Level 3 Mathematics and Statistics (Statistics) (91586) 2017

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© Mana Tohu Mātauranga o Aotearoa, 2017. 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, 2017

91586M Te whakahāngai i ngā tuari tūponotanga hei whakaoti rapanga

9.30 i te ata Rāhina 27 Whiringa-ā-rangi 2017 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 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 mō ō tuhinga, whakamahia ngā whārangi wātea kei muri o tēnei pukapuka, ka āta tohu ai i ngā 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.

HOATU TE PUKAPUKA NEI KI TE KAIWHAKAHAERE HEI TE MUTUNGA O TE WHAKAMĀTAUTAU.

Te Pāngarau me te Tauanga (Tauanga) 91586M, 2017

MĀ TE KAIMĀKA

ANAKE

TŪMAHI TUATAHI

(a) Kataeaterahingawaimōtehoroihīreretewhakatauiramātētahitaurangimatapōkereewhaiuaraanaiwaengaite20ritamete200rita.Mekīkoterahingawaikatinowhakamahiapeainahoroihīrereanahe50rita.

(i) Mātewhakamahiitētahitauiratuaritūponotangatōtika,tātaihiahewhakatautatamōteōrauongāhoroihīrereheitiihoite50ritatewaikawhakamahia.

(ii) Mātewhakamahiitētahitauiratuaritūponotangatōtika,tātaihiahewhakatautatamōteōrauongāhoroihīrerehenuiakeite40ritatewaikawhakamahia.

(b) KataeaengākaihautūwakangātaupāngawhakaterengaGPSwaeapūkororerekētewhakamahiheiwhiwhiwhakatautatamōteroaotewākitetaekitētahiwāhi.IwhakahaerehiaherangahauheitūhurahepēheatetōtikaongāwhakatautatawāhaerengamaiitētahitaupāngawhakaterengaGPSake.Mōiahaerengairotoiterangahau,iwhakatauriteatewāhaerengawhakatautatakitewāhaerengatika,ā,itātaihiatererekētangapūrawa(tirohiatetūtohiiraro).

Haerenga Wā haerenga whakatau tata Wā haerenga tika

Te rerekētanga pūrawa i waenga i te wā haerenga whakatau tata me te wā

haerenga tika1 10.4meneti 11.3meneti 0.9meneti2 6.5meneti 5.2meneti 1.3meneti3 3.9meneti 3.9meneti 0meneti… … … …

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MĀ TE KAIMĀKA

ANAKE

(i) Mekī,kote15%ongāhaerengaitutukiirotoiterangahaukakīia“kāoreitetika”mātewhakamahiitererekētangapūrawa.

Mēnātekaungāhaerengaitīpakohiamatapōkerehiaiterangahau,mātewhakamahiitetauiratōtika,tātaihiatetūponotangaewhāitenuirawaongāhaerengaikīia“kāoreitetika”.

(ii) Parahautiatewhakamahiitetuaritūponotangamōtōtuhingakitewāhanga(i).

(iii) Kotererekētangapūrawatautohariteiwaengaitewāhaerengawhakatautatametewāhaerengatikamōngāhaerengairotoitēneirangahauhe3.5meneti,meteinemahoraote2.8meneti.Ikiteaanōeterangahaue87%ongāhaerengaheitiihoiterimamenetingārerekētangapūrawa.

MatapakitiangāpūtakeeRUAikoreaietikakitewhakamahiitētahituarimāoriheiwhakatauiraingārerekētangapūrawaiwaengaitewāhaerengawhakatautatametewāhaerengatikamōngāhaerenga.

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QUESTION ONE

(a) Theamountofwaterusedwhentakingashowercanbemodelledbyarandomvariablethattakesonvaluesbetween20litresand200litres.Themostlikelyamountofwaterusedwhentakingashoweris50litres.

(i) Usinganappropriateprobabilitydistributionmodel,calculateanestimateforthepercentageofshowersthatuselessthan50litresofwater.

(ii) Usinganappropriateprobabilitydistributionmodel,calculateanestimateforthepercentageofshowersthatusemorethan40litresofwater.

(b) CardriverscanusevariousmobilephoneGPSnavigationappstogetanestimateofthetimeitwilltaketotraveltoadestination.AstudywascarriedouttoinvestigatehowaccuratethetraveltimeestimateswerefromoneparticularGPSnavigationapp.Foreachtripinthestudy,theestimatedtraveltimewascomparedtotheactualtraveltime,andtheabsolutedifferencecalculated(seethetablebelow).

