te Pāngarau me te tauanga (tauanga), kaupae 3, 2014 · 2014-11-20 · Te Pāngarau me te Tauanga (Tauanga) 91586M, 2014 MĀ T KAIMĀKA ANAK(Pātai tuataHi (a) Ka taea te whakatauira

See back cover for an English translation of this cover

915865

3SuperviSor'S uSe only

9 1 5 8 6 M

© Mana Tohu Mātauranga o Aotearoa, 2014. Pūmau te mana. Kia kaua rawa he wāhi o tēnei tuhinga e tāruatia ki te kore te whakaaetanga a te Mana Tohu Mātauranga o Aotearoa.

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taPeke

te Pāngarau me te tauanga (tauanga), kaupae 3, 2014

91586M te whakahāngai i ngā tuari tūponotanga hei whakaoti rapanga

9.30 i te ata Rāpare 20 Whiringa-ā-rangi 2014 Whiwhinga: Whā

Paetae kaiaka kairangiTe whakahāngai i ngā tuari tūponotanga hei whakaoti rapanga.

Te whakahāngai i ngā tuari tūponotanga mā te whakaaro whaipānga hei whakaoti rapanga.

Te whakahāngai i ngā tuari tūponotanga mā te whakaaro waitara hōhonu hei whakaoti rapanga.

Tirohia mehemea e ōrite ana te Tau Ākonga ā-Motu (nsn) kei tō pepa whakauru ki te tau kei runga ake nei.

Me whakautu e koe ngā pātai katoa kei roto i te pukapuka nei.

Whakaaturia ngā mahinga KATOA.

Me mātua riro mai i a koe te pukaiti o ngā Tikanga Tātai me ngā Papatau L3–STATMF.

Ki te hiahia koe ki ētahi atu wāhi hei tuhituhi whakautu, whakamahia te (ngā) whārangi kei muri i te pukapuka nei, ka āta tohu ai i ngā tau pātai.

Tirohia mehemea kei roto nei ngā whārangi 2 – 15 e raupapa tika ana, ā, kāore hoki he whārangi wātea.

Hoatu te PukaPuka nei ki te kaiwHakaHaere Hei te Mutunga o te wHakaMātautau.

Te Pāngarau me te Tauanga (Tauanga) 91586M, 2014

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Pātai tuataHi

(a) Kataeatewhakatauiraterahiotematūwhakakori(caffeine)irotoitētahikawhe“inukotahi”mātētahituarimāori,metetohariteote115mg meteinemahoraote10mg.

Mekīkaotahiaetētahikiritakikiatorungākawhe“inukotahi”.

Tātaihiatetūponotangakeiwaengaite108mgmete122mgtematūwhakakorikeirotoingākawhekatoa e toru.

Hōmaingāwhakapaengaheiwhakaputa.

(b) Kotewākapāmaingāpāngakitetangataotematūwhakakoriirotoitanainukawheimuriitanainumangakatāeatewhakatauiramātētahitaurangimatapōkereewhaiuaraiwaengaite0menetimete40meneti.Koteāhuaneikotewākatinopāmaingāpāngakitetangataotematūwhakakoriirotoitanainukawhehe10meneti.

Mātewhakamahiitētahitauiratōtika,tātaihiatetūponotangakaitiakeiterimameneti,kanuiakeRĀNEIite10menetierongoaitetangataitepāngaotematūwhakakoriirotoitanainukawhe.

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Question one

(a) Theamountofcaffeineina“singleshot”coffeecanbemodelledbyanormaldistribution,withmean115mg andstandarddeviation10mg.

Supposethatacustomerordersthree“singleshot”coffees.

Calculatetheprobabilitythatall threecoffeescontainbetween108mgand122mgofcaffeine.

Giveanyassumption(s)thatneedtobemade.

(b) Thetimeittakesforapersontofeeltheeffectsofthecaffeineintheircoffeeaftertheydrinkitcanbemodelledbyarandomvariablethattakesonvaluesbetween0minutesand40minutes.Themostlikelytimeittakesapersontofeeltheeffectsofthecaffeineintheircoffeeis10minutes.

Usinganappropriatemodel,calculatetheprobabilitythatitwilltakelessthanfiveminutesORmorethan10minutesforapersontofeeltheeffectsofthecaffeineintheircoffee.

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Mathematics and Statistics (Statistics) 91586, 2014

ASSESSOR’S USE ONLY


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(c) Tatakite35%ongākawhelatte“rahinganui”nekeatuite405mLmirakakeiroto.

