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1 Statistics Walter Antoniotti 21st Century Learning Products Edition ISBN 1-929850-01-8 Copyright 2001 by 21st Century Learning Products All rights reserved. QUICK NOTES is a registered trademark of 21st Century Learning Products Fred and Lulu have been provided by Corel Draw and Image Club Graphics (403-262-8008) respectively. 21st Century Learning Products 227 Baboosic Lake Road Merrimack, NH 03054 havebeenprivilegedtolearnfrommanysuchtea chers,themostpertinentofwhomis , , , ,Jr., , , ,Rindge,NewHampshire, ,graduatestatisticsstudentatGeorgeMasonU niversity,Washington,DC, ,ChairoftheBusinessDepartment,PlymouthSt ateCollege,Plymouth,NewHampshire, ,Nashua,NewHampshire, ,asDirectorandthenDeanofContinuingEducat ionforFranklinPierceCollege,Rindge,NewHa mpshire,WalterhelpedbuildoneofNewEngland ' ,asFranklinPierceCollege'sSpecialAssista ntforProfessionalStudiesProgramDevelopme nt,Walterenjoysteaching,writing, ,Marietta,Ohio,anda MastersofBusinessAdministrationdegreefro mNortheasternUniversity,Boston, , , , , TOPIC ofconcern,a , ' ' , ,whicharerequiredtodoschoolworkandhomewo rk, ,butit , ,a 's (problemsolving,fixorrepair,program) (dance,sports,drivinga bus) (acting,mime,sports) (composing,playingmusic,clapping) (reading,usingwords,publicspeaking,story telling) (socialskills,readingotherpeople,working ina group) (introspection,self-assessment,goalmakin g,vision,planning)

2 (abletodistinguishamong,classify,andusee nvironmentalfeatures) , , WorldofMultipleIntelligenceDetermininged ucationalrequirementsbeginsbymatchinga person' 'sappropriatelevelrequireshonestanalysis ofintelligence,motivation, ,thehealthindustryrequiresdoctorsandnurs es,hospitaldirectorsandfloorsupervisors, , ,theappropriateeducation,consideringmini mumcoreintelligenceandspecialintelligenc erequirements, , ,peopleoftenwanttocompeteata result, , given careerlevel, ,gender, ,Document(understandingforms),andQuantit ativeskillsconductedbytheDepartmentofEdu cationin1992reportedthat15to20% :totalityunderstUdysuchasthestudentsatte ndinga :subsetofa populationsuchasthestudentsinoneclassof :a characteristicofa populationsuchastheaverageageofstudentsa ttendinga :a characteristicofasamplesuchastheaveragea geofstudentsina classofa ,organizing,presenting,analyzing, classcanbeusedtoestimatetheaverageageofs tudentsattendinga ,concise, ,theories, AboutUsingDatainDecisionMakingArray:53, 57,64, 66,68, 70, 73, 76,76, 77,82, 85, 88,93, :High-Low=97-53= 'sfirststepwastomakea range(highnumberminusthelownumber) frequencydistributionwitheachofthefivegr adesrepresentinga ' ,LindaSmithopenedLinda'sVideoShowcase,a ,88,53,66,97,73, 64,82,77,57,93,85,70, 76,68IA continuousexampleofhowLindaSmithcalculat esstatisticsandusesthemwhenmakingbusines sdecisionsforLinda'sVideoShowcaseis , 'sVideoShowcasePracticeSetsProvideReinfo rcementEachQuickNoteschapterisfollowedby a PracticeSetproblem,justturnbacktwopagesa ndlookat 'sMusicEmporiumPracticeSetsdealwithhowDa rinJonescalculatesandusesstatisticswhenm anagingDarin' ' ,DarinJonesopenedDarin'.