Trip Estimated travel time Actual travel timeAbsolute difference

between estimated travel time and actual travel time

1 10.4minutes 11.3minutes 0.9minutes2 6.5minutes 5.2minutes 1.3minutes3 3.9minutes 3.9minutes 0minutes… … … …

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Mathematics and Statistics (Statistics) 91586, 2017

ASSESSOR’S USE ONLY

(i) Suppose15%oftripsmadeduringthestudywereclassifiedas“notaccurate”usingtheabsolutedifference.

Iftentripsfromthestudywerechosenatrandom,usinganappropriatemodel,calculatetheprobabilitythatatmostfourofthetripswereclassifiedas“notaccurate”.

(ii) Justifytheuseoftheprobabilitydistributionforyouranswerinpart(i).

(iii) Themeanabsolutedifferencebetweentheestimatedtraveltimeandtheactualtraveltimefortripsinthisstudywas3.5minutes,withastandarddeviationof2.8minutes.Thestudyalsofoundthat87%oftripshadabsolutedifferencesoflessthanfiveminutes.

DiscussTWOreasonswhyitwouldbeinappropriatetouseanormaldistributiontomodeltheabsolutedifferencesbetweentheestimatedtraveltimeandtheactualtraveltimefortrips.

1.

2.

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MĀ TE KAIMĀKA

ANAKE

TŪMAHI TUARUA

(a) Ewhakaatuanatetūtohiiraroneiitetuaritūponotangaotetaurangimatapōkere,X.

x 0 1 2 3 4

P(X = x) 0.11 0.21 0.24 0.25 0.19

(i) E(X)=2.2.

TātaihiateVAR(X ).

(ii) KotetaurangimatapōkereYkuawhaiVAR(Y)=1.5376.

VAR(X + Y )=5.5696.

HewehekēaX me Y?

Tautokonatōtuhingakingātauākītauangaetōtikaana.

(b) KataeatepaemahanatohariteotētahirūmanohoiAotearoaitētahipōtakuruatewhakatauiramātētahituarimāori,metetautohariteote17.8°Cmeteinemahoraote2.1°C.

(i) Mātewhakamahiitēneitauira,iwaengaiēheauaraeruakotōtūmanakomōte95%owaenganuiongāpaemahanatoharitemōngārūmanohoiAotearoaitētahipōtakurua?

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MĀ TE KAIMĀKA

ANAKE

(ii) MatapakitiakiaKOTAHIteāhuatangaheiwhakaaroaroinawhakatauirahiatepaemahanatohariteitētahirūmanohoiAotearoaitētahipōtakurua?

(iii) MēkīerimangāwhareitīpakohiamatapōkerehiaiAotearoa,ā,ikiteaitewhāoēneiwhareirarotepaemahanatohariteoterūmanohoitetahipōtakuruaite16°C.

Heitipeatetūponotangakakiteaewhānekeaturāneingāwhareirotoiterimaheitiakeite16°Ctepaemahanatohariteoterūmanoho,eaikitetauiratuaritūponotangaewhakaahuatiaanairunga?

Whakamahiahetātaitangaheitautokoitōwhakautu.

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QUESTION TWO

(a) ThetablebelowshowstheprobabilitydistributionoftherandomvariableX.

x 0 1 2 3 4

P(X = x) 0.11 0.21 0.24 0.25 0.19

(i) E(X)=2.2.

CalculateVAR(X ).

(ii) TherandomvariableYhasVAR(Y)=1.5376.

VAR(X + Y )=5.5696.

AreXandYindependent?

Supportyouranswerwithappropriatestatisticalstatements.

(b) TheaveragetemperatureinaNewZealandlivingroomonawintereveningcanbemodelledbyanormaldistribution,withmean17.8°Candstandarddeviation2.1°C.

(i) Usingthismodel,betweenwhattwovalueswouldyouexpectthemiddle95%ofaveragetemperaturesforNewZealandlivingroomsonawintereveningtobe?

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(ii) DiscussONEfactorthatshouldbeconsideredwhenmodellingtheaveragetemperatureinaNewZealandlivingroomonawinterevening.

(iii) SupposethatfiveNewZealandhouseswereselectedatrandom,anditwasfoundthattheaveragetemperatureofthelivingroomonawintereveningwasbelow16°Cforfourofthesehouses.

Wouldfindingfourormorehousesoutoffivewithanaveragetemperatureofthelivingroombelow16°Cbeunlikelyundertheprobabilitydistributionmodeldescribedabove?

Supportyouranswerwithacalculation.

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MĀ TE KAIMĀKA

ANAKE

TŪMAHI TUATORU

IkohiaetētahirangahaungāraraungamōtewhakamahiwaiirotoingākāingaiAotearoaheiāwhinaingākaunihera,ngātarikāwanatanga,mengākaiwhakaratowaikitewhakaurumaiingāwhakaritengamōtewhakamahitōtikaitewai.

