inthissetting,achildwithtypeObloodisa“success”(S)andachildwithanotherbloodtypeisa“failure”(F).Fortheparentshavingn=5children,eachwithprobabilityp=0.25oftypeOblood:

P(X=3)=binompdf(5,0.25,3)=0.08789TofindP(X>3),weusedthecomplementrule:P(X>3) =1-P(X≤3) =1-binomcdf(5,0.25,3) =0.01563Ofcourse,wecouldalsohavedonethisasP(X>3) =P(X=4)+P(X=5) =binompdf(5,0.25,4)+binompdf(5,0.25,5) =0.01465+0.00098=0.01563

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GeometricprobabilityonthecalculatorTherearetwohandycommandsontheTI-84forfindinggeometricprobabilities:

geometpdf(p,k)computesP(Y=k)

geometcdf(p,k)computesP(Y≤k)Thesetwocommandscanbefoundinthedistributionsmenu(2nd/VARS)ontheTI-84.Wewillillustratetheuseofthesecommandsusingthefollowingexample:Yourteacherisplanningtogiveyou10problemsforhomework.Asanalternative,youcanagreetoplaytheBirthDayGame.Here’showitworks.Astudentwillbeselectedatrandomfromyourclassandaskedtoguessthedayoftheweek(forinstance,Thursday)onwhichoneofyourteacher’sfriendswasborn.Ifthestudentguessescorrectly,thentheclasswillhaveonlyonehomeworkproblem.

Ifthestudentguessesthewrongdayoftheweek,yourteacherwillonceagainselectastudentfromtheclassatrandom.Thechosenstudentwilltrytoguessthedayoftheweekonwhichadifferentoneofyourteacher’sfriendswasborn.Ifthisstudentgetsitright,theclasswillhavetwohomeworkproblems.Thegamecontinuesuntilastudentcorrectlyguessesthedayonwhichoneofyourteacher’s(many)friendswasborn.Yourteacherwillassignanumberofhomeworkproblemsthatisequaltothetotalnumberofguessesmadebymembersofyourclass.FortheBirthDayGame,withprobabilityofsuccessp=1/7oneachtrial:

P(Y=10)=geometpdf(1/7,10)=0.0357

TofindP(Y<10),usegeometcdf:

P(Y<10)=P(Y≤9)=geometcdf(1/7,9)=0.7503

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ConfidenceintervalforapopulationproportionTheTI-84canbeusedtoconstructaconfidenceintervalforanunknownpopulationproportion.We’lldemonstrateusingthefollowingexample:TheGallupYouthSurveyaskedarandomsampleof439U.S.teensaged13to17whethertheythoughtyoungpeopleshouldwaittohavesexuntilmarriage.Ofthesample,246said“Yes.”Constructandinterpreta95%confidenceintervalfortheproportionofallteenswhowouldsay“Yes”ifaskedthisquestion.Ofn=439teenssurveyed,X=246saidtheythoughtthatyoungpeopleshouldwaittohavesexuntilaftermarriage.Toconstructaconfidenceinterval:

• PressSTAT,thenchooseTESTSandA:1-PropZInt.• Whenthe1-PropZIntscreenappears,enterx=246,n=439,andconfidencelevel0.95.

• Highlight“Calculate”andpress .The95%confidenceintervalforpisreported,alongwiththesampleproportionp-hatandthesamplesize,asshownhere.

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InversetonthecalculatorMostnewerTI-84calculatorsallowyoutofindcriticalvaluest*usingtheinversetcommand.Aswiththecalculator’sinverseNormalcommand,youhavetoentertheareatotheleftofthedesiredcriticalvalue.Wewillusethefollowingexampletoillustratethisprocess.Supposeyouwanttoconstructiona95%confidenceintervalforthemeanμofaNormalpopulationbasedonanSRSofsizen=12.Whatcriticalvaluet*shouldyouuse?Press2ndVARS(DISTR)andchoose4:invT(.ThencompletethecommandinvT(.975,11)andpressENTER.

