906425

3SUPERVISOR’S USE ONLY

9 0 6 4 2 M

© Mana Tohu Mātauranga o Aotearoa, 2011. 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.

Te Tauanga me te Whakatauira, Kaupae 3, 201190642 Te tātai i ngā āputa māia o ngā tawhā taupori

2.00 i te ahiahi Rāhina 14 Whiringa-ā-rangi 2011 Whiwhinga: Toru

Tirohia mehemea e ōrite ana te Tau Ākonga ā-Motu 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 ngā whārangi kei muri i te pukaiti nei.

Tirohia mehemea kei roto nei ngā whārangi 2 – 11 e raupapa tika ana, ā, kāore hoki he whārangi wātea.

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

MĀ TE KAIMĀKA ANAKE Paearu Paetae

Paetae Paetae Kaiaka Paetae KairangiTe tātai i ngā āputa māia o ngā tawhā taupori.

Te whakaatu māramatanga ki ngā āputa māia.

Te whakaatu māramatanga o ngā ariā kei muri i ngā āputa māia.

Whakakaotanga o te tairanga mahinga

See back cover for an English translation of this cover

Kia 35 meneti hei whakautu i ngā pātai o tēnei pukapuka.

PĀTAI TUATAHI

E tūhuratia ana e Waka Kotahi (NZTA) te roa me te whānui o ngā tauaru e tōia e ngā waka i Aotearoa.

(a) I inea te whānui o ngā tauaru i ngā waka tōtō tauaru e 45 i tīpako matapōkeretia. He 1.48 mita te whānui toharite o ngā tauaru tīpako me te ine mahora o te 0.22 mita.

Kimihia he āputa māia 90% mō te whānui toharite o ngā tauaru waka katoa i Aotearoa.

(b) Whakamāramahia te take e kore e taea te tātai i tētahi āputa māia 100% mō te whānui toharite o ngā tauaru waka katoa i Aotearoa.

(c) Kei te hiahia te NZTA ki te whakatau tata i te roa toharite o ngā tauaru waka katoa ki roto i te 15 cm o te roa toharite tūturu me te taumata māia o te 99%. Ko ngā tātai o mua a te NZTA e whakaatu ana he 41 cm te ine mahora o te roa o nga tauaru.

Tātaihia te rahi tīpako mōkito ka hiahiatia.

2

Te Tauanga me te Whakatauira 90642, 2011

MĀ TE KAIMĀKA

ANAKE

3

Statistics and Modelling 90642, 2011

ASSESSOR’S USE ONLY

You are advised to spend 35 minutes answering the questions in this booklet.

QUESTION ONE

The New Zealand Transport Agency (NZTA) is investigating the lengths and widths of trailers towed by cars in New Zealand.

(a) A random sample of 45 cars towing trailers was stopped, and the widths of their trailers were measured. The widths of the sampled trailers had a mean of 1.48 m and standard deviation of 0.22 m.

Find a 90% confidence interval for the mean width of all car trailers in New Zealand.

(b) Explain why it would not be possible to calculate a 100% confidence interval for the mean width of all car trailers in New Zealand.

(c) The NZTA wants to estimate the mean length of all car trailers to within 15 cm of their true mean length with 99% confidence. Past studies by the NZTA have shown that the standard deviation of the lengths of trailers is 41 cm.

Calculate the minimum sample size required.

(d) Ka hangaia e te NZTA tētahi ara hipa mēnā ko te roa toharite o ngā tauaru katoa e tōia ana e ngā waka i tētahi wāhi ake he nui ake i te 2.95 m.

I taua wāhi, i kitea i roto i te tīpakotanga o ngā tauaru 100 he 3.06 m te roa toharite me te ine mahora o te 0.39 m.

Tūhuratia te tūponotanga ka waihangahia e te NZTA tētahi ara hipa i taua wāhi.

4

Te Tauanga me te Whakatauira 90642, 2011

MĀ TE KAIMĀKA

ANAKE

5

Statistics and Modelling 90642, 2011

ASSESSOR’S USE ONLY

(d) The NZTA will construct a passing lane if the mean length of all trailers towed by cars at a particular location is greater than 2.95 m.

At that location, a sample of 100 trailers was found to have a mean length of 3.06 m with a standard deviation of 0.39 m.

Investigate the likelihood that the NZTA will construct a passing lane at that location.

PĀTAI TUARUA

E whakapono ana te NZTA he rerekē te tere o ngā waka e haere ana i te Huarahi Matua 1 i waenga i Te-Ika-a-Māui me Te Waipounamu. I inea te tere o ngā waka i ngā wāhi i whiriwhiri matapōkeretia i te Huarahi Matua 1 me te pae tere o te 100 kiromita i te hāora i Te-Ika-a-Māui me Te Waipounamu.

Kei te papatau ngā otinga mō ngā motu e rua.

