Te Pāngarau me te Tauanga (Tauanga), Kaupae 3, 2015 · 20 724 17 807 19 221 ... Kua tuhia e...
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915855 3 SUPERVISOR'S USE ONLY 91585M © Mana Tohu Mātauranga o Aotearoa, 2015. Pūmau te mana. Kia kaua rawa he wāhi o tēnei tuhinga e whakahuatia ki te kore te whakaaetanga tuatahi a te Mana Tohu Mātauranga o Aotearoa. MĀ TE KAIMĀKA ANAKE TAPEKE Te Pāngarau me te Tauanga (Tauanga), Kaupae 3, 2015 91585M Te whakahāngai ariā tūponotanga hei whakaoti rapanga 2.00 i te ahiahi Rāpare 19 Whiringa-ā-rangi 2015 Whiwhinga: Whā Paetae Kaiaka Kairangi Te whakahāngai ariā tūponotanga hei whakaoti rapanga. Te whakahāngai ariā tūponotanga mā te whakaaro whaipānga hei whakaoti rapanga. Te whakahāngai ariā tūponotanga mā te whakaaro waitara hōhonu hei whakaoti rapanga. Tirohia mēnā e rite ana te Tau Ākonga ā-Motu (NSN) kei runga i tō puka whakauru ki te tau kei runga i tēnei whārangi. Me whakamātau koe i ngā tūmahi KATOA kei roto i tēnei pukapuka. Tuhia ō mahinga KATOA. Tirohia mēnā kei a koe te pukapuka Tikanga Tātai me ngā Tūtohi L3–STATMF. Mēnā ka hiahia whārangi atu anō koe mō ō tuhinga, whakamahia ngā whārangi wātea kei muri o tēnei pukapuka, ka āta tohu ai i te tau tūmahi. Tirohia mēnā e tika ana te raupapatanga o ngā whārangi 2 – 15 kei roto i tēnei pukapuka, ka mutu, kāore tētahi o aua whārangi i te takoto kau. ME HOATU RAWA KOE I TĒNEI PUKAPUKA KI TE KAIWHAKAHAERE Ā TE MUTUNGA O TE WHAKAMĀTAUTAU.
Transcript of Te Pāngarau me te Tauanga (Tauanga), Kaupae 3, 2015 · 20 724 17 807 19 221 ... Kua tuhia e...
915855
3SUPERVISOR'S USE ONLY
9 1 5 8 5 M
© Mana Tohu Mātauranga o Aotearoa, 2015. Pūmau te mana. Kia kaua rawa he wāhi o tēnei tuhinga e whakahuatia ki te kore te whakaaetanga tuatahi a te Mana Tohu Mātauranga o Aotearoa.
MĀ TE KAIMĀKA ANAKE
TAPEKE
Te Pāngarau me te Tauanga (Tauanga), Kaupae 3, 2015
91585M Te whakahāngai ariā tūponotanga hei whakaoti rapanga
2.00 i te ahiahi Rāpare 19 Whiringa-ā-rangi 2015 Whiwhinga: Whā
Paetae Kaiaka KairangiTe whakahāngai ariā tūponotanga hei whakaoti rapanga.
Te whakahāngai ariā tūponotanga mā te whakaaro whaipānga hei whakaoti rapanga.
Te whakahāngai ariā tūponotanga mā te whakaaro waitara hōhonu hei whakaoti rapanga.
Tirohia mēnā e rite ana te Tau Ākonga ā-Motu (NSN) kei runga i tō puka whakauru ki te tau kei runga i tēnei whārangi.
Me whakamātau koe i ngā tūmahi KATOA kei roto i tēnei pukapuka.
Tuhia ō mahinga KATOA.
Tirohia mēnā kei a koe te pukapuka Tikanga Tātai me ngā Tūtohi L3–STATMF.
Mēnā ka hiahia whārangi atu anō koe mō ō tuhinga, whakamahia ngā whārangi wātea kei muri o tēnei pukapuka, ka āta tohu ai i te tau tūmahi.