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One-sampletintervalsforμonthecalculatorConfidenceintervalsforapopulationmeanusingtprocedurescanbeconstructedontheTI-84,thusavoidingtheuseofaz-table.Hereisabriefsummaryofthetechniqueswhenyouhavetheactualdatavaluesandwhenyouhaveonlynumericalsummaries.Wewillusethefollowingexample(s)toillustratetheprocess.1.Usingrawdata:Amanufacturerofhigh-resolutionvideoterminalsmustcontrolthetensiononthemeshoffinewiresthatliesbehindthesurfaceoftheviewingscreen.Toomuchtensionwilltearthemesh,andtoolittlewillallowwrinkles.Thetensionismeasuredbyanelectricaldevicewithoutputreadingsinmillivolts(mV).Somevariationisinherentintheproductionprocess.Herearethetensionreadingsfromarandomsampleof20screensfromasingleday’sproduction

Constructandinterpreta90%confidenceintervalforthemeantensionμofallthescreensproducedonthisday.Enterthe20videoscreentensionreadingsdatainL1.Fromthehomescreen,

• PressSTAT,arrowovertoTESTS,andchoose8:TInterval….• OntheTIntervalscreen,adjustyoursettingsasshownandchooseCalculate.

2.Usingsummarystatistics:

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Environmentalists,governmentofficials,andvehiclemanufacturersareallinterestedinstudyingtheautoexhaustemissionsproducedbymotorvehicles.

Themajorpollutantsinautoexhaustfromgasolineenginesarehydrocarbons,carbonmonoxide,andnitrogenoxides(NOX).ResearcherscollecteddataontheNOXlevels(ingrams/mile)forarandomsampleof40light-dutyenginesofthesametype.ThemeanNOXreadingwas1.2675andthestandarddeviationwas0.3332.Constructandinterpreta95%confidenceintervalforthemeanamountofNOXemittedbylight-dutyenginesofthistype.Thistime,wehavenodatatoenterintoalist.ProceedtotheTIntervalscreenasinStep1,butchooseStatsasthedatainputmethod.WhenyougettotheTIntervalscreen,entertheinputsshownandcalculatetheinterval.

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One-proportionztestonthecalculatorTheTI-84canbeusedtotestaclaimaboutapopulationproportion.We’lldemonstrateusingthefollowingexample:Apotato-chipproducerhasjustreceivedatruckloadofpotatoesfromitsmainsupplier.Iftheproducerdeterminesthatmorethan8%ofthepotatoesintheshipmenthaveblemishes,thetruckwillbesentawaytogetanotherloadfromthesupplier.Asupervisorselectsarandomsampleof500potatoesfromthetruck.Aninspectionrevealsthat47ofthepotatoeshaveblemishes.Carryoutasignificancetestatthe =0.10significancelevel.Whatshouldtheproducerconclude?Inarandomsampleofsizen=500,thesupervisorfoundX=47potatoeswithblemishes.Toperformasignificancetest:PressSTAT,thenchooseTESTSand5:1-PropZTest.Onthe1-PropZTestscreen,enterthevaluesshown:p0=0.08,x=47,andn=500.Specifythealternativehypothesisas“prop>p0.”Note:xisthenumberofsuccessesandnisthenumberoftrials.Bothmustbewholenumbers!Ifyouselectthe“Calculate”choiceandpress ,youwillseethattheteststatisticisz=1.15andtheP-valueis0.1243.Ifyouselectthe“Draw”option,youwillseethescreenshownhere.Comparetheseresultswiththoseintheexample.

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ComputingP-valuesfromtdistributionsonthecalculatorYoucanusethetcdfcommandontheTI-84tocalculateareasunderatdistributioncurve.Thesyntaxistcdf(lowerbound,upperbound,df).Toaccessthiscommand:Press2ndVARS([DISTR])andchoosetcdf(.Let’susethetcdfcommandtocomputetheP-valuesfromthefollowingexamples:Betterbatteries

ThebatterycompanywantstotestH0:μ=30versusHa:μ>30basedonanSRSof15newAAAbatterieswithmeanlifetime hoursandstandarddeviationsx=9.8hours.Two-sidedtest

WhatifyouwereperformingatestofH0:μ=5versusHa:μ 5basedonasamplesizeofn=37andobtainedt=−3.17?Sincethisisatwo-sidedtest,youareinterestedintheprobabilityofgettingavalueoftlessthan−3.17orgreaterthan3.17.

• Betterbatteries:TofindP(t≥1.54),executethecommandtcdf(1.54,100,14).• Two-sidedtest:TofindtheP-valueforthetwo-sidedtestwithdf=36andt=−3.17,executethe

command2*tcdf(−100,−3.17,36).