Kawhakamahiaetētahirangatirawharekawhetētahituarimāorimetetoharite400mLkitewhakatauiraiterahingaotemirakairotoingālatte“rahinganui”.

(i) Mātewhakamahiitēneitauira,tātaihiateōrautangaongālatte“rahinganui”katāeatetūmanakohenuiakeite410mLotemirakakeiroto.

(ii) Matapakihiakiakotahiteaukatingakatāeamēnākawhakamahiatetuarimāoriheiwhakatauiraiterahiotemirakakeirotoitētahilatte“rahinganui”.

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(c) Around35%of“largesize”lattecoffeescontainmorethan405mLofmilk.

Acaféownerusesanormaldistributionwithmean400mLtomodeltheamountofmilkusedin“largesize”lattes.

(i) Usingthismodel,calculatethepercentageof“largesize”lattesthatcouldbeexpectedtocontainmorethan410mLofmilk.

(ii) Discussonepotentiallimitationwithusinganormaldistributiontomodeltheamountofmilkusedina“largesize”latte.

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Pātai tuarua

(a) Kotewhakatautataatētahirangatirawharekawhehe30%ongāinukawhelattekamahiamaiitemirakamōmonaiti.Irotoitekotahiwiki,ituhiaeterangatirawharekawheehiangālatteimahiakitemirakamōmonaiti,kitētahiatumomomirakarānei,mōngā“huinga”ongāotalattepiritataerima.

(i) Parahautiatewhakamahingaotetuarihuaruaheiwhakatauiraitemahaongālatteirotoitētahihuingaote5kamahiamaiitemirakamōmonaiti.

(ii) Kuatīmataterangatirawharekawhekitewhakaputaitētahikauwhataewhakatauriteanaingāraraungaikohia(tetuariwhakamātauewhakaaturiakaurukitiaana)metetauiratuarihaurua(tetuariariāewhakaaturiaanakitekikorangi).

Whakaotihiatekauwhatamātewhakaatuingāuaraetoeanamōtetauiratuarihuarua.

00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

1 2 3 4 5

Number of lattes with trim milk per set of 5

Pro

port

ion o

f se

ts o

f 5

Temahaongālattewhaimirakamōmonaitiiiahuingaoterima

Teōwehengaongāhuingaoterima

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(iii) Matapakitiaheahate/ngāwhakataukaputakiterangatirawharekawhemaiitekauwhatakuaotiitewhārangi6.

(b) Kotetautohariteongāotaā-waeamōtekawheitehāoraiwhiwhiitewharekawhehe4.6.

(i) Mātewhakamahiitētahitauiratuaritūponotangatōtika,tātaihiatetūponotangaeruatemōrahiongāotaā-waeamōtekawhekawhiwhiitewharekawheirotoitētahihawhehāora.

(ii) Mōtewhakamahiitetuaritangakawhakamahiaitewāhanga(b)(i),meputakiakotahitewhakapaenga,nekeaturānei.

Tautohuakiakotahitewhakapaengakeitehēpea,ā,matapakihiatetakeepēneiai.

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Question two

(a) Acaféownerestimatesthat30%oflattecoffeesaremadewithtrimmilk.Overaperiodofaweek,thecaféownerhasrecordedhowmanylattesweremadewithtrimmilkorwithothermilkfor“sets”offiveconsecutivelatteorders.

(i) Justifytheuseofthebinomialdistributiontomodelthenumberoflattesinasetof5thataremadewithtrimmilk.

(ii) Thecaféownerhasbeguntoproduceagraphcomparingthedatacollected(theexperimentaldistributionshownshaded)andthebinomialdistributionmodel(thetheoreticaldistributionshowninblue).

Completethegraphbyshowingtheremainingvaluesforthebinomialdistributionmodel.

00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

1 2 3 4 5

Number of lattes with trim milk per set of 5

Pro

port

ion o

f se

ts o

f 5

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(iii) Discusswhatconclusion(s)thecaféownercoulddrawfromthecompletedgraphonpage8.

(b) Themeannumberofphoneordersforcoffeeperhourreceivedbythecaféis4.6.

(i) Usingasuitableprobabilitydistributionmodel,calculatetheprobabilitythatthecaféreceivesatmosttwophoneordersforcoffeeoveranyhalf-hourperiod.

(ii) Toapplythedistributionusedinpart(b)(i),atleastoneassumptionneedstobemade.

Identifyonesuchassumptionthatmaybeinvalid,anddiscusswhythisisthecase.

