Business Statistics A DecisionMaking Approach, 6th edition.ppt

businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 1 businessstatistics adecision makingapproach6thedition chapter7estimatingpopulationvalues businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 2 chaptergoals aftercompletingthischapter youshouldbeableto distinguishbetweenapointestimateandaconfidenceintervalestimateconstructandinterpretaconfidenceintervalestimateforasinglepopulationmeanusingboththezandtdistributionsdeterminetherequiredsamplesizetoestimateasinglepopulationmeanwithinaspecifiedmarginoferrorformandinterpretaconfidenceintervalestimateforasinglepopulationproportion businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 3 confidenceintervals contentofthischapterconfidenceintervalsforthepopulationmean whenpopulationstandarddeviation isknownwhenpopulationstandarddeviation isunknowndeterminingtherequiredsamplesizeconfidenceintervalsforthepopulationproportion p businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 4 pointandintervalestimates apointestimateisasinglenumber aconfidenceintervalprovidesadditionalinformationaboutvariability pointestimate lowerconfidencelimit upperconfidencelimit widthofconfidenceinterval businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 5 pointestimates wecanestimateapopulationparameter withasamplestatistic apointestimate mean proportion p p x businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 6 confidenceintervals howmuchuncertaintyisassociatedwithapointestimateofapopulationparameter anintervalestimateprovidesmoreinformationaboutapopulationcharacteristicthandoesapointestimatesuchintervalestimatesarecalledconfidenceintervals businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 7 confidenceintervalestimate anintervalgivesarangeofvalues takesintoconsiderationvariationinsamplestatisticsfromsampletosamplebasedonobservationfrom1samplegivesinformationaboutclosenesstounknownpopulationparametersstatedintermsoflevelofconfidencenever100 sure businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 8 estimationprocess mean isunknown population randomsample meanx 50 sample businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 9 generalformula thegeneralformulaforallconfidenceintervalsis pointestimate criticalvalue standarderror businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 10 confidencelevel confidencelevelconfidenceinwhichtheintervalwillcontaintheunknownpopulationparameterapercentage lessthan100 businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 11 confidencelevel 1 supposeconfidencelevel 95 alsowritten 1 95arelativefrequencyinterpretation inthelongrun 95 ofalltheconfidenceintervalsthatcanbeconstructedwillcontaintheunknowntrueparameteraspecificintervaleitherwillcontainorwillnotcontainthetrueparameternoprobabilityinvolvedinaspecificinterval continued businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 12 confidenceintervals populationmean unknown confidence intervals populationproportion known businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 13 confidenceintervalfor known assumptionspopulationstandarddeviation isknownpopulationisnormallydistributedifpopulationisnotnormal uselargesampleconfidenceintervalestimate businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 14 findingthecriticalvalue considera95 confidenceinterval z 025 1 96 z 025 1 96 pointestimate lowerconfidencelimit upperconfidencelimit zunits xunits pointestimate 0 businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 15 commonlevelsofconfidence commonlyusedconfidencelevelsare90 95 and99 confidencelevel confidencecoefficient zvalue 1 281 6451 962 332 573 083 27 80 90 95 98 99 998 999 80 90 95 98 99 99 8 99 9 businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 16 intervalandlevelofconfidence confidenceintervals intervalsextendfromto 100 1 ofintervalsconstructedcontain 100 donot samplingdistributionofthemean x x1 x2 businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 17 marginoferror marginoferror e theamountaddedandsubtractedtothepointestimatetoformtheconfidenceinterval example marginoferrorforestimating known businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 18 factorsaffectingmarginoferror datavariation eas samplesize n easnlevelofconfidence 1 eif1 businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 19 example asampleof11circuitsfromalargenormalpopulationhasameanresistanceof2 20ohms weknowfrompasttestingthatthepopulationstandarddeviationis 35ohms determinea95 confidenceintervalforthetruemeanresistanceofthepopulation businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 20 example asampleof11circuitsfromalargenormalpopulationhasameanresistanceof2 20ohms weknowfrompasttestingthatthepopulationstandarddeviationis 35ohms solution continued businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 21 interpretation weare98 confidentthatthetruemeanresistanceisbetween1 9932and2 4068ohmsalthoughthetruemeanmayormaynotbeinthisinterval 98 ofintervalsformedinthismannerwillcontainthetruemeananincorrectinterpretationisthatthereis98 probabilitythatthisintervalcontainsthetruepopulationmean thisintervaleitherdoesordoesnotcontainthetruemean thereisnoprobabilityforasingleinterval businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 22 confidenceintervals populationmean unknown confidence intervals populationproportion known businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 23 ifthepopulationstandarddeviation isunknown wecansubstitutethesamplestandarddeviation sthisintroducesextrauncertainty sincesisvariablefromsampletosamplesoweusethetdistributioninsteadofthenormaldistribution confidenceintervalfor unknown businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 24 assumptionspopulationstandarddeviationisunknownpopulationisnormallydistributedifpopulationisnotnormal uselargesampleusestudent stdistributionconfidenceintervalestimate confidenceintervalfor unknown continued businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 25 student stdistribution thetisafamilyofdistributionsthetvaluedependsondegreesoffreedom d f numberofobservationsthatarefreetovaryaftersamplemeanhasbeencalculatedd f n 1 businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 26 ifthemeanofthesethreevaluesis8 0 thenx3mustbe9 i e x3isnotfreetovary degreesoffreedom df idea numberofobservationsthatarefreetovaryaftersamplemeanhasbeencalculatedexample supposethemeanof3numbersis8 0letx1 7letx2 8whatisx3 here n 3 sodegreesoffreedom n 1 3 1 2 2valuescanbeanynumbers butthethirdisnotfreetovaryforagivenmean businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 27 student stdistribution t 0 t df 5 t df 13 t distributionsarebell shapedandsymmetric buthave fatter tailsthanthenormal standardnormal twithdf note tzasnincreases businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 28 student sttable uppertailarea df 25 10 05 1 1 000 3 078 6 314 2 0 817 1 886 2 920 3 0 765 1 638 2 353 t 0 2 920 thebodyofthetablecontainstvalues notprobabilities let n 3df n 1 2 10 2 05 2 05 businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 29 tdistributionvalues withcomparisontothezvalue confidencetttzlevel 10d f 20d f 30d f 801 3721 3251 3101 28 901 8121 7251 6971 64 952 2282 0862 0421 96 993 1692 8452 7502 57 note tzasnincreases businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 30 example arandomsampleofn 25hasx 50ands 8 forma95 confidenceintervalfor d f n 1 24 sotheconfidenceintervalis 46 698 53 302 businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 31 approximationforlargesamples sincetapproacheszasthesamplesizeincreases anapproximationissometimesusedwhenn 30 technicallycorrect approximationforlargen businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 32 determiningsamplesize therequiredsamplesizecanbefoundtoreachadesiredmarginoferror e andlevelofconfidence 1 requiredsamplesize known businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 33 requiredsamplesizeexample if 45 whatsamplesizeisneededtobe90 confidentofbeingcorrectwithin 5 alwaysroundup sotherequiredsamplesizeisn 220 businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 34 if isunknown ifunknown canbeestimatedwhenusingtherequiredsamplesizeformulauseavaluefor thatisexpectedtobeatleastaslargeasthetrue selectapilotsampleandestimate withthesamplestandarddeviation s businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 35 confidenceintervals populationmean unknown confidence intervals populationproportion known businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 36 confidenceintervalsforthepopulationproportion p anintervalestimateforthepopulationproportion p canbecalculatedbyaddinganallowanceforuncertaintytothesampleproportion p businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 37 confidenceintervalsforthepopulationproportion p recallthatthedistributionofthesampleproportionisapproximatelynormalifthesamplesizeislarge withstandarddeviationwewillestimatethiswithsampledata continued businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 38 confidenceintervalendpoints upperandlowerconfidencelimitsforthepopulationproportionarecalculatedwiththeformulawherezisthestandardnormalvalueforthelevelofconfidencedesiredpisthesampleproportionnisthesamplesize businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 39 example arandomsampleof100peopleshowsthat25areleft handed forma95 confidenceintervalforthetrueproportionofleft handers businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 40 example arandomsampleof100peopleshowsthat25areleft handed forma95 confidenceintervalforthetrueproportionofleft handers 1 2 3 continued businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 41 interpretation weare95 confidentthatthetruepercentageofleft handersinthepopulationisbetween16 51 and33 49 althoughthisrangemayormaynotcontainthetrueproportion 95 ofintervalsformedfromsamplesofsize100inthismannerwillcontainthetrueproportion businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 42 changingthesamplesize increasesinthesamplesizereducethewidthoftheconfidenceinterval example ifthesamplesizeintheaboveexampleisdoubledto200 andif50areleft handedinthesample thentheintervalisstillcenteredat 25 butthewidthshrinksto 19 31 businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 43 findingtherequiredsamplesizeforproportionproblems solveforn definethemarginoferror pcanbeestimatedwithapilotsample ifnecessary orconservativelyusep 50 businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 44 whatsamplesize howlargeasamplewouldbenecessarytoestimatethetrueproportiondefectiveinalargepopulationwithin3 with95 confidence assumeapilotsampleyieldsp 12 businessstatistics adecision makingapproach 6e 2005prentice hall inc chap7 45 whatsamplesize solution for95 confidence usez 1 96e 03p 12 souseth