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One-samplettestonthecalculatorYoucanperformaone-samplettestusingeitherrawdataorsummarystatisticsontheTI-84.Let’suse

thecalculatortocarryoutthetestofH0:μ=5versusHa:μ<5fromthefollowingdissolvedoxygenexample:Thelevelofdissolvedoxygen(DO)inastreamorriverisanimportantindicatorofthewater’sabilitytosupportaquaticlife.AresearchermeasurestheDOlevelat15randomlychosenlocationsalongastream.Herearetheresultsinmilligramsperliter(mg/l):Adissolvedoxygenlevelbelow5mg/lputsaquaticlifeatrisk.StartbyenteringthesampledatainL1.Then,todothetest:

• PressSTAT,chooseTESTSand2:T-test.

• Adjustsettingsasshown.Ifyouselect“Calculate,”thefollowingscreenappears:

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Theteststatisticist=−0.94andtheP-valueis0.1809.

Ifyouspecify“Draw,”youseeatdistributioncurve(df=14)withthelowertailshaded.

Ifyouaregivensummarystatisticsinsteadoftheoriginaldata,youwouldselecttheoption“Stats”insteadof“Data.”

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ConfidenceintervalforadifferenceinproportionsTheTI-84canbeusedtoconstructaconfidenceintervalforp1−p2.We’lldemonstrateusingthefollowingexample:AspartofthePewInternetandAmericanLifeProject,researchersconductedtwosurveysinlate2009.Thefirstsurveyaskedarandomsampleof800U.S.teensabouttheiruseofsocialmediaandtheInternet.Asecondsurveyposedsimilarquestionstoarandomsampleof2253U.S.adults.Inthesetwostudies,73%ofteensand47%ofadultssaidthattheyusesocial-networkingsites.Usetheseresultstoconstructandinterpreta95%confidenceintervalforthedifferencebetweentheproportionofallU.S.teensandadultswhousesocial-networkingsites.Ofn1=800teenssurveyed,X=584saidtheyusedsocial-networkingsites.Ofn2=2253adultssurveyed,X=1059saidtheyengagedinsocialnetworking.Toconstructaconfidenceinterval:PressSTAT,thenchooseTESTSandB:2-PropZInt.Whenthe2-PropZIntscreenappears,enterthevaluesshown.Highlight“Calculate”andpress .

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SignificancetestforadifferenceinproportionsTheTI-84canbeusedtoperformsignificancetestsforcomparingtwoproportions.Here,weusethedatafollowingexample:Highlevelsofcholesterolinthebloodareassociatedwithhigherriskofheartattacks.Willusingadrugtolowerbloodcholesterolreduceheartattacks?TheHelsinkiHeartStudyrecruitedmiddle-agedmenwithhighcholesterolbutnohistoryofotherseriousmedicalproblemstoinvestigatethisquestion.Thevolunteersubjectswereassignedatrandomtooneoftwotreatments:2051mentookthedruggemfibroziltoreducetheircholesterollevels,andacontrolgroupof2030mentookaplacebo.Duringthenextfiveyears,56meninthegemfibrozilgroupand84menintheplacebogrouphadheartattacks.Istheapparentbenefitofgemfibrozilstatisticallysignificant?Performanappropriatetesttofindout.ToperformatestofH0:p1−p2=0:PressSTAT,thenchooseTESTSand6:2-PropZTest.

• Whenthe2-PropZTestscreenappears,enterthevaluesx1=56,n1=2051,x2=84,n2=2030.Specifythealternativehypothesisp1<p2,asshown.

• Ifyouselect“Calculate”andpress ,youaretoldthatthezstatisticisz=−2.47andtheP-valueis0.0068,attopright.Theseresultsagreewiththosefromthepreviousexample.Doyouseethecombinedproportionofheartattacks?

• Ifyouselectthe "Draw"option,youwillseethescreenshown here.

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Two-sampletintervalsonthecalculatorYoucanusethetwo-sampletintervalcommandontheTI-84toconstructaconfidenceintervalforthedifferencebetweentwomeans.We’llshowyouthestepsusingthesummarystatisticsfromthefollowingexample:TheWadeTractPreserveinGeorgiaisanold-growthforestoflong-leafpinesthathassurvivedinarelativelyundisturbedstateforhundredsofyears.Onequestionofinteresttoforesterswhostudytheareais“Howdothesizesoflongleafpinetreesinthenorthernandsouthernhalvesoftheforestcompare?”Tofindout,researcherstookrandomsamplesof30treesfromeachhalfandmeasuredthediameteratbreastheight(DBH)incentimeters.ComparativeboxplotsofthedataandsummarystatisticsfromMinitabareshownbelow.PressSTAT,thenchooseTESTSand0:2-SampTInt….

