Level 3 Mathematics and Statistics (Statistics) (91585) 2018 · 2018-10-29 · (d) The 349 secondary schools are classified basedon the ages of the students on their roll. There are

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© Mana Tohu Mātauranga o Aotearoa, 2018. 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, 2018

91585M Te whakahāngai ariā tūponotanga hei whakaoti rapanga

9.30 i te ata Rāpare 22 Whiringa-ā-rangi 2018 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 – 23 kei roto i tēnei pukapuka, ā, 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, 2018

MĀ TE KAIMĀKA

ANAKE

TŪMAHI TUATAHI

(a) Imātaihiaetētahirangahaumōtewhakatairangakaiitetau2017iAotearoangāpaetukutuku70angātinokamupenekaimeteinu.E24oēneipaetukutukuitearotahikingātaiohi,ā,koērāatuiarotahikēatukitetauporiwhānui.Ipaingāwhakapaemōtehauoraote21ongāpaetukutuku,ā,oauapaetukutuku,ewaruiarotahiatukingātaiohi.

Katīpakomatapōkeretiatētahioēneipaetukutuku.

(i) Whakamāramahiamaitetakeeharangāpāpono“hepaitewhakapaemōtehauoraatētahipaetukutuku”,ā,“kaarotahitētahipaetukutukukitetauporiwhānui”itetāuketētahiitētahi.

Tautokonatōtuhingamātētahitātaitangakotahiiteitirawa.

(ii) Etautokoanangāraraungamaiitēneirangahauitewhakapaeeruawhakareangaakepeatetūponotangakaputaitētahipaetukutukuhewhakapaepaimōtehauoramēnāearotahianatepaetukutukukingātaiohikauakēkitetauporiwhānui?

Tautokonatōtuhingamāngātātaitanga.

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QUESTION ONE

(a) A2017foodmarketingstudyfromNewZealandexamined70websitesbelongingtothemostpopularfoodanddrinkbrands.24ofthesewebsitesweretargetedatteenagers,whiletheothersweretargetedatthegeneralpopulation.21ofthewebsitesmadeapositivehealthclaim,andofthesewebsites,eightweretargetedatteenagers.

Oneofthewebsitesischosenatrandom.

(i) Explainwhytheevents“awebsitemakesapositivehealthclaim”and“awebsiteistargetedatthegeneralpopulation”arenotmutuallyexclusive.

Supportyouranswerwithatleastonecalculation.

(ii) Doesthedatafromthisstudysupporttheclaimthatitismorethantwiceaslikelythatawebsitemakesapositivehealthclaimifthewebsiteistargetedatteenagersratherthanthegeneralpopulation?

Supportyouranswerwithcalculations.

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Mathematics and Statistics (Statistics) 91585, 2018

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MĀ TE KAIMĀKA

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(b) Ihangaiahetauiraematapaeanaitemomopēnihokahokonaetētahitangataeaikingāāhuatangarerekēotetangata,mengāmeataurangipērāitetaipakeke,temoniwhiwhi,metetūngamārena.Heiwhakamātauitetauira,100ngātāngataiuiuitia,ā,iwhakatauriteatemomopēnihoakeihokonamaieiatangatakitemomoimatapaetiaetetauira.

Ewhakaatuanatetūtohiiraroingāotingaotēneiwhakamātau.

Momo akeMomo i matapaetia

A B CA 18 3 5B 5 38 9C 12 2 8

(i) Tātaihiateōrauongāmatapaeitetika(heōritetemomoimatapaetiakitemomoake).

(ii) HomaikiaKOTAHIterarurarukataeamōtetōtikatangaotetauiramōngāmomorerekē,eaikingāraraungaitukunairungaake.

Tautokonatōwhakautumāngātātaitangatāpirieruaiteitirawakitewāhanga(i).

(iii) Etorungātāngataitīpakohiamaiiterangahau.

Tātaihiatetūponotangaihētematapaeatetauiraitemomopēnihomōtetangatakotahianakeoēneitokotoru.