Tirohia mēnā e tika ana te raupapatanga o ngā whārangi 2 – 15 kei roto i tēnei pukapuka, ka mutu, kāore tētahi o aua whārangi i te takoto kau.
ME HOATU RAWA KOE I TĒNEI PUKAPUKA KI TE KAIWHAKAHAERE Ā TE MUTUNGA O TE WHAKAMĀTAUTAU.
Te Pāngarau me te Tauanga (Tauanga) 91585M, 2015
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TŪMAHI TUATAHI
(a) EwhakaatuanatepapatauewhaiakeitemahaongāwakaipūrongohiakiNgāPirihimanaoAotearoakuatāhaetiamaiitetau2011kite2013,ā,metemahaongāwakaerēhitatiaanakiTeWakaKotahiiauatauanō.
2011 2012 2013Te maha o ngā waka i pūrongohia kua tāhaetia 20 724 17 807 19 221
Te maha o ngā waka i rēhitatia 4 210 511 4 248 612 4 315 539
(i) KoēheaoēneitauinuiaketemōreareatangawhānuikatāhaetiatētahiwakaiAotearoa?
Tautokonatōtuhingakingātātaingatōtika.
(ii) HomaikiaKOTAHItepūtakeheahainohoaingāmōreareatangaitātaitiaitewāhanga(i)heiwhakatautatanoaihootemōreareatangawhānuitūturukatāhaetiahewakaitauatau.
(iii) Keitehiahiatētahirangatiraotētahiwakakitewhakamahiitemōreareatangawhānui katāhaetiahewakaiAotearoaitetau2013heiwhakatautataitemōreareatangakatāhaetiatōnaakewakaitetau2015.
Matapakitiaheahaatuanōngāmeaheiwhakaaroaromāterangatiraotewakaheiwhakatautataitēneimōreareatanga.
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QUESTION ONE
(a) ThefollowingtableshowsthenumberofvehiclesreportedtotheNZPoliceasstolenover2011to2013,andthenumberofvehiclesregisteredwiththeNZTransportAgencyineachoftheseyears.
2011 2012 2013Number of vehicles reported as stolen 20724 17807 19 221
Number of vehicles registered 4210511 4248612 4 315 539
(i) WhichoftheseyearshadthegreatestoverallriskofavehiclebeingstoleninNew
Zealand?
Supportyouranswerwithappropriatecalculations.
(ii) GiveONEreasonwhytheriskscalculatedinpart(i)areonlyestimatesofthetrueoverallriskofavehiclebeingstoleninthatyear.
(iii) AcarownerwantstousetheoverallriskofacarbeingstoleninNewZealandduring2013toestimatetheriskoftheirowncarbeingstolenduring2015.
Discusswhatelsethecarownershouldconsidertoestimatethisrisk.
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(b) KuatuhiaetētahikaiwhakauruwakatukuruakiAotearoamēnākeitetahamauī,tetahamataurāneiotewakatetaupokipenehīnioiawaka,ituaatukiētahiatukōreromōngāwaka.
Mōtekawengawhakamutungaongāwakatukuruaikaweamai,i1321 ongāwakakeitetahamauītetaupokipenehīni,ā,22.8%ongāwakahehiriwa.
(i) Kakōwhirihiamatapōkeretiatētahiwakamaiitauakawengaongāwakatukurua.
Whakatauhiatetūponotangahehiriwatauawakakamutukeitetahamauītetaupokipenehīni.
Tuhiatewhakapaeehiahiatiaanaheiwhakatauitēneitūponotanga.
(ii) Kuakitetētahikiritakiitētahiteihanapenehīniongāwakatekauetikipenehīnianaitēneiwā,ewhituoauawakakeitetahamauītetaupokipenehīni.
MewhakamāramaatukitekiritakikiakauaiaewhakaputakīangawhānuikotenuingaongāwakaiAotearoakeitetahamauīngātaupokipenehīni,eaikingākitengaatekiritaki.