• ChooseStatsastheinputmethodandenterthesummarystatisticsasshown.

• Entertheconfidencelevel:C-level:.90.ForPooled:choose“No.”We’lldiscusspoolinglater.• HighlightCalculateandpress .

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Two-samplettestswithcomputersoftwareandcalculatorsTechnologygivessmallerP-valuesfortwo-sampletteststhantheconservativemethod.That’sbecausecalculatorsandsoftwareusethemorecomplicatedformulaonpage637ofyourtextbooktoobtainalargernumberofdegreesoffreedom.ThebelowfiguregivescomputeroutputfromFathomandMinitabforthetwo-samplettestfromthecalciumexperiment.TheP-valuesdifferslightlybecauseFathomuses15.59degreesoffreedomwhileMinitabtruncatestodf=15.Let’slookatwhatthecalculatordoes.

• EntertheGroup1(calcium)datainlist1andtheGroup2(placebo)datainlist2.• Toperformthesignificancetest,gotoSTATandchoose4:2-SampTTest.

• Inthe2-SampTTestscreen,specify“Data”andadjustyourothersettingsasshown.

• Highlight “Calculate”andpressENTER.

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Theresultstellusthatthetwo-sampletteststatisticist=1.604,andtheP-valueis0.0644.ThereisenoughevidenceagainstH0torejectitatthe10%significancelevel,butnotatthe5%or1%significancelevels.

Ifyouselect“Draw”insteadof“Calculate,”theappropriatetdistributionwillbedisplayed,showingtheteststatisticandtheshadedareacorrespondingtotheP-value.Note:Thecalculator’s90%confidenceintervalforthetruedifferenceis(-0.4767,11.022).Thisisquiteabitnarrowerthan(-0.754,11.300).

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FindingP-valuesforchi-squaretestsonthecalculatorTofindtheP-valueinthefollowingM&M’Sexamplewithyourcalculator:ThetableshowstheobservedandexpectedcountsforJerome’srandomsampleof60M&M’SMilkChocolateCandies.Usetheχ2cdfcommand.You’llfindthiscommandinthedistributions(DISTR)menuontheTI-84.Weaskfortheareabetweenχ2=10.180andaverylargenumber(we’lluse1000)underthechi-squaredensitycurvewith5degreesoffreedom.Thecommandthatdoesthisisχ2cdf(10.180,1000,5).Asthecalculatorscreenshotsshow,thismethodgivesamorepreciseP-valuethanthechisquaretable.

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Chi-squaregoodness-of-fittestonthecalculatorYoucanusetheTI-84toperformthecalculationsforachi-squaregoodness-of-fittest.We’llusethedatafromthefollowingexampletoillustratethesteps:ThetableshowstheobservedandexpectedcountsforJerome’srandomsampleof60M&M’SMilkChocolateCandies.1.Entertheobservedcountsandexpectedcountsintwoseparatelists.

• ClearL1andL2.• EntertheobservedcountsinL1.CalculatetheexpectedcountsseparatelyandentertheminL2.

2.Performachi-squaregoodness-of-fittest.PressSTAT,arrowovertoTESTSandchooseD:χ2GOF-Test….Entertheinputsshown.IfyouchooseCalculate,you’llgetascreenwiththeteststatistic,P-value,anddf.IfyouchoosetheDrawoption,you’llgetapictureoftheappropriatechi-squaredistributionwiththeteststatisticmarkedandshadedareacorrespondingtotheP-value.

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Chi-squaretestsfortwo-waytablesonthecalculatorYoucanusetheTI-89toperformcalculationsforachi-squaretestforhomogeneity.We’llusethedatafromthefollowingexampletoillustratetheprocess:Marketresearcherssuspectthatbackgroundmusicmayaffectthemoodandbuyingbehaviorofcustomers.Onestudyinasupermarketcomparedthreerandomlyassignedtreatments:nomusic,Frenchaccordionmusic,andItalianstringmusic.Undereachcondition,theresearchersrecordedthenumbersofbottlesofFrench,Italian,andotherwinepurchased.Hereisatablethatsummarizesthedata:1.Entertheobservedcountsinmatrix[A].

• Press2ndX-1(MATRIX),arrowtoEDIT,andchoose1:A.• Enterthedimensionsofthematrix:3x3.

• Entertheobservedcountsfromthetwo-waytableinthesamelocationsinthematrix.

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2.Specifythechi-squaretest,thematrixwheretheobservedcountsarefound,andthematrixwheretheexpectedcountswillbestored.