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Pātai tuatoru

(a) EwhakaatuanatetūtohiiraroneiitetuaritūponotangaotetaurangimatapōkereN,temahaongāotamōngākawheheikaweatumātētahikiritaki.

n 1 2 3 4 5P(N = n) 0.49 0.31 0.1 0.08 0.02

(i) Tātaihiatetautohariteongākawheheikaweatuiotahiaengākiritaki.

(ii) Kawhiwhingākiritakikatoaitētahipaepaepepamārōheikaweatuiārātoukawhe.

Koteutumōtemahiiiakawhe,taeatukingārauemimetemahi,he$1.80.

Koteutuotētahipaepaepupuriingākawhetaeatukiteruahe$0.20.

Koteutuotētahipaepaepupuriingākawheetorutaeatukiterimahe$0.40.

Tātaihiateutuetūmanakohiaanamōiaotakawhekaweatu.

(b) Hepātarakeitekiripaepaeotewharekawhemāngākiritakiheitukuwhakaarokitētahikaupapaarohaoterohe.Kāoretewharekawheitemōhioehiatemonikatukunamaiiiawāeiakiritaki,ēngarikotenuietūmanakohiakeiwaengaite50henetimeterimatāra.

(i) Mātewhakamahiitētahitauiratōtika,whakatauhiaterahingamōrahikatūmanakohiaetetoakawhekāoreeekehiaengāwhakaaroangākiritaki80%.

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(ii) Parahautiatōtīpakotangaotetauiratuaritūponotangatōtika.

(c) Ituhiaetētahikamupenehokopīnikawhetemahaongātorongaitanapaetukutukuirotoingāwāhangawā15menetiirotoingāmaramaeruakuahipa.Ikiteaetekamupeneoauawāhangawāe96%,kotahitetorongakitepaetukutukuiteitirawa.

Tātaihiatetūponotanganekeatuite10ngātorongakitepaetukutukuirotoitētahiwāhangawā30meneti.

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Question tHree

(a) ThetablebelowshowstheprobabilitydistributionoftherandomvariableN,thenumberoftakeawaycoffeesorderedbyacustomer.

n 1 2 3 4 5P(N = n) 0.49 0.31 0.1 0.08 0.02

(i) Calculatethemeannumberoftakeawaycoffeesorderedbycustomers.

(ii) Allcustomersaregivenacardboardtraytocarrytheirtakeawaycoffees.

Thecosttomakeeachcoffee,includingthecostofthematerialsandlabour,is$1.80.

Thecostofatraythatcanholduptotwocoffeesis$0.20.

Thecostofatraythatcanholdthreetofivecoffeesis$0.40.

Calculatetheexpectedcostofeachtakeawaycoffeeorder.

(b) Acaféhasajaronthefrontcounterforcustomerstogivemoneyforalocalcharity.Thecaféisnotsurehowmucheachcustomerwillgiveeachtime,butexpectsanamountbetween50centsandfivedollars.

(i) Usinganappropriatemodel,determinethemaximumamountthatthecaféwouldexpect80%ofcustomerstodonatelessthan.

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(ii) Justifyyourselectionofanappropriateprobabilitydistributionmodel.

(c) Acompanythatsellscoffeebeansrecordedthenumberofvisitstoitswebsitein15-minuteperiodsoverthelasttwomonths.Thecompanyfoundthatin96%ofsuchperiods,therewasatleastonevisittothewebsite.

Calculatetheprobabilitythatthewebsitewillreceivemorethan10visitsinanygiven30-minuteperiod.

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anaKetau pātai

He puka anō mēnā ka hiahiatia.tuhia te (ngā) tāu pātai mēnā e hāngai ana.

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QUEStiON NUmbER

Extra paper if required.Write the question number(s) if applicable.

© New Zealand Qualifications Authority, 2014. All rights reserved.No part of this publication may be reproduced by any means without the prior permission of the New Zealand Qualifications Authority.

Level 3 Mathematics and statistics (statistics), 2014

91586 apply probability distributions in solving problems

9.30 am Thursday 20 November 2014 Credits: Four

achievement achievement with Merit achievement with excellenceApply probability distributions in solving problems.

Apply probability distributions, using relational thinking, in solving problems.

Apply probability distributions, 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 space for any answer, use the page(s) 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

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te Pāngarau me te tauanga (tauanga), kaupae 3, 2014 · 2014-11-20 · Te Pāngarau me te Tauanga (Tauanga) 91586M, 2014 MĀ T KAIMĀKA ANAK(Pātai tuataHi (a) Ka taea te whakatauira