Confidence Intervals for Means

1. ConfidenceIntervalsforMeans Chapter8,Section1 StatisticalMethodsII QM3620

2. EstimationandGuessing SupposeIwantedtoguessyourage.Iwouldlookyouovercarefully and thenbasedontheinformationI have, Iwouldventureaguess.OddsarethatIwouldbewrong.Imightactuallybe close, butIwouldmostlikelybewrong. IfitwasreallyimportantthatIgetthatguesscorrect, Icouldwidenmyguessabitto includearangeofvalues.Ifyougotoacarnivalwheretheywillattempttoguessyour ageforaprize, theyconsiderthemselvescorrectiftheyguesswithinacoupleyears. Theygivethemselvesa “marginoferror”sotospeak. Thatmarginoferrorcompensatesforthelimitedinformationtheyhaveonyou.If theycouldgleanmoreinformation,likeyourparent’sage, whenyougraduatedfrom highschool,etc., thentheycouldaccuratelydetermineyourpreciseage.More informationallowsyoutoguessmoreaccuratelyandmoreprecisely.

3. EstimationandGuessing Sohowdoesthatfigureintowhatwearelearning? Animportantpartofstatisticsistheabilitytoguessthevalueofcertainnumbers.Okay, guessisadubiousword.Let’susethephraseestimate; itssoundsmorescientific. Sothisisaclassonguessing…er, Imeanestimating? Partially, butitisnotassimpleasthat.Therearebadwaystoestimateandgoodways to estimate, so wearegoingtolearnhowtogoaboutittherightway.

4. AGoodEstimate Sowhatmakesagoodestimate?  Agoodestimateisaccurate. Agoodestimateisprecise.   Accuracy?Precision? Whatdoyoumean?  Accuracyistheabilitytoestimatecorrectly. Precisionisthespecificityoftheestimate. Thinkofitthisway:Icouldbe100%accurateifIsaidthatyouarebetween0and 115yearsofage, butIwouldbeextremelyimprecise. Alternatively, Icouldguessthatyouareexactly21.6yearsold.Iwouldbe extremelyprecise, butwhataremychancesofguessingcorrectly?     Thereisatrade-offbetweenaccuracyandprecision.Preciseestimateslead toinaccuracy, whereasaccurateestimatesrequirelooserprecision. SohowdoIwinthis“game”ifthereisatrade-off?   Thatiswherestatisticscomesin. Statisticsallowsustocontrolthetradeoffof precisionandaccuracysoyoucanmakethe “best”estimateforyourspecific situation. 

5. HowDid TheyDoIt? Statisticiansrealizedthattheyhadtostartwiththeirbestonenumberestimate, the point estimate.Thisistheonenumberthatyouwouldguessifyouwereonlyallowed onenumber.  So, whatistheonenumber?  Well, thatdependsonwhatyouareestimating.Generallyspeaking, ifyouhaveasampleofdata, the bestonenumberestimateofthestatisticinapopulationisthesamestatisticinthesample.Inother words, thebestonenumberestimateofthemeanofapopulationwouldbethemeanofthesample. Ifyouwanttoestimatetheproportionofapopulation, youstartwiththeproportioninthesample. Ifyouwanttoestimate…well,yougetthepicture.    Sowhyaren’twedoneoncewehavethepointestimate?  That’swhatIwasexplainingearlierwhenwetalkedaboutaccuracyandprecision. Thepointestimateisreallyprecise; it’sonenumber.But, it’salsoboundtobeinaccurate.Youonly havesomeofthedata(i.e. asample)ratherthanallofthedata(i.e. thepopulation)soanythingbased onlimitedinformationisboundtobeinaccurate. Okay, givemeahammer.Let’spoundthatlastpointin.UNLESS YOUHAVE ALLOF THEDATA,YOU ARE WORKING WITHLIMITEDINFORMATION.LIMITEDINFORMATIONLEADS TO INACCURATEESTIMATES.SO, CHOOSE YOURPOISON…INACCURACYORIMPRECISION. IMPRECISIONCANBEDEALT WITHBUTINACCURACYJUSTMAKES YOU WRONG. Wearegoingtobuildarangeormarginoferrorforourestimatetoimproveouraccuracy…and sacrificeabitofprecisionintheprocess.Thatisthebestwaytoapproachsituationswithlimited information.    