• PressSTAT,arrowtoTESTS,andchooseC:χ2Test.• Adjustyoursettingsasshown.

3.Choose“Calculate”or“Draw”tocarryoutthetest.Ifyouchoose“Calculate,”youshouldgettheteststatistic,P-value,anddfshownbelow.Ifyouspecify“Draw,”thechi-squaredistributionwith4degreesoffreedomwillbedrawn,theareainthetailwillbeshaded,andtheP-valuewillbedisplayed.4.Toseetheexpectedcounts,gotothehomescreenandaskforadisplayofthematrix[B].

• Press2ndx-1(MATRIX),andchoose2:[B].

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RegressioninferenceonthecalculatorLet’susethedatafromthefollowingexampletoillustratesignificancetestsandconfidenceintervalsontheTI-84.

Enterthex-values(NEAchange)intoL1andthey-values(Fatgain)intoL2.Todoasignificancetest:

• PressSTAT,thenchooseTESTSandF:LinRegTTest….• IntheLineRegTTestScreen,adjusttheinputasshown.Thenhighlight“Calculate”andpress

ENTER.Thelinearregressionttestresultstaketwoscreenstopresent.

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ComparetheseresultswiththeMinitabregressionoutputbelow.

Toconstructaconfidenceinterval:

• PressSTAT,thenchooseTESTSandG:LinRegTInt….• IntheLinRegTIntscreen,adjusttheinputsasshown.Thenhighlight“Calculate”andpress

ENTER.Thelinearregressiontintervalresultstaketwoscreenstopresent.Weshowonlythefirstscreen.

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TransformingtoachievelinearityonthecalculatorWe’llusedatafromthefollowingexampletoillustrateageneralstrategyforperformingtransformationswithlogarithmsontheTI-84.Asimilarapproachcouldbeusedfortransformingdatawithpowersandroots.OnJuly31,2005,ateamofastronomersannouncedthattheyhaddiscoveredwhatappearedtobeanewplanetinoursolarsystem.TheyhadfirstobservedthisobjectalmosttwoyearsearlierusingatelescopeatCaltech’sPalomarObservatoryinCalifornia.OriginallynamedUB313,thepotentialplanetisbiggerthanPlutoandhasanaveragedistanceofabout9.5billionmilesfromthesun.(Forreference,Earthisabout93millionmilesfromthesun.)Couldthisnewastronomicalbody,nowcalledEris,beanewplanet?

Atthetimeofthediscovery,therewerenineknownplanetsinoursolarsystem.Herearedataonthedistancefromthesunandperiodofrevolutionofthoseplanets.Notethatdistanceismeasuredinastronomicalunits(AU),thenumberofearthdistancestheobjectisfromthesun.

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(a) plotsthenaturallogarithmofperiodagainstdistancefromthesunforall9planets.(b) Plotsthenaturallogarithmofperiodagainstthenaturallogarithmofdistancefromthesun

forthe9planets.

• EnterthevaluesoftheexplanatoryvariableinL1andthevaluesoftheresponsevariableinL2.Makeascatterplotofyversusxandconfirmthatthereisacurvedpattern.

• DefineL3tobethelogarithm(base10ore)ofL1andL4tobethelogarithm(samebase)ofL2.Toseewhetheranexponentialmodelfitstheoriginaldata,makeaplotofL4versusL1andlookforlinearity.Toseewhetherapowermodelfitstheoriginaldata,makeaplotofL4versusL3andlookforlinearity.(Weusedlntomatchtheexample.)

• Ifalinearpatternispresent,calculatetheequationoftheleast-squaresregressionlineandstoreitinY1.Fortheplanetdata,weexecutedthecommandLinReg(a+bx)L3,L4,Y1.

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• Constructaresidualplottolookforanydeparturesfromthelinearpattern.ForXlist,enterthelistyouusedastheexplanatoryvariableinthelinearregressioncalculation.ForYlist,usetheRESIDliststoredinthecalculator.Fortheplanetdata,weusedL3astheXlist.

• Tomakeapredictionforaspecificvalueoftheexplanatoryvariablex=k,modifytheregressionequationinY1bychangingxtologxorlnx,ifappropriate.ThenuseY1(k)toobtainthepredictedvalueoflogyorlny.Togetthepredictedvalueofy,use10ˆAnsoreˆAnstoundothe

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logarithmtransformation.Here’sourpredictionoftheperiodofrevolutionforEris,whichisatadistanceof102.15AUfromthesun: