Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
2/3/2017 Section 7-3 Estimating a Population Mean SAMPLE MEANS 1. Formanypopulations,thedistributionof samplemeans ̅ tendstobemore consistent(withlessvariation)thanthe distributionsofothersamplestatistics. 2. Forallpopulations,thesamplemean ̅ is anunbiasedestimatorofthepopulation meanµ,meaningthatthedistributionof samplemeanstendstocenteraboutthe valueofthepopulationmeanµ. POINT ESTIMATE PointEstimate: Thesamplemean ̅ isthe bestpointestimate (orsinglevalueestimate) ofthepopulationmean . 1 2/3/2017 COMMENT Itisrarethatwewanttoestimatethe unknownvalueofapopulationmeanbutwe somehowknowthevalueofthepopulation standarddeviation .Therealisticsituationis that isnotknown.(Webeginthissectionby consideringthismorerealisticscenario.) When isnotknown,weconstructthe confidenceintervalbyusingtheStudent distributioninsteadofthestandardnormal distribution. ASSUMPTIONS FOR CONFIDENCE INTERVAL OF MEAN WITH σ NOT KNOWN 1. Thesampleisasimplerandomsample. 2. Eitherorbothofthefollowingconditions aresatisfied: • Thepopulationisnormally distributed • n >30 THE STUDENT t DISTRIBUTION Ifapopulationhasanormaldistribution,then thedistributionof ̅ isaStudentt distribution forallsamplesofsize .TheStudent distributionisoftenreferredto asthe distribution. 2 2/3/2017 DEGREES OF FREEDOM Findingacriticalvalue / requiresavaluefor thedegreesoffreedom (ordf).Ingeneral, thenumberofdegreesoffreedomfora collectionofsampledataisthenumberof samplevaluesthatvaryaftercertainrestraints havebeenimposedonthedatavalues.Forthe methodsofthissection,thenumberofdegrees offreedomisthesamplesizeminus1;thatis, degreesoffreedom 1 FINDING THE CRITICAL VALUE Acriticalvalue / canbefoundusingTableA‐3which isfoundonpage586,insidethebackcover,andonthe FormulasandTablescard.Ifthetabledoesnotinclude thenumberofdegreesoffreedomthatyouneed,you could • usetheclosestvalue • beconservativeandusingthenextlowernumber ofdegreesoffreedom • interpolate.Forexample,ifyouhave55degrees offreedom,youcouldfindthemeanofthe criticalvaluesfor50and60. Tokeepthingssimple,wewillusetheclosestvalue. MARGIN OF ERROR ESTIMATE OF µ (WITH σ NOT KNOWN) / · where(1 )istheconfidenceleveland has 1 degreesoffreedom. / 3 2/3/2017 CONFIDENCE INTERVAL ESTIMATE OF THE POPULATION MEAN μ (WITH σ NOT KNOWN) ̅ where ̅ / · CONSTRUCTING A CONFIDENCE INTERVAL FOR μ (σ NOT KNOWN) 1. 2. Verifythatthetworequiredassumptionsaremet. With unknown(asisusuallythecase),use 1 degreesoffreedomandrefertoTableA‐3tofindthe criticalvalue / thatcorrespondstothedesired confidenceinterval.(Fortheconfidencelevel,refer to“AreainTwoTails.”) 3. Evaluatethemarginoferror 4. Findthevaluesof ̅ and ̅ .Substitutethese inthegeneralformatoftheconfidenceinterval: ̅ ̅ . Roundtheresultusingthesameround‐offruleon thefollowingslide. 5. / · ROUND-OFF RULE FOR CONFIDENCE INTERVALS USED TO ESTIMATE μ 1. Whenusingtheoriginalsetofdata to constructtheconfidenceinterval,round theconfidenceintervallimitstoonemore decimalplace thanisusedfortheoriginal dataset. 2. Whentheoriginalsetofdataisunknown andonlythesummarystatistics , ̅ , areused,roundtheconfidenceinterval limitstothesamenumberofplaces as usedforthesamplemean. 