Tuhiangāwhakapaekaputametewhakamāramaheahatetakemeputaēnei.

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(b) Amodelhasbeendevelopedthatpredictsthebrandoftoothpastesomeonewillpurchasebasedonvariouscharacteristicsoftheperson,includingvariablessuchasage,income,andmaritalstatus.Totestthemodel,100peopleweresurveyed,andtheactualbrandoftoothpastepurchasedbyeachpersonwascomparedtothepredictedbrandfromthemodel.

Thetablebelowshowstheresultsofthistest.

Actual brandPredicted brand

A B CA 18 3 5B 5 38 9C 12 2 8

(i) Calculatethepercentageofthepredictionsthatwerecorrect(thepredictedbrandwasthesameastheactualbrand).

(ii) GiveONEpotentialissuewiththeappropriatenessofthemodelforthedifferentbrands,basedonthedataprovidedabove.

Supportyouranswerwithatleasttwoadditionalcalculationstopart(i).

(iii) Threeofthepeoplefromthesurveywereselected.

Calculatetheprobabilitythatthemodelincorrectlypredictedthebrandoftoothpasteforonlyoneofthesethreepeople.

Stateanyassumptionsmadeandexplainwhytheseneedtobemade.

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MĀ TE KAIMĀKA

ANAKE

TŪMAHI TUARUA

E349ngākuratuaruaiAotearoaitetīmatangaotetau2018.ItautapatiangākuratuaruakatoaiAotearoaheikurakeiTeIka-a-Māui,keiTeWaipounamurānei.

• 12.9%oēneikurahekuratāne

• 71.1%ongākuratānekeiTeIka-a-Māui

• 23.0%ongākurakōtiro,tāne-kōtirorāneikeiTeWaipounamu.

Katīpakomatapōkeretiatētahioēneikuratuarua.

(a) TātaihiatetūponotangakeiTeIka-a-Māuitauakuratuarua.

(b) MekīkoBtepāpono“hekuratāne”,ā,koNtepāpono“hekurakeiTeIka-a-Māui”.

MetātaimetewhakamahiiaP(N | B)meP(N)kitewhakamāramamēnāhewehekēngāpāponoBmeN.

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QUESTION TWO

Therewere349secondaryschoolsinNewZealandatthestartof2018.AllsecondaryschoolsinNewZealandweredesignatedasbeingintheNorthIslandorSouthIsland.

• 12.9%oftheseschoolsareboys’schools

• 71.1%oftheboys’schoolsareintheNorthIsland

• 23.0%ofthegirls’orco-educationalschoolsareintheSouthIsland

Oneofthesesecondaryschoolsisselectedatrandom.

(a) CalculatetheprobabilitythatthesecondaryschoolisintheNorthIsland.

(b) LetBbetheevent“aboys’school”andletNbetheevent“aNorthIslandschool”.

CalculateanduseP(N | B)andP(N)toexplainwhethereventsBandNareindependent.

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MĀ TE KAIMĀKA

ANAKE

(c) Kaputaingātātaritangaatuanōongākura349ngāmōhiohioewhaiake:

• IwhakatauhiakongākuratuaruakatoaiAotearoahekuratūmataiti,hekurakāwanatanga(keirotongākurahouruaitēnei)rānei.

• 16hekuratāne-kōtirotūmataiti.

• Ongākuratāne,kotahianakehetūmataiti.

• 331hekurakāwanatanga.

• 71.0%ongākurakāwanatangahekuratāne-kōtiro.

Tātaihiatetūponotangamōtētahikuratuaruakatīpakomatapōkeretiahekurakāwanatangakōtiro.

Ka haere tonu te Tūmahi Tuarua i te whārangi 10

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(c) Furtheranalysisofthe349schoolsprovidesthefollowinginformation:

• AllsecondaryschoolsinNewZealandweredesignatedaseitherprivateorstate(whichincludespartnershipschools).