3 17,22,17,8,12,15, 14, 16,21,29, # + placeforeveryoutcomeF."ProbabilityFormul aReviewReviewsandTestsThefirst4 a :P(A)=~ :P(A)=~ofQuickNotesStatistics.~ normallydistributedtoPracticeSets,QuickQ uestions,withameanof41hoursanda ( < )IntroducingFredLookAheadandLuluReviewI' remindertolookoverthemainpointsof a :I' 'mheretoremindyoutoreviewonceina ,andwewillreviewtogether:( greatwaytoreviewbasicconcepts,especially whenstudyingfora test! andinallofchapter6 hasbeenprovidedinchapters3 and4 !Thisinformationmayberequiredbythosetaki nga "groupedmeasures" , ,practicesetinstructions, purchased, "compdir"fordirectionsonhowtouseyoursoft warewithQuickNotesStatistics . andin allofchapter6 !Thisinformationmayberequiredforthosetak inga "groupedmeasures" ,Excel,Accounting,Economics,Management, GroupedData6 MeasuringDispersionof GroupedDataDescriptiveStatisticsFormulaR eviewandTestPage241016222834789101112 PartIIProbability.)

4 TheBasisforInferentialStatisticsUndersta ndingProbabilityProbabilityPart\IMultipl icationRulesDiscreteProbabilityDistribut ionsContinuousNormalProbabilityDistribut ionsSamplingandtheSamplingDistributionof theMeansSamplingDistributionsPart"Probab ilityFormulaReviewandTestPartIIII nferentialStatistics4046525866707613 LargeSampleHypothesisTesting8414 LargeSampleHypothesisTestingPartII8815 HypothesisTestingofPopulationProportions 9416 SmallSampleHypothesisTestingUsingStudent 'st Test9817 StatisticalQualityControl10218 Analysisof Variance10819 Two-FactorAnalysisof Variance11420 NonparametricHypothesisTestingofNominalD ata12021 NonparametricHypothesisTestingof OrdinalDataPartI12622 NonparametricHypothesisTestingof OrdinalDataPartII132 InferentialStatisticsExecutiveSummary,Fo rmulaReview,andTest135 PartIVCorrelationandRegression23 CorrelationAnalysis14624 SimpleLinearRegressionAnalysis152 CorrelationandRegressionFormulaReviewand Test158 PartVCumulativeReview25 TaxonomyofStatistics16226 TaxonomyofParametricStatistics16327 ProblemReview164 PartVITheProfessor'sAnswerBookAppendixI CompleteSolutionstoPracticeSetsPS5 AppendixIICompleteSolutionstoQuickQuesti onsQQ3 AppendixIIIC ompleteSolutionstoTestsT35 PartVIIS tatisticalTablesST1 PartVIIII ndexI 1 Chapter1 StatisticsIs AboutUsingDatainDecisionMakingRememberto lookatthekeypointsofa ,thenatureofmeasurement, :totalityunderstudysuchasthestudentsatte ndinga :subsetofa populationsuchasthestudentsinoneclassofa :a characteristicofa populationsuchastheaverageageofstudentsa ttendinga :a characteristicofa samplesuchastheaverageageofstudentsina classofa ,organizing,presenting,analyzing, classcanbeusedtoestimatetheaverageageofs tudentsattendinga.

5 Anactivitysubjecttovariation, ,gradesona :expressednumerically, ,a gradeof85anda :notexpressednumerically, ,a gradeofB :onlyfinitevalues,suchasthecountablenumb ers,canexistonthex-axis, ,defectsina tireandthenumbercorrectona :measurementmayassumeanyvalueassociatedw ithanuninterruptedscale, ,a 2 3 4 5 ,asshownhere,represents1 of4 (t). (levels)determine data' ,providingtheleast productis poor,average, ,noattemptis ,theirmeasureofzerois ,theuseofproportionsis ,becauseoffixedcosts, (datacollectedbytheBureauoftheCensus). publishedbya noncollector(BureauoftheCensusdataprinte dina newspaper). form(questionnaire)iscompletedbytherespo ndent(individual,company,etc.). prescribedlocation(registeringa car). )Samplingerroroccursbecausea sampleistakenratherthana )Nonsamplingerror,whichoccurswithanysurv ey, sampleissmallerthana census, samplemoreaccuratethana ' ,LindaSmithopenedLinda'sVideoShowcase,a ,88,53,66,97,73, 64,82,77,57,93,85,70, 76, 'sfirststepwastomakea range(highnumberminusthelownumber) :53,57,64, 66, 68, 70, 73, 76, 76, 77,82, 85, 88,93,97 Range:High-Low=97-53= frequencydistributionwitheachofthefivegr adesrepresentinga groupingis calleda pieceofdata(outcome) (collectivelyexhaustive)in thattheremustbea classwidthshouldbeeasilydivisible, ,5,10,50,100,500, usedwitha numbersuchasfivetodetermineanapproximate classwidth.