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(b) Animporterofsecond-handcarsintoNewZealandhasrecordedwhethereachcarhasthepetrolcapontheleft-handsideortheright-handsideofthecar,inadditiontootherinformationaboutthecars.
Forthelastshipmentofsecond-handcarsimported,1321 ofthecarshadthepetrolcapontheleft-handsideand22.8%ofthecarsweresilver.
(i) Onecarischosenatrandomfromthisshipmentofimportedsecond-handcars.
Determinetheprobabilitythatthiscarissilverandhasthepetrolcapontheleft-handside.
Statetheassumptionyouneedtomaketodeterminethisprobability.
(ii) Acustomeratapetrolstationhasobservedthatofthetencarscurrentlygettingpetrol,sevenofthesecarshavepetrolcapsontheleft-handside.
ExplaintothecustomerwhyageneralisationshouldnotbemadethatcarsinNewZealandaremorelikelytohavepetrolcapsontheleft-handside,basedonwhatthecustomerhasobserved.
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TŪMAHI TUARUA
(a) EwhakapaetiaanatētahikaiwhakauruwakakiAotearoakeitewhakaawhiwhiiaingātatauine-tawhiti(teineongākiromitatapekekuahaerehiaetewaka)kite10kiromitatūtatamōētahiongāpānuitangakeitanapaetukutuku.
E20ngāwakaepānuitiaanaitēneiwāetekaiwhakauruheihokoatuirungaitanapaetukutuku.
Keiraroerārangihaereanangātatauine-tawhitimōēneiwaka.
1 485 25 384 25 499 26 890 29 56835 279 47 872 49 200 64 788 68 05072 690 75 730 84 457 91 575 92 29793 033 109 532 113 395 137 209 142 980
(i) Heahateōwehengaongāwakaepānuitiaanaetekaiwhakauruhe0tetauwhakamutungaotetatauine-tawhiti?
(ii) Mātewhakaarokotetauwhakamutungaotētahitatauine-tawhitimōtētahiwakakawhakatauhiamātetūponotangaanake,homaihetauira(ariā)whakatautatamōtetūponotangahe0tetauwhakamutungaotetatauine-tawhiti.
(iii) Iwhakahaerehiaetētahikiritakiāwangawangahewhakatarunaheitūhuraitetaurangitangairotoiteōwehengaongāwakakingātōpū20he0tetauwhakamutungaotetatauine-tawhiti,irungaitewhakapaekotetauwhakamutungaotētahitatauine-tawhitimōtētahiwakakawhakatauhiaetetūponotangaanake.
Ewhakaaturiaanairarotētahiwhakarāpopototangaongāotingawhakataruna(1000whakamātautau).
Ōwehenga he 0 te tau whakamutunga
020
120
220
320
420
520
620
720
820
neke atu rānei
Auautanga 130 260 289 187 92 32 9 1 0
Irungaiēneiotingawhakataruna,heahatewhakatauatekiritakimēnākariromātetūponotangaanakeewhakariteaitetauwhakamutungaotētahitatauine-tawhitimōngāwakaepānuitiaana?
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QUESTION TWO
(a) AnimporterofcarsintoNewZealandissuspectedofroundingtheodometerreading(themeasureofthetotalkilometresthecarhasdriven)tothenearest10kilometresforsomeoftheadvertisementsontheirwebsite.
Thecarimportercurrentlyhas20carslistedforsaleontheirwebsite.
Theodometerreadingsforthesecarsarelistedbelow.
1485 25384 25499 26890 2956835279 47872 49200 64788 6805072690 75730 84457 91575 9229793033 109532 113395 137209 142980
(i) Whatproportionofcarsadvertisedbytheimporterhas0asthelastdigitoftheodometerreading?
(ii) Assumingthatthelastdigitofanodometerreadingforacarisdeterminedbychancealone,giveamodel(theoretical)estimatefortheprobabilitythatthelastdigitofanodometerreadingis0.
(iii) Aconcernedcustomerconductedasimulationtoinvestigatethevariabilityintheproportionofcarsinsetsof20thathave0asthelastdigitoftheodometerreading,basedonanassumptionthatthelastdigitofanodometerreadingforacarisdeterminedbychancealone.