• 16areprivateco-educationalschools.

• Oftheboys’schools,onlyoneisprivate.

• 331arestateschools.

• 71.0%ofstateschoolsareco-educational.

Calculatetheprobabilitythatasecondaryschoolselectedatrandomisastategirls’school.

Question Two continues on page 11.

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MĀ TE KAIMĀKA

ANAKE

(d) Kawhakatauhiangākuratuarua349eaikingātaipakekeongāākongaotārātaurārangiingoa.E236ngākuraiwhakatauhiaheTau9-13,ā,kote113kawhakatauhiahekura momokē.

ItonoatētahiakomangaongāākongaTauangaTau13etōrātoukaiakokiatīpakoitetīpakongamatapōkereongākuratuarua50,metewhakakapi,maiingākuratuarua349.Kongāotingaiputaitētahiākongakotahihe25ngākuraheTau9kite13irotoitanatīpakoongākuratuarua50.

(i) Whakamāramahiamaitetakeemāharaharaanapeatekaiakokāoreitīpakomatapōkeretiaeteākongangākuratuarua50.

Tautokonatōwhakautumāngātātaitangaewhakatauriteanaingātatauitūmanakohia,meērāikitea.

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(d) The349secondaryschoolsareclassifiedbasedontheagesofthestudentsontheirroll.Thereare236schoolsclassifiedasYear9to13,and113otherschools.

AclassofYear13Statisticsstudentswasaskedbytheirteachertoeachselectarandomsampleof50secondaryschools,withreplacement,fromthe349secondaryschools.Theresultsproducedbyonestudentcontained25schoolsthatwereYear9to13inhersampleof50secondaryschools.

(i) Explainwhytheteachermightbesuspiciousthatthestudentdidnotrandomlyselectthe50secondaryschools.

Supportyouranswerwithcalculationscomparingexpectedandobservedcounts.

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MĀ TE KAIMĀKA

ANAKE

(ii) Ihoahoatia,iwhakahaerehiaetekaiakotētahiwhakatarunaheitūhuramēnākataeangāotingaateākonga,inaitīpakomatapōkeretiaeteākongangākuratuarua50.Mōiawhakamātau,itīpakomatapōkeretiaeiangākura50mengāwhakakapimaiingākuratuarua349,metetuhiehiaoēneikura50hekuraTau9kite13.

Ewhakaaturiaanairarotētahiwhakarāpopototangaongāotingawhakataruna(1000ngāwhakamātautau).

20Te maha o ngā kura Tau 9 ki te 13 mai i te 50 (whakataruna) n = 1000

25 30 35 40 45 50

E 9 o ngā otinga 1000 kua tātuhia ki te whero

Eaikiēneiotingawhakataruna,heahatewhakataumutungakataeaetekaiako?

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(ii) Theteacherdesignedandcarriedoutasimulationtoinvestigatewhetherthestudent’sresultswouldhavebeenunlikely,giventhestudentdidrandomlyselectthe50secondaryschools.Foreachtrial,sherandomlyselected50schoolswithreplacementfromthe349secondaryschools,andrecordedhowmanyofthese50schoolswereYear9to13schools.

Asummaryofthesimulationresultsisshownbelow(1000trials).

20Number of Year 9 to 13 schools out of 50 (simulated) n = 1000

25 30 35 40 45 50

9 out of 1000 results shown in red

Basedonthesesimulationresults,whatconclusioncouldtheteachermake?

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MĀ TE KAIMĀKA

ANAKE

TŪMAHI TUATORU

IwhakamahiangāwhakatautataaTeKotahitangaoNgāWhenuaoTeAo(UN)mōtetau2018eaikingātataurangawhānuiā-motumaiingāraraungahourawamōiawhenuakitehangaiteKauwhata1meteTūtohi1iraro.