6 # possible,=;4= ,a , ,, tallyis a classareitsfrequency(rateofoccurrence). ,expressedasa decimal,is ,relativefrequency, 'sVideoShowcaseDailyRentalsBeginning1/2/ 98 StatedClassRealClassTallyFrequencyRelati veCumulativeFrequencyLimitsLimits(f)Freq uencyf+nMore-thanLess-than50- ++ histogramisa verticalbarchartdepictinga frequencypolygon(amany-sidedfigure)is a linegraphdepictinga 'ddB h .IdhX1+X250+ relativefrequencypolygonis similartoa frequencypolygonexceptit (Ogives) more-thancumulativefrequencydistribution beginswiththenumberof is associatedwitha smallerandsmallernumberof ,andat theupperclasslimit,y " ~..---:.._" "'":"'"""--'----:.._"--~. :.. ,..j""". ~.. >.100%oc:Q)6-80%~u:~60%:;:;. !!&40%.~~::]20%E::]()More-thanOgive16~14 5512::]0"10~Q)8>~6"3E4::]() ""' '--'--' andPS7ofAppendixI 'sMusic ,DarinJonesopenedDarin' :17,22,17,a,12,15,14,16,21, 29, ' :Thex-axismaybelabeledwiththelowerstated orrealclasslimits,theclassmidpoints, less-thancumulativerelativefrequencypoly gon(Ogive)anda :A less-thancumulativerelativefrequencydist ributioncanbeusedtoestimatethepercentile sdefinedin andPS7 ofAppendixI (collectivelyexhaustive) # + placeforeveryoutcomeData:38,48,27,14,31, 23,46,38,54,26,44,33,17,34,6, frequencydistribution,relativefrequencyd istribution, (f)FrequencyMore-thanLess-than5 MeasuringCentralTendencyofUngroupedDataD on' is usedtomeasuregrades,successinsports,busi nesssuccess, (X) substantial, ,shecountedlastweek' samplebecausesheonlyincludedpartoflastye ar' :3,7, 7,4,1,8, ,readx bar, ( )E!

7 'sdata, weightedmean(xw) datasethasa numberofduplicatevalues,a occurringmorethanonceisassigneda variablenameconsistingofcapitalx witha subscriptanda weight(W)witha 'sVideoShowcaserentstapesfor$2,$3,and$ dayof36,18,and6 respectiverentalsiscalculatedasfollows: (36)($2)+(18)($3)+(6)($4)=$72+$54+$24=$1 50=$250w-36+18+66060' ( )= ,3, :L( )=(1-4)+(3-4)+(8-4)=(-3)+(-1)+(4)= measureofcentraltendencyconcernsit beingseverelyaffectedbya topofthispageasanexample,themeanissmallb ecausea bigsnowstormresultedina daywithonly1 notbeinginfluencedbya ~ 'spositionusingthisexpression.~ ' ,7, 7,4,1,8, was7 (bimodal)ormore(multimodal) ,themodeis nota goodrepresentationofthedata'smiddlevalue ,.Asa result,it is ,knowingthevaluethatoccurredmostoftenis :1,3,4,5,7,7, , (7)10+.5=1"0+.5= ~ ,themedian, ~+.5~+.5= + .5 = ~ .25ofthedistancebetween3,thesecondnumber ,and4, ~+.5~+ .5 = +.5= 4~5 Thisdatais is a ~+52;+.