4 2/3/2017 FINDING A CONFIDENCE INTERVAL FOR µ WITH TI-83/84 1. 2. 3. 4. 5. 6. 7. 8. 9. Select STAT. Arrow right to TESTS. Select 8:TInterval…. Select input (Inpt) type: Data or Stats. (Most of the time we will use Stats.) Enter the sample mean, x. Enter the sample standard deviation, Sx. Enter the size of the sample, n. Enter the confidence level (C-Level). Arrow down to Calculate and press ENTER. PROPERTIES OF THE STUDENT t DISTRIBUTION 1. TheStudent distributionisdifferentfor differentsamplesizes(seeFigurebelow forthecases 3 and 12). PROPERTIES OF THE STUDENT t DISTRIBUTION (CONTINUED) 2. 3. 4. 5. TheStudent distributionhasthesamegeneral symmetricbellshapeasthenormaldistribution butitreflectsthegreatervariability(withwider distributions)thatisexpectedwithsmallsamples. TheStudentt distributionhasameanof 0 (justasthestandardnormaldistributionhasa meanof 0). ThestandarddeviationoftheStudentt distributionvarieswiththesamplesizeandis greaterthan1(unlikethestandardnormal distribution,whichhasa 1). Asthesamplesize getslarger,theStudent distributiongetsclosertothenormaldistribution. 5 2/3/2017 ESTIMATING A MEAN WHEN σ IS KNOWN Requirements: 1. Thesampleisasimplerandomsample. 2. Eitherorbothoftheseconditionsaresatisfied:The populationisnormallydistributedor 30. ConfidenceInterval: ̅ ̅ wherethemarginoferror isfoundfromthefollowing: / ⋅ Note:Thecriticalvalue / isfoundfromTableA‐2(the standardnormaldistribution). FINDING A CONFIDENCE INTERVAL FOR µ WITH TI-83/84 1. 2. 3. 4. 5. 6. 7. 8. 9. SelectSTAT. ArrowrighttoTESTS. Select7:ZInterval…. Selectinput(Inpt)type:Data orStats.(Mostof thetimewewilluseStats.) Enterthestandarddeviation,σ. Enterthesamplemean, . Enterthesizeofthesample,n. Entertheconfidencelevel(C‐Level). ArrowdowntoCalculate andpressENTER. SAMPLE SIZE FOR ESTIMATING µ ⁄ · where zα/2 = criticalz scorebasedondesired confidencelevel E = desiredmarginoferror σ = populationstandarddeviation 6 2/3/2017 ROUND-OFF RULE FOR SAMPLE SIZE n Whenfindingthesamplesize ,iftheuseof theformulaonthepreviousslidedoesnot resultinawholenumber,alwaysincrease the valueof tothenextlarger wholenumber. FINDING THE SAMPLE SIZE WHEN σ IS UNKNOWN 1. Usetherangeruleofthumb(seeSection3‐3) toestimatethestandarddeviationasfollows: range/4. 2. Startthesamplecollectionprocesswithout knowing and,usingthefirstseveralvalues, calculatethesamplestandarddeviations and useitinplaceofσ.Theestimatedvalueof canthenbeimprovedasmoresampledataare obtained,andtherequiredsamplesizecanbe adjustedasyoucollectmoresampledata. 3. Estimatethevalueofσ byusingtheresultsof someotherearlierstudy. CHOOSING THE APPROPRIATE DISTRIBUTION 7 2/3/2017 CHOOSING BETWEEN z AND t Conditions Method σ notknownandnormallydistributed population or σ notknownand 30 UseStudent distribution σ knownandnormallydistributed population or σ knownand 30 Usenormal( ) distribution Populationisnotnormallydistributed and 30. Useanonparametric methodor bootstrapping FINDING A POINT ESTIMATE AND E FROM A CONFIDENCE INTERVAL Pointestimateofµ: ̅ upperconfidencelimit lowerconfidencelimit 2 Marginoferror: upperconfidencelimit lowerconfidencelimit 2 8