Kauwhata 1: Ōrau o te taupori ā-ao me te ōrau o te horahanga whenua o te ao, mā te paparahi

Amerika ki te Tonga

Te Ao-o-Kiwa Amerika ki te Raki

Uropi Āhia Te Kōpakatanga ki te Tonga

Āwherika

% o te taupori o te ao

Ōrau

% o te horahanga whenua o te ao

5.7%

11.9%

0.5%

6.0%7.8%

16.4%

9.9%6.8%

59.7%

29.3%

9.2%

0.0%

16.4%

20.3%

0

10

20

30

40

50

60

70

Tūtohi 1: Ōrau o te taupori e noho ana i ngā wāhi tāone me te ōrau o te taupori e noho ana i ngā wāhi tuawhenua, mā te paparahi

Paparahi % o te taupori e noho ana i ngā wāhi tāone

% o te taupori e noho ana i ngā wāhi tuawhenua

AmerikakiteTonga 80% 20%TeAo-o-Kiwa 69% 31%AmerikakiteRaki 81% 19%Uropi 74% 26%Āhia 49% 51%Āwherika 41% 59%

(a) WhakamahiangāraraungaheitātaiitētahiwhakatautatamōteōrauotetauporioteaoenohoanaingāwāhitāoneoĀhia.

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MĀ TE KAIMĀKA

ANAKE

(b) Kotewhakaarohēotētahiākongamēnāemōhiotiaanakeitētahiwāhitāonetetangataenohoana,koteāhuaneikeiAmerikakiteRaki,kiteTongarāneitauatangataenohoana.

(i) Whakamāramahiamaiipēheateputaoēneiwhakaarohēotēneiākonga.

(ii) Eaikiēneiraraunga,54.53%otetauporioteaokeingāwāhitāoneenohoana. Tātaihiatētahiwhakatautatamōtetūponotangaenohoanatētahitangataitīpako

matapōkeretiaiAmerikakiteRaki,kiteTongarānei,inakeitētahiwāhitāone iaenohoana.

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QUESTION THREE

UnitedNationsestimatesfor2018basedonnationalcensusesfromthemostrecentdatafromeachcountrywereusedtoconstructGraph1andTable1below.

Graph 1: Percentage of the world’s population and the percentage of the world’s land area, by continent

South America Oceania North America Europe Asia Antarctica Africa

% of world’s population

Percentage

% of world’s land area

5.7%

11.9%

0.5%

6.0%7.8%

16.4%

9.9%6.8%

59.7%

29.3%

9.2%

0.0%

16.4%

20.3%

0

10

20

30

40

50

60

70

Table 1: Percentage of the population that live in urban areas and the percentage of the population that live in rural areas, by continent

Continent % of population that live in urban areas

% of population that live in rural areas

SouthAmerica 80% 20%Oceania 69% 31%NorthAmerica 81% 19%Europe 74% 26%Asia 49% 51%Africa 41% 59%

(a) Usethedataprovidedtocalculateanestimateforthepercentageoftheworld’spopulationthatliveinanurbanareaofAsia.

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(b) Astudentincorrectlyreasonsthatifyouknowthatapersonlivesinanurbanarea,thenitislikelythatthispersonlivesinNorthorSouthAmerica.

(i) Explainhowthestudenthasmadethisincorrectreasoning.

(ii) Basedonthisdata,54.53%oftheworld’spopulationliveinurbanareas. CalculateanestimatefortheprobabilitythatarandomlyselectedpersonlivesinNorth

orSouthAmerica,giventhattheyliveinanurbanarea.

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MĀ TE KAIMĀKA

ANAKE

(c) EwhakaaturiaanaetētahipaetukutukungāwhakaahuaGoogleStreetViewitīpakomatapōkeretiamaiiteaowhānui.Kongākaupapairotoiēneiwhakaahuakakīiahewhakaahuaongāwhare(mengānohoanga),ngāwhakaahuawhakakitekiterānei.Keirotoiiawhakaahuakotewāhiitangohiaaitewhakaahua.