6. Populations?Samples? Forgotaboutsamplesandpopulation, eh? Okay, quickreview.Supposewewereinterestedindeterminingtheaveragepricepaid foratickettoabaseballgame.   Whenwesaypopulation, whatwereallymeanis ALLoftheobservationsthatwewanttotalk about.  DowereallywanttomakeanestimateforALLbaseballgames? Wecan’tseriouslybeincludingLittleLeaguegames.Perhapsitisbettertosayallticketstoprofessionalgames. Oh,notinterestedinminorleaguegames.Thepopulationwouldthenbemoreproperlydefinedasallprofessionalmajor leaguegames,orwecouldsayallMLBgames. Whatyear?Thanks!Youhadtothrowsomethingintothemix,didn’tyou.Let’ssaythisyear’sMLBgames.     Nowthatwehaveapopulationdefined, let’stakealookattheobservations.  Itmightbepossibletodeterminethepriceofeveryticketbybadgeringthefrontofficeofeachbaseballteam,butthat doesnotaddressthediscountsoffered.Noteveryticketsgoesforfullprice. Andhowtheheckarewegoingtofindouthowmuchsomeonepaidforaticketwhenthegamehasalreadybeenplayed? Howarewegoingtofindallthosepeople? Bottomlineisthatlookingateveryobservationinapopulationispossible…sometimes…butusuallywehavetolook atthoseobservationsforwhichwehaveaccess. Thisdoesn’tevenstarttoaddresstheproblemslookingatobservationswhenwehavetodestroytheobservationtoget ameasurement.Sayforexample,theneedforacompanylikeGeneralElectrictodeterminetheaveragelifespanoftheir lightbulbs.Ifwedestroyalloftheobservations,thenwherearewe?     Asampleisjustasubsetofthepopulation; onethatwecangetourhandsonandhopefully onethatrepresentsthepopulationwell.Randomsamplesarealwaysthebest. 

7. MarginofError We’vegotasample, andwe’vecalculatedthepointestimate.Inthiscase, let’ssaywehaveasample of100observationsoftheamountspentonaMLBticketthisyear.  Theaveragepriceofthese100ticketswas$23.52.Thiswouldmakeagoodstarting(point)estimatefortheaverage priceforallofthetickets. DoyousupposethatthisisreallytheaveragepricepaidforALLMLBticketsthisyear?   Didn’tthinkso.Youaretoosmarttothinkitwasthateasy.  Nowwehavetobuildamarginoferroraroundourpointestimate.  Ourmarginoferrordependsonhowmuchtheticketpricesvary.  Thinkofitthisway.Ifalloftheticketsin the samplewereexactlythesameprice,thenwemightreasonablythinkthatall ticketsinthepopulationcostthesame.Ifthatisthecase,wecoulduseoursampleaveragetoestimatethepopulation averagewithnomarginoferror.Simple…butunrealistic. Pointoffact: Ifthereisalargevariationinthevaluesofthesampleobservations, wecanpresumethatthereisalsoa largevariationinthevaluesinthepopulation.Thatwouldimplythatoursampleaveragemightchangequiteabit fromsampletosampleastheobservationscouldbequitedifferentineachsample. Anotherpointoffact: Thefewerobservationsinthesample, thelessinformationweevenhaveaboutthepoint estimateandthegreaterwehavetocompensateforalackofcompleteinformation. BottomLine: Themarginoferrorhaseverythingtodowiththenumberofandvarianceintheobservations, andnot thevalueofthepointestimate.

8. Whyit Works Hopefullyyounowunderstandthebasicthoughtsbehindestimatingwitharangeusing amarginoferror.Creatingoneisnotthatdifficult.Justremindyourselfthatboring statisticiansformulatedthissoyoudon’thaveto. TheMechanics   Toactuallycreatetherangeusingamarginoferror, wefocusonhowmuchasamplemeanislikelytovary fromsampletosample. Thistellsushowclosewecanexpectthesamplemeantobefrom hepopulationmean weareinterestedin. Ifthesamplemeansareexpectedtovary substantiallyfromsampletosample, thepopulationmeancouldbefarawayfromthesamplemeanandwe wouldhavetocompensatewithalargemarginoferror.Ifthesamplemeansvarylittle ,thenthepopulation meanwilllikelybeclosetothesamplemeanandourmarginoferrorwillbesmall.