Te-Ika-a-Māui Te WaipounamuTe rahi o te tīpako 301 198Te toharite tīpako (kiromita i te hāora) 89.2 94.5Te ine mahora o te tīpako (kiromita i te hāora)

5.6 6.7

(a) Waiho ko μ1 te tere toharite o ngā waka i Te-Ika-a-Māui, ka waiho ko μ2 te tere toharite o ngā waka i Te Waipoumanu, i ngā wāhi he 100 kiromita i te hāora te pae tere.

Kimihia he āputa māia 95% mō te μ1 – μ2, me te whakamārama mēnā e whakaatu ana te rangahau a te NZTA i te rerekētanga o te tere o ngā waka o te Huarahi Matua 1 i Te-Ika-a-Māui me te tere o ngā waka o te Huarahi Matua 1 i Te Waipounamu.

(b) Homai kia RUA neke atu rānei ngā whakamārama he aha te āputa māia i tātaitia ai i te wāhanga (a) kei runga ake e kore pea e tuku taunakitanga rawaka mā te NZTA hei whakatau otinga mō te tere o ngā waka e haere ana i te Huarahi Matua 1, i Te-Ika-a-Māui me Te Waipounamu.

6

Te Tauanga me te Whakatauira 90642, 2011

MĀ TE KAIMĀKA

ANAKE

QUESTION TWO

The NZTA believes that the speed of cars travelling on State Highway 1 differs between the North Island and South Island. It measured the speed of cars at randomly selected locations on State Highway 1, where the speed limit is 100 km per hour in each of the North Island and the South Island.

The results from both islands are shown in the table.

North Island South IslandSample size 301 198Sample mean (km per hour) 89.2 94.5Sample standard deviation (km per hour) 5.6 6.7

(a) Let μ1 be the mean speed of cars in the North Island, and let μ2 be the mean speed of cars in the South Island, in regions where the speed limit is 100 km per hour.

Find a 95% confidence interval for μ1 – μ2, and hence explain whether the NZTA study shows there is a difference between the speeds of cars on State Highway 1 in the North Island and the speeds of cars on State Highway 1 in the South Island.

(b) Give at least TWO reasons why the confidence interval calculated in part (a) above may not provide sufficient evidence for the NZTA to make a conclusion about speeds of cars travelling on State Highway 1, in the North Island and in the South Island.

7

Statistics and Modelling 90642, 2011

ASSESSOR’S USE ONLY

(c) I kohia anō e te NZTA he raraunga mō ngā motupaika. I kitea e 85 o ngā motupaika 400 i rangahautia e rua ngā tāngata runga.

Tātaitia he āputa māia 90% mō te ōwehenga o ngā motupaika katoa whai tāngata tokorua.

(d) I tīpakohia e te NZTA te tere o ngā motupaika 120 i whiriwhiri matapōkeretia i tētahi kokonga ake. I whakaputahia e ngā raraunga tētahi āputa māia 97% mō te µ, arā, te tere toharite o ngā motupaika katoa i taua kokonga he:

58.3 < µ < 67.1 Kimihia te tau toharite me te ine mahora o te tīpako.

8

Te Tauanga me te Whakatauira 90642, 2011

MĀ TE KAIMĀKA

ANAKE

9

Statistics and Modelling 90642, 2011

ASSESSOR’S USE ONLY

(c) The NZTA also collected data about motorbikes. It found that 85 out of 400 motorbikes surveyed were carrying two people.

Calculate a 90% confidence interval for the proportion of all motorbikes that were carrying two people.

(d) The NZTA sampled the speed of 120 randomly selected motorbikes at a particular corner. The data generated a 97% confidence interval for µ, the mean speed of all motorbikes at that corner as:

58.3 < µ < 67.1 Find the mean and standard deviation of the sample.

10

Te Tauanga me te Whakatauira 90642, 2011

MĀ TE KAIMĀKA

ANAKETAU PĀTAI

He wāhi anō mēnā ka hiahiatia.Tuhia te (ngā) tau pātai mēnā e hāngai ana.

11

Statistics and Modelling 90642, 2011

ASSESSOR’S USE ONLY

QUESTION NUMBER

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

© New Zealand Qualifications Authority, 2011. 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 Statistics and Modelling, 201190642 Calculate confidence intervals for population

parameters

2.00 pm Monday 14 November 2011 Credits: Three

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 extra space provided at the back of this booklet.

Check that this booklet has pages 2 – 11 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.

ASSESSOR’S USE ONLY Achievement Criteria

Achievement Achievement with Merit Achievement with ExcellenceCalculate confidence intervals for population parameters.

Demonstrate an understanding of confidence intervals.

Demonstrate an understanding of the theory behind confidence intervals.

Overall level of performance

90

64

2M

English translation of the wording on the front cover