GAC-010-Test-3-Sample-Questions-2

GAC010Test3-ExamSampleQuestionsQuestion1[11marks]a.)Thefollowingdatashowtheages,tothenearestyear,of60employeeswhentheystartedworkingatacertainfactory.AgesNoOfEmployees10-19.9320-29.91630-39.91840-49.91050-59.9860-69.95Total60i.)Graphthefrequencyhistogram[3marks]ii.)Howmanywereunder40whentheystartedworkingthere?[1mark]iii.)Howmanywereaged30andoverwhentheyfirststartedworkingthere?[1mark]b.)Usethetablebelowtodetermine:i.)Taxableincomeon$57678whenAllowabledeductionsfortheyearamountto$1745ii.)HowmuchannualtaxapersonwouldpaywhenhisGrossincomeis$57678?TaxableincomeTaxonthisincome$1-$6000Nil$6001-$2000017centsforeach$1over$6000$20001-$50000$2380+30centsforeach$1over$20000$50001-$60000$11380+42centsforeach$1over$50000$60001andover$15580+47centsforeach$1over$60000Source:2001ATOTaxGuideSolutions1.a.)i.)a.)ii.)3+16+18=37a.)iii.)18+10+8+5=41b.)i.)TaxableIncome=gross-deductions=$57678-$1745=$55933b.)ii.)TaxPayableon$57678is$57678-$1745=$55933on$50000=$11380on$55933-$50000=$5933*.42=$2491.86TaxPayableon$55933=$11380+$2491.86=$13871.86Question2[16marks]Thestemandleafdiagramdisplaysthescoresobtainedby40studentsintheirfinalMathematicsexam.StemLeaf4575357862456897111234455666688983557889933588a.)Calculatethemean,medianandmode.[5marks]b.)Calculatetherange,andthelowerandupperquartiles.[5marks]c.)Constructaboxandwhiskerdiagram.[3marks]d.)Hence,commentonthesymmetryorotherwiseofthedata.[1mark]e.)Findthevarianceandstandarddeviation.[2marks]Solutionsa.)Mean=SxnMedian=X(n+1)=X41/2=X20.52Mode=76b.)Range=HighestValue-Lowestvalue=98-45=53Q1=X(n+1)4Q1=X(41)4=X10.25Q3=X3(n+1)4Q3=X3(41)4=X30.75c.)BoxandWhiskerdiagram______________________________________________________3040506070809045Q1MdQ398d.)Thedataisskewedtotheleft(negativelyskewed)duetothegreaternumberofsmallervalues.e.)VarianceisS^2=S(x-^2*fn-1Standarddeviationiss=√s^2Question3[15marks]a.)HouseholdsinacertaintownweresurveyedtodeterminewhethertheywouldsubscribetoaPayTVChannel.Thehouseholdswereclassifiedaccordingto“high”,“medium”,and“low”incomelevels.Theresultsofthesurveyweresummarizedinthefollowingtable.IncomeLevelWillSubscribeWillnotsubscribeTotalHigh165016503300Medium160063407940Low52019002420Total3770989013660Ahouseholdischosenatrandom.Findtheprobabilitythatthehousehold:i.)willsubscribe.[1mark]ii.)hasalowincomelevel.[1mark]iii.)willsubscribeandahasalow-incomelevel.[2marks]iv.)willsubscribeorhasalowincomelevel.[2marks]v.)willsubscribegiventhatthehouseholdhasalowincomelevel.[2marks]b.)i.)12peoplearetobedividedintotwogroupswith8inoneand4intheother.Inhowmanywayscanthisbedone?[2marks]ii.)Atelevisionfirmhasinstock9televisionsetswith32inchtubesand6setswith28inchtubes.Foursetsareremovedatrandom.Whatistheprobabilitythattheyareallofthesamesize?[3marks]iii.)AparticularfirmhasintypeofTVtubeisalmostcertaintofailwithin3500hours.Ifthelifeofatubefollowsanormaldistributionwithastandarddeviationof150hours,findthemeanlifeofthesetubes.[2marks]Solutions3.a)i.)(willsubscribe)=3770=0.2713660ii.)P(lowlevelincome)=2420=0.1713660iii.)P(willsubscribeandhaslowlevelincome)=520=0.0313660iv.)P(willsubscribeorhaslowlevelincome)=3770+190013660=0.41v.)P(willsubscribegivenhouseholdhaslowlevelincome)=520=0.212420b.)i.)NumberofarrangementsnCr=n!r!(n-r)!12C8*4C4=12!*4!8!(12-8)!4!(4-4)!=12!*4!8!(4!)4!(0)!=12*11*10*9*8!*18!(4)!=12*11*10*94*3*2*1=11880=49524ii.)P(samesize)=P(LLLL)+P(SSSS)=(9*8*7*6)+(6*5*4*3)1514131215141312=0.10iii.)Almostcertaintofailis3standarddeviationsabovethemean.therefore,meanisμ+3σ=3500μ=3500-3σ=3500-3*150=3500-450=3050or,3500-150=33503350-150=32003200-150=30503050320033503500Question4[13marks]Mostmovietrailersarenormallydistributedwithameanof2minutesandastandarddeviationof30seconds.a.)Findtheprobabilitythatamovietrailerselectedatrandomwillbei.)lessthan160secondsii.)lessthan80secondsiii.)over180secondsiv.)over100secondsbutlessthan140seconds.b.)Ifyouwatch100ofthesemovietrailers,howmanywillbeunder140seconds?Solutions4.a.)i.)LetX=timeinsecondsμ=120,σ=30X:N(120,30^2)z=X-μ=X-120σ30P(X160)=P(Z160-120)=P(Z140)3030=P(Z4.66)=.5+P(0≤Z≤1.66)=.5+.4515=.9515120160________________________________________04.66Za.)ii.)P(X80)=P(Z80-120)=P(Z-40)3030=P(Z-1.33)=.5-P(-1.33Z0)=.5-P(0Z1.33)=.5-.4082=.091880120_____________________________________________-1.330Za.)iii)P(X180)=P(Z180-120)=P(Z60)3030=P(Z2)=.5-P(0Z2)=.5-.4772=.0228120180__________________________________________02Za.)iv)P(100X140)=P(100-120Z150-120)3030=P(-20Z30)3030=P(-.66Z1)=P(-.66Z0)+P(0Z1)=P(0Z.66)+P(0Z1)=.2454+.3413=.5867100120150_____________________________________________-.6601Zb.)P(X140)=P(Z140-130)30=P(Z.33)=.5+P(0Z.33)=.5+.1293=.6293So,for100trailers:100*.6293=62.93=about63trailerswillbeunder140seconds.120140______________________________________________0.33Question5[14marks]a.)Frankrecentlyboughtasmallboat.Heput$8000asdownpaymentandborrowedtherestofthemoney,withpaymentsof$400attheendofevery4monthsfor4years.Iftheinterestis8%p.a.compoundedthreetimesayear,whatisthecurrentvalueoftheboat?[4marks]Unit7F,Ex7.7,p.