Imuaiterangahau,kotewhakatautatahe4.7ngāwātohariteehīreretiaanateputungaparaeiatangataiAotearoairotoitētahiwā24haora.

(a) (i) MātewhakamahiitetauiratuaritangaPoisson,tātaihiahewhakatautatamōtetūponotangaheitiihoite5ngāwāehīreretiaanaetetangatateputungaparaitētahiwā24haora.

(ii) HomaikiaKOTAHItepūtakeheahaaiikoreetikapeatewhakamahiitētahituaritangaPoissonheiwhakatauiraitemahaongāhīrereputungaparamō tētahi wā 4 haora.

(b) Ikohiangāraraungamōtemahaongāwāihīreretiaeiatangatateputungaparairotoitētahiwā24haora.Kuawhakarāpopototiairaroitetūtohingāraraungamaiingātāngata200putanoaingākāinga84irotoiterangahau.

Te maha o ngā hīrere putunga para i tētahi wā 24 haora

0 1 2 3 4 5 6 7 8 9 10

Hautanga 0 0.01 0.1 0.17 0.26 0.14 0.12 0.11 0.05 0.01 0.03

(i) Tātaihiatetautohariteongāhīrereputungaparaiotiirotoitētahiwā24haoramōngātāngatairotoitēneirangahau.

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MĀ TE KAIMĀKA

ANAKE

(ii) Hewhāiteitirawatehīrereputungaparaatenuingaongātāngatairotoitēneirangahauitētahiwā24haora?

Whakamahiahetātaitangaheitautokoitōwhakautu.

(iii) Ewhakaatuanatekauwhatairaroitetuaritangawhakamātautau(ngāpaekauruku)metētahituaritangaPoissonoteλ=4.7(tetauiratuaritangaewhakaaturiaanakitepango).

0 1

Number of toilet flushes during a 24-hour period

Prop

ortio

n

2 3 4 5 6 7 8 9 100

0.05

0.10

0.15

0.20

0.25

0.30

MatapakitiakiaRUAngāpūtakeikoreaipeatetuaritangaPoissonoteλ=4.7itetauirapaimōtemahaongāhīrereputungaparamōtētahiwā24haora.

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2.

Hautanga

Temahaongāhīrereputungaparaitētahiwā24haora

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QUESTION THREE

AstudycollecteddataonwaterusewithinNewZealandhomesforthepurposeofassistingcouncils,governmentagencies,andwatersupplierstointroducewaterefficiencymeasures.

Priortothestudy,itwasestimatedthateachpersoninNewZealandflushesthetoiletonaverage4.7timesper24-hourperiod.

(a) (i) UsingaPoissondistributionmodel,calculateanestimatefortheprobabilitythatapersonflushesthetoiletlessthanfivetimesinany24-hourperiod.

(ii) GiveONEreasonwhyitmaynotbeappropriatetouseaPoissondistributiontomodelthenumberoftoiletflushesforany 4-hour period.

(b) Datawascollectedonthenumberoftimeseachpersonflushedthetoiletduringa24-hourperiod.Thedatafrom200peoplefromacross84homesinthestudyissummarisedinthetablebelow.

Number of toilet flushes during 24-hour period

0 1 2 3 4 5 6 7 8 9 10

Proportion 0 0.01 0.1 0.17 0.26 0.14 0.12 0.11 0.05 0.01 0.03

(i) Calculatethemeannumberoftoiletflushesmadeper24-hourperiodforpeopleinthisstudy.

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(ii) Didmostpeopleinthisstudyflushthetoiletatleastfourtimesduringa24-hourperiod?

Supportyouranswerwithacalculation.

(iii) Thegraphbelowshowstheexperimentaldistribution(shadedbars)andaPoissondistributionwithλ=4.7(themodeldistributionshowninblack).

0 1

Number of toilet flushes during a 24-hour period

Prop

ortio

n

2 3 4 5 6 7 8 9 100

0.05

0.10

0.15

0.20

0.25

0.30

DiscussTWOreasonswhyaPoissondistributionwithλ=4.7maynotbeagoodmodelforthenumberoftoiletflushesforany24-hourperiod.

1.

2.

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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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QUESTION NUMBER

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

91

58

6M

Level 3 Mathematics and Statistics (Statistics), 2017

91586 Apply probability distributions in solving problems

9.30 a.m. Monday 27 November 2017 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–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 – 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

Level 3 Mathematics and Statistics (Statistics) (91586) 2017

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