8 5= + .5 = ~7 Note:7 is .75ofthedistancebetween7 thedifferencebetweenQ3andQ1' percentilewouldbestatedasfollows:100+. !!2.+.5=90(7)+.5=630+.5= ~ result, dailyWalkmansales:8,12, 14, 15,16, 16, 17,17,21, 22, pagestoaboutthesamepagelocationin QuickNotestoseehowLindasolveda ;onefor$149,onefor$159,anda thirdfor$ ;43weretheleastexpensive,90weremoderatel ypriced, ( datafromquestionI,provethatthesumofthede viationsfroma , :5,7,3,8,6,10,9, ;+.54.~+.5 LXL(WxXx) +. measureofcentraltendency? ,provethatthesumofthedeviationsaroundana rithmeticmeanis_Data:3,7, weightedmeanofparkingticketscosting$25, $35,and$45withcorrespondingweightsof10,2 0, $35? \.Lx-~Ix-\.Ll35-227522752245- :N is a (AD) , 'sandtheotherhavinghalfA'sandhalfC' , [inchapter3 ,7, 7,4,1,8,5 (H)minusthelowestvalue(L). -1= , ,a snowstormcausedLindatocloseearlylimiting thatday' ~ ( )2x235-24975244975244945-111615-41618539 64Q50---<225 Total=35 Total=38 Total=213 AlternativeFormula0'2==L:2_(LNX)2= (a2)andstandarddeviation(a) is '2==L(X-~) ("=j;;2=.))]

9 [ (S2)andstandarddeviation(s) [S2=L~~,X)' , whencalculatingthesamplestandarddeviatio n,it willbebias(notrepresentativeof0'). (Assumedataonpage16is sampledata.)orI2(~>lsLX--n-=Vn-1=2213-(3 ~)__J2136-175= , , , , '=100, 2(100)500 200~ smallstandarddeviationarounda testmeanof95soeveryonereceivesa ' methodtoestimatetheminimumproportionofit emsthatarewithina designatednumberofstandarddeviationsfrom themeanfornonsymmetricaldistributionswit hmeansgreaterthan1.~Aswiththeempiricalru le,theestimateworksforboth1 -J< standarddeviationsofthemeanisatleast1 minus1 overK squaredprovidedKisa constantgreaterthan1.[ standard1 -.~=1 -~=1 -~=~deviationsofa meaniscalculatedasfollows:.,,-(2))+ (CV) accomplishedusingthecoefficientof $40millionin saleswillhavea muchlargerstandarddeviationthanDepartmen tB whichhasonly$ 's<rwas$4millionandDepartmentB's<rwas$40 0, ,whichexpressesthestandarddeviationasa percentof themean, ~(100) $4,000,000(100)-10 1.]]]

10 -~(100)-$40,000,000-10 ForDepartmentBCVCJ$400,00001.. =~(100)=$3,000,000(100)= :DepartmentA hadlesssalesdollarvariabilityeventhoughi t hada ,thestandarddeviation, :8,12, 14, 15,16,16,17, 17,21,22, :17~ ' rangeforWalkmansalesforone,two, 'sruletodeterminea tradepublicationthattheaverageWalkmansal esandstandarddeviationfora storehissizeandtypeare18and3 ,areDarin'sWalkmansalesmoreorlessvariabl ethanthoseofhisindustry? theparameterorstatisticit <X-X), ' :5,7,3,8,6,10,9,8x= (usealternativeformula) 'sruletocalculatethepercentageofquestion IIoutcomesthatwillbewithin3 datasetofgradesis normallydistributedandhasa meanof84anda ,it maybenecessarytodetermineparametersandst atisticsusinga :Don'tforgettolookaheadSymbolDefinitionS ymbolDefinitionXthesamplemeanfxfrequency timestheclassmidpointXthemidpointofa class'Lfxsummationoffxfthefrequencyofa [X=Lnbe, 'staperentalsfora [x=X,;x,).. (f) ~Totalsn= ,fx=1, , (52)(7)( )=27, 'sclassCFscumulativefrequencybeforetheme dian'sfrequencyiclassinterval(width) ~todeterminethelocationof themiddlefrequency.]]