9. Whyit Works • Howdoweknowhowmuchthesamplemeanswillvarywhenweonlyhaveonesample? • Goodquestion!Thegoodnewsisthatthevariationinthesamplemeanscanbedirectlycalculatedfromthevariationin theindividualobservations. • Thereisamathematicalrelationship…thestandarderrorofthesamplemeancanbe estimatedbydividingthestandarddeviationoftheobservationsbythesquarerootofthesamplesize. • StandardError?Yes,toeliminateconfusion,statisticiansusethetermstandarddeviationtorefertothevariationin individualdatapointsandthetermstandarderrortorefertothevariationincalculationslikeameanormedian. • Sothe standarderrorofthesamplemeansisreallyjustthestandarddeviationofthesamplemeans…orhowmuchsample meansvaryfromsampletosample.

10. TheEquation Ournextstepistousethemarginoferror(theamountweaddandsubtractfromourpointestimate)toformup Our “rangeguess”(ormoretechnically,ourconfidence interval.)Thisconfidenceintervalwillgiveusthebest accuracyandprecisioncombination…butweneedastatisticalmultiplefromthestatisticians.Thismultipletakes intoaccountthesamplesizeandtheconfidence(accuracy)level(aswasmentionedinthe“MarginofError”slide). Thegeneralformulaforallconfidenceintervalsis:  Point Estimate StandardErrorof PointEstimate   Multiple Theversionforaconfidenceintervalforapopulationmeanis: Sample Mean StandardErrorof SampleMean   t-value Takeacloselookatthatsecondformula.  Thecalculationtotherighthandsideoftheplus-minussign(±)isthemarginoferror. Thestandarderrorofthesamplemeancanbeestimateddirectlyfromthesampleyoutook(seetheslide“Why It Works”againifyoudon’tbelieveme). Westatedthatthemarginoferrorneedstobecomegreaterifthesamplesizeissmaller(hencewehaveless information)oriftheconfidencelevelishigher(toincreaseaccuracyweloseprecision)…THEREFOREwecan expectagreatermultipleforasmallersamplesizeandagreatermultipleforahigherconfidencelevel.Thet-value hasallofthisbuiltin.   

11. Thet-value • Youaregoingtoneedtobeabletodeterminethet-value (Stats I). • Whydoweuseatvalue?Letmeexplainitthisway.Thetvaluetakesintoaccountthatwe havehadtoestimateeverythingfromourdata. • Weestimatethestandard deviationandthestandarderrorofasamplemeanfromourdata…andthatwasbeforewe evengottoourmainpurposeofestimatingtheoverallmean.Youcanonlyestimatesomany thingsbeforeyoubetterstartcompensatingforit.Thetvaluehasabuiltincompensatorfor thoseintermediateestimates. • Thetextbookreviewshowtolookupthetinthetableonpages340-342, ifyouareso inclined.Personally, Ithinkatablesellsthetvalueshort.Therearetoomanyvaluesthatare neededthatdonotshowuponthetables.Usethecomputer.Itcanfillintheblankswhere notvalueapparentlyexists.ThenextslideshowshowwedoitinExcel. • Bytheway, inthefuturethemultipleisnotalwaysatvalue.Italldepends.Ifyouare estimatingsomeotherstatisticbesidesthepopulationmean, youwilllikelybeusingan altogetherdifferentmultiple. • Thebottomlineisthatthemultipleisinvariablylinkedtotheaccuracyandprecisionofthe estimate.Highconfidence(highaccuracy)leadstoabigmultipleandlessprecision(wide marginoferror), andviceversa.

12. FindingatvalueusingExcel ThetDistribution TofindatvalueinExcel,youuse theTINVfunction.TheTINV functionneedstwobitsof information: 1)Theconfidencelevelyouwant;and 2)thesamplesize(minus1). -tvalue TheTINVfunctiontakestheconfidencelevelbackwards.Itwantstoknow thelevelof “unconfidence”orthe“probabilitythatyouarewrong”.Tobe 95%confidentwouldimplythatyouare5%“unconfident”.The TINV functionlooksuptheareaoutsidetheblueareasinthegraphabove. Thesamplesizehastobereducedbyonewhenlookingupthetvalueto compensatefortheestimationwedidofthestandarddeviation.Trustme onthis…youdon’twanttoseethederivations. Thetdistributionlooksjustlikea normaldistribution,exceptitisabit flatter(whichdependsonthe amountofinformationyouhave– i.e.thesamplesize).Aninfinite samplesizemakesthet distributionidenticaltothenormal distribution. +tvalue So,atvalue fora95% confidence(or.95indecimal format)withasamplesizeof 100wouldbe: =TINV(1-0.95,100-1)