Asummaryofthesimulationresultsisshownbelow(1000trials).
Proportion with 0 last digit
020
120
220
320
420
520
620
720
820
orhigher
Frequency 130 260 289 187 92 32 9 1 0
Basedonthesesimulationresults,whatconclusioncouldthecustomermakeinrespecttowhetherornotthelastdigitofanodometerreadingforthecarsadvertisedisdeterminedbychancealone?
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(b) Itetau2013,63.9%ongāwakaikaweamaiitāwāhiirēhitatiakiTeWakaKotahiimahiamaiiHapani.OēneiwakaimahiaiHapani,80.3%hewakatukurua.
MewhakaaroikōwhirihiamatapōkeretiatētahiongāwakaikaweamaiitāwāhiirēhitatiakiTeWakaKotahiitetau2013.
(i) Whakamāramahiamaitetakeeharaitetāuketētahiitētahiēneipāpono“ImahiamaiiHapani”me“Hewakatukuruatewaka”.
Whakauruangāwhakaarowhaitakeotetauangakitōwhakamāramatanga.
(ii) WhakamāramahiamaitetakekataeatewhakataumaiiēneikōreroanakekotewakaitīpakohiakoteāhuaneiimahiamaiiHapani,itemeahewakatukuruatewakaitīpakohia.
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(b) In2013,63.9%ofimportedcarsregisteredwiththeNewZealandTransportAgencyweremanufacturedinJapan.OfthesecarsmanufacturedinJapan,80.3%wereusedcars.
SupposethatoneoftheimportedcarsregisteredwiththeNewZealandTransportAgencyin2013wasselectedatrandom.
(i) Explainwhytheevents“ThecarwasmanufacturedinJapan”and“Thecarisausedcar”arenotmutuallyexclusive.
Includestatisticalreasoninginyourexplanation.
(ii) ExplainwhyitcanbededucedfromthisinformationalonethatthecarselectedismorelikelytohavebeenmanufacturedinJapanthannot,giventhecarselectedisausedcar.
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TŪMAHI TUATORU
(a) KaheriaengātāngataōrātouwakakingāpokapūwhakamātautaumōtētahiWarrantofFitness(WOF).
Etorungāpokapūwhakamātautauiarotakehiainātataneiirotoitekotahimarama:tepokapūwhakamātautauA,tepokapūwhakamātautauB,metepokapūwhakamātautauC.Itauawā,ituhiangāotingakatoamōngāwhakamātautauimahiaiiapokapūwhakamātautau.
40%ongāwhakamātautauiarotakehiaimahiaitepokapūwhakamātautauA,ā,25%ongāwhakamātautauiarotakehiaimahiaitepokapūwhakamātautauB.
OngāwhakamātautauimahiaitepokapūwhakamātautauA,82%imomoho(ihipaitewakateWOF). OngāwhakamātautauimahiaitepokapūwhakamātautauB,96%imomoho. OngāwhakamātautauimahiaitepokapūwhakamātautauC,94%imomoho.
(i) Heahateōrauongāwhakamātautauimahiaitewāotearotakeimomoho?
(ii) Ongāwhakamātautaukāoreimomoho,heahateōwehenga imahiaitepokapūwhakamātautauC?
Kahiahiapeakoekitewhakapae10000ngāwhakamātautauimahiaitewāotearotakeongāpokapūwhakamātautauetoru.
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QUESTION THREE
(a) PeopletaketheircarstotestingcentresforaWarrantofFitness(WOF).
Threetestingcentreswererecentlyreviewedoveraone-monthperiod:testingcentreA,testingcentreB,andtestingcentreC.Duringthistime,allresultsfortestscompletedbyeachofthetestingcentreswererecorded.
40%ofthetestsreviewedwerecompletedbytestingcentreA,and25%ofthetestsreviewedwerecompletedbytestingcentreB.
OfthetestscompletedbytestingcentreA,82%weresuccessful(thecarpassedtheWOF). OfthetestscompletedbytestingcentreB,96%weresuccessful. OfthetestscompletedbytestingcentreC,94%weresuccessful.