KongāraraungamaiingāwhakaahuaGoogleStreetViewmahaiwhakaaturiaetēneipaetukutukuiwhakamahiakitewaihangaitētahitauiratūponotanga.Hemeawhakaatutēneitauiraheirākautūponotangakāoreiotiiraro.

Uropi

Ngā whare

Whakakitekite

Ngā whare

Whakakitekite

Ngā whare

Whakakitekite

Amerika ki te Raki

Wāhi kē

0.402

0.301

0.297

0.88

0.87

0.64

Paparahi Kaupapa o te whakaahua

(i) WhakamāramahiamainātewāhiongāwhakaahuaGoogleStreetViewkeitēneipaetukutukumengāraraungaateUNkeiteKauwhata1itewhārangi14,ewhakaatuaikāoreaGoogleStreetViewitewhiwhiwhakaahuamaiingāwhenuakatoaoteao.

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MĀ TE KAIMĀKA

ANAKE (ii) Mekīkotētahiwhakaahuaewhakaaturiaanakitepaetukutukuhewhakaahua

whakakitekite.

TātaihiatewhakatautatamōtetūponotangakotewāhiihopukinaaitewhakaahuakoAmerikakiteRaki.

(iii) Kawhakaaturiaetētahikaiwhakamahiotepaetukutukungāwhakaahua50aGoogleStreetView,ā,kakitea15ongāwhakaahuaiwhakaaturiahemeahopuiUropi.

Whakamahiaēneimōhiohio,mengāmōhiohioiwhakaaturiaitewāhanga(c)otēneitūmahi,kitematapakiingārerekētangaiwaengaitētahitūponotangapono,hewhakatautatatauira,metētahiwhakatautatawhakamātautau.

Tautokonatōtuhingakingāōwehengahāngai.

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(c) AwebsitedisplaysrandomlyselectedGoogleStreetViewphotosfromaroundtheworld.Thesubjectsofthesephotoscanbedescribedaseitherphotosofbuildings(includinghouses)orphotosofscenery.Eachincludesthelocationofwherethephotowastaken.

DatafromalargenumberofGoogleStreetViewphotosdisplayedbythiswebsitewereusedtodevelopaprobabilitymodel.Thismodelhasbeenpresentedasapartiallyconstructedprobabilitytreebelow.

Europe

Buildings

Scenery

Buildings

Scenery

Buildings

Scenery

North America

Elsewhere

0.402

0.301

0.297

0.88

0.87

0.64

Continent Photo subject

(i) ExplainhowthelocationoftheGoogleStreetViewphotosdisplayedbythiswebsiteandtheUNdatashowninGraph1onpage16,couldshowthatGoogleStreetViewdoesnothavephotosfromallcountriesintheworld.

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(ii) Supposethataphotodisplayedonthewebsiteisaphotoofscenery.

CalculateanestimatefortheprobabilitythatthelocationofwherethephotowastakenisNorthAmerica.

(iii) Auserofthewebsitedisplays50GoogleStreetViewphotosandfindsthat15ofthephotosdisplayedweretakeninEurope.

Usethisinformation,andtheinformationpresentedearlierinpart(c)ofthisquestion,todiscussthedifferencesbetweenatrueprobability,amodelestimate,andanexperimentalestimate.

Supportyouranswerwithrelevantproportions.

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MĀ TE KAIMĀKA

ANAKETAU TŪMAHI

He whārangi anō ki te hiahiatia.Tuhia te (ngā) tau tūmahi mēnā e tika ana.

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23



QUESTION NUMBER

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

Level 3 Mathematics and Statistics (Statistics), 2018

91585 Apply probability concepts in solving problems

9.30 a.m. Thursday 22 November 2018 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–STATF.

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 – 23 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.

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5M

English translation of the wording on the front cover

Documents

Level 3 Mathematics and Statistics (Statistics) (91585) 2018 · 2018-10-29 · (d) The 349 secondary schools are classified basedon the ages of the students on their roll. There are