13. Nowthatwehaveeverything Puttingitaltogether…  Wehaveapointestimate…themeanwecalculatedfromthesample…referredtoasx Wehavethemeasureofthevariationinthesample…thestandarddeviationwe calculatedfromthesample…whichisreferredtoass Wehavethesamplesize…thenumberofobservationsinthesample…whichis referredtoasn Wehavethemultiple…thetvaluefromExcel…whichisreferredtoast     Whatdoesthatspell?  xsnt No, itactuallyspellsconfidenceintervalcalculationtime  

14. TheCalculation Rememberourconfidenceintervalequationfromanearlierslide, Sample Mean StandardError ofSampleMean  t-value s n x t Algebraically,theequationlookslike: WherexSampleMean s n t StandardErrorof SampleMean t-value and

15. SomethingtoKeepinMind s n t Marginof Error Variationinthesample: Samplesize: ConfidenceLevel(CL): ass asn asCL Marginof Error Marginof Error Marginof Error    …andmarginoferrorisdirectlyrelatedtoprecision.Asmallermarginoferrorisa morepreciseestimate.

16. So WhatwasthePointof All This? Remember, ourwholereasonforthisseriesof thoughtsandcalculationswastohelpusbest estimatethemeanofsomevariableinalargegroup byusingonlythelimitedinformationprovidedbya samplefromthatgroup.

17. ApplicationTime Let’strythisforreal

18. Business ApplicationHighlights Readthediscussionofintervalestimationwhenthestandarddeviationforthe populationisunknown(page340)andtheexplanationofthetdistributionand degreesoffreedom(pages340-342). Readthebusinessapplicationonpages342-343. HeritageSoftwareoperatesaservicecenterin Tulsa, Oklahomatorespondto servicecallsontheireducationalandbusinesssoftware. Timespenthelpingacustomerisanimportantmeasureofefficiencyofthese operations.Moretimepercustomersmeansthatmoreserviceoperatorsmustbe onstafftohandletheload. Managementwouldliketheaveragecalltimefortheseserviceoperatorstobe estimated. Asampleof25callswascollectedandrecordedwiththeintentofestimatingthe averagetimeforallcallstakenbytheserviceoperators.      

19. The Approach The25callsthatwehavedataonwillserveasabasisforourestimate.Wecancalculatea meanfromthesampleanduseitasapointestimate(onenumberestimate)forthemean lengthofallcalls.Theistheanchorofourintervalestimate. Wealsoneedthestandarddeviationoftheobservations.This, alongwiththesamplesize, willbeusedtocalculatethestandarderrorofthesamplemean. Thetvalue,whichisourmultiple, willbedeterminedusingExcel.Wetrynottousetablesin thisclass. We’llusetwodifferentapproachesonExceltogetustheinformationweneed.Onewilldo mostoftheworkforus.Theotherrequiresustowalkthroughtheprocessstepbystep. KNOW THEMBOTH. Thekeytothiswholeprocessistoremindyourselfthatyouareworkingwithasmallbitof informationfromasample.Youaretryingtoestimatesomethingthatyouhavenowayof verifying.Usingalogicaldata-drivenapproach, wecanderiveinformationfromthesampleto giveusa “bestguess”intervalthatalsoprovidesuswithameansofmeasuringour “accuracy”viatheconfidencelevel.Itisbettertoknowhowcertainyouarethantobe shootinginthedark.     

20. AReiterationofDefinitions ThePointEstimate  Thepointestimateforthepopulationmeanisalwaysthesamplemean.It’s ouronenumberbestguess.  TheConfidenceLevel  Astandardconfidencelevelis95%,whichmeansouroddsare19outof20 (95%)thattheconfidenceintervalcapturesthemeanlengthofaservice centercallforallcalls.Remember, thisisbasedsolelyontheinformationwe havefromasmallsampleofcalls. If95%isn’tgoodenough(notyourdecisioninthiscase), thenyoucanbemore certainbyusinga99%confidenceinterval.Thebadnews: rememberwhat happenstoprecisionwhenwewanttobemoreaccurate.   TheStandardErrorofthePointEstimate  Themeanofasample, whichisourpointestimate, isgoingtochangefrom sampletosample.Takingintoaccountthisvariationisakeypartofformingan intervalestimate.Ifthepointestimateswouldvaryagreatdeal, thenwewill havetoformupaprettywideintervaltotakethatintoaccount. 