(i) Whatpercentageoftestscompletedduringthereviewweresuccessful?
(ii) Oftheteststhatwereunsuccessful,whatproportionwere completedattestingcentreC?
Youmaywishtoassumethattherewere10000testscompletedduringthereviewofthethreetestingcentres.
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(iii) Irungaingāotingaotearotake,kuawhakatautētahirangatirawakakiaheriaatueiatōnawakakitepokapūwhakamātautauBkianuiaketōnatūponotangakamomohotanawhakamātautauWOF.
Kataeatēneiwhakatauteparahau?
(b) EwhakaaturiaanakitepapatauirarongākōreromōtetawhitoongāwakamengāmotupaikairēhitatiakiTeWakaKotahi(NZTA)itepitootetau2013.Ewhakaatuanatēneipapatauingākōreroanakemōngāwaka,motupaikarāneieitiakeite5tautetawhitomaiitepitootetau2013.
Te tawhito o ngā waka i rēhitatia ki te NZTA i te pito o te tau 2013
0 tau te
tawhito
1tau te
tawhito
2 tau te
tawhito
3 tau te
tawhito
4 tau te
tawhitoŌwehenga o ngā waka 0.238 0.223 0.188 0.186 0.165
Ōwehenga o ngā motupaika
0.215 0.181 0.177 0.183 0.244
KakōwhirihiamatapōkeretiatētahiwakametētahimotupaikamaiingāwakairēhitatiakiteNZTAitepitootetau2013.
Itemeaheitiakeiterimatautetawhitootewakametemotupaika,whakatautatahiatetūponotangaheruatauiteitirawatetawhitoakeotemotupaikaitewaka.
Tautokonatōtuhingakingātauākītauangamengātātaitaietōtikaana.
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(iii) Basedontheresultsofthereview,acarownerhasdecidedthattheyshouldtaketheircartotestingcentreBtoincreasetheirchancesofhavingasuccessfulWOFtest.
Isthisdecisionjustified?
(b) InformationabouttheagesofcarsandmotorcyclesregisteredwiththeNewZealandTransportAgency(NZTA)attheendof2013ispresentedinthetablebelow.Thistableshowsinformationaboutonlycarsormotorcycleslessthan5yearsoldattheendof2013.
Age of vehicles registered with NZTA at the end of 2013
0 years old
1year old
2 years old
3 years old
4 years old
Proportion of cars 0.238 0.223 0.188 0.186 0.165
Proportion of motorcycles 0.215 0.181 0.177 0.183 0.244
OnecarandonemotorcyclearechosenatrandomfromvehiclesregisteredwithNZTAattheendof2013.
Giventhatbothvehiclesarelessthanfiveyearsold,estimatetheprobabilitythatthemotorcycleisatleasttwoyearsolderthanthecar.
Supportyouranswerwithappropriatestatisticalstatementsandcalculations.
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He whārangi anō ki te hiahiatia.Tuhia te (ngā) tau tūmahi mēnā e tika ana.
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Level 3 Mathematics and Statistics (Statistics), 2015
91585 Apply probability concepts in solving problems
2.00 p.m. Thursday 19 November 2015 Credits: Four
Achievement Achievement with Merit Achievement with ExcellenceApply probability concepts in solving problems.
Apply probability concepts, using relational thinking, in solving problems.
Apply probability concepts, using extended abstract thinking, in solving problems.
Check that the National Student Number (NSN) on your admission slip is the same as the number at the top of this page.
You should attempt ALL the questions in this booklet.
Show ALL working.
Make sure that you have the Formulae and Tables Booklet L3–STATMF.
If you need more room for any answer, use the space provided at the back of this booklet and clearly number the question.
Check that this booklet has pages 2 – 15 in the correct order and that none of these pages is blank.
YOU MUST HAND THIS BOOKLET TO THE SUPERVISOR AT THE END OF THE EXAMINATION.
English translation of the wording on the front cover