21. TheImplied Assumptions Allstatisticalestimatesaregoingtocomewithsomeassumptions.  Thefirstassumptionisthatyouarenottryingtocookthedatatomake theestimatecomeoutinsomepredeterminedway.We’llsaythatyouare supposedtobeunbiased.Thisisnotanexplicitassumption, butitisthere nonetheless. Thesecondbasicassumptionisthatwetookarandomsample   Mostallstatisticalcalculationsassumethatyouarenotchoosingthemembersofthe samplebasedonopinion.Forthemathematicstoworkinthissituation, youneedtobe allowingeachobservationtohaveanequalchancetobeinthesample.Thatiscalleda randomsample…likerollingdicetoseewhichobservationtoinclude. ThinkbacktotheMLBticketpriceexample.Itisprobablyimpossibletorandomly choosefromtheticketsthatweresold.Thuswemayhavetriedtousearandomsample, butitisdoubtfulthatweactuallyachievedrandomness.Thatmakesanyresults questionable.Whensomeonesaystheysampledrandomly, askthemhowtheydidit. Questioneverything. NOTE:Statisticalcalculationsarebuilttodealwithrandomsamples.Anyconfidence intervalsyoucalculatewillneveradjustforbiasorpoorsamplingtechniques.Themargin oferrorisbuilttohandlenaturalvariationinthedata, notincompetence.   

22. TheFormal Assumptions Thetextindicatesthewehavetoassumethattheobservationvalues(calllengthsin thiscase)aremound-shapedornormallydistributed, specificallyifthesamplesizeis small.  Thebrutaltruthisthatevenwithsmallsamplesizes, theconfidenceintervalwearecalculatingwill work, andstatisticianstendtospendtoomuchtimeonthelittledetails. Themain problemhereisnotgoingtobethedistributionoftheobservations, butthemeansbywhichwe choosethem.Ifwecouldreallyrandomlyselectfromallobservations, thenitislikelythatthose observationsareallinthecomputer…andthenwedon’treallyneedtousestatisticstoestimate anything.Wejustcalculatetherealnumberbyusingalloftheobservations. Forreallysmallsamplesizes, youdoneedtomakesurethattheobservationsarenotstronglynon- mound-shaped. Didyoufollowthat?Thatmeans, ifthesampleisreallysmall, say 20orless, thenthe distributionoftheobservationsshouldreallybereasonablymound-shaped, withstrongemphasison theword “reasonably”.Theproblemsonlyseemtooccurinasamplewithastrongbi-modal distribution(whichmeansthatobservationsformwhatlooksmoreliketwomoundsratherthan one.) Howdoyoucheckthedistributionoftheobservations?Iamgladyouasked.Plotahistogram.They arenotdifficulttodowithExcelandyoumightbesurprisedontheamountofinformationyoucan gleanaboutsomevariablebylookingatahistogram.

23. Problem 1 HeritageSoftwareoperatesaservicecenterin Tulsa, Oklahomatorespondto servicecallsontheireducationalandbusinesssoftware. Timespenthelpingacustomerisanimportantmeasureofefficiencyofthese operations.Moretimepercustomersmeansthatmoreserviceoperatorsmustbe onstafftohandletheload. Managementwouldliketheaveragecalltimefortheseserviceoperatorstobe estimated. Asampleof25callswascollectedandrecordedwiththeintentofestimatingthe averagetimeforallcallstakenbytheserviceoperators.

24. Problem 2 Medlin & Associates is a CPA firm that is conducting an audit of a discount chain store. Management would like to have some measure of the amount of error that is occurring during checkout operations. A sample of 20 transactions are taken and the amount and direct of a mischarge is noted. Positives values indicate overcharges to the customers; negative values are undercharges. The problem asks for a 90% confidence interval. That means that our interval will be narrower, but our degree of certainty is less. We are trading accuracy for precision.