Te Pāngarau me te Tauanga, Kaupae 1, 2011

Transcription

1 See back cover for an English translation of this cover 91037M SUPERVISOR S USE ONLY Te Pāngarau me te Tauanga, Kaupae 1, Te whakaatu māramatanga mō te tūponotanga, raraunga hoki 9.30 i te ata Rāhina 14 Whiringa-ā-rangi 2011 Whiwhinga: Whā Paetae Paetae Kaiaka Paetae Kairangi Te whakaatu māramatanga mō te tūponotanga, raraunga hoki. Te whakaatu māramatanga mō te tūponotanga, raraunga hoki me te parahau i ngā kīanga me ngā kitenga. Te whakaatu māramatanga mō te tūponotanga, raraunga hoki me te whakaatu i te matatau ki te tauanga. 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. 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 25 e raupapa tika ana, ā, kāore hoki he whārangi wātea. HOATU TE PUKAPUKA NEI KI TE KAIWHAKAHAERE HEI TE MUTUNGA O TE WHAKAMĀTAUTAU. TAPEKE MĀ TE KAIMĀKA ANAKE Mana Tohu Mātauranga o Aotearoa, 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.

2 2 Kia 60 meneti hei whakautu i ngā pātai o tēnei pukapuka. PĀTAI TUATAHI I kī atu te koroua o Tuahu ki a ia he ōrite pea te roa o te whanganga tangata ki te tāroa o te tangata (ko te whanganga o te tangata ka totoro mai i ngā matimati o te ringa mauī ki ngā matimati o te ringa matau i te wā e roharoha ana ō ringa). Ka whakaaro a Tuahu mēnā he pono tēnei. I kohia e ia ngā ine o ngā tāne me ngā kōtiro o te tau 10 e 100 nāna i tīpako matapōkere. Ka tāia e ia tētahi kauwhata marara o ngā hua. Kā tāpirihia e ia tētahi rārangi e uru tino pai ana ki te kauwhata. E whakaaturia ana ngā hua o te kauwhata i raro nei, ā, kei roto i te papatau ētahi tauanga. Whanganga (henemita) Tāroa (henemita) Tauanga Tāroa Whanganga te tau toharite mōkito hauwhā raro tau waenga hauwhā runga mōrahi whānuitanga whānuitanga i waenga hauwhātanga Te Pāngarau me te Tauanga 91037, 2011

3 3 (a) (i) He aha te tāroa o te tangata tāroa rawa o te kauwhata? MĀ TE KAIMĀKA ANAKE (ii) He aha te tāroa o te tangata poto rawa tōna whanganga? (iii) E hia ngā tāngata ko ō rātau whanganga kei waenganui i te 120 me te 135 henemita? (b) He aha i tōtika ai te kauwhata marara hei whakaatu i ngā raraunga i kohia e Tuahu? (c) Kei runga i te kauwhata ētahi pūwāhi kāre pea e ōrite ana ki ngā ine mō tētahi tauira tau 10. Homai te tāroa me te whanganga mō ngā pūwāhi e TORU kāre pea i te tika. Whakamāramahia mai ki ōu whakaaro kāre pea aua ine nō tētahi tauira tau 10. Te Pāngarau me te Tauanga 91037, 2011

4 4 You are advised to spend 60 minutes answering the questions in this booklet. QUESTION ONE Tuahu s grandfather told him that a person s arm-span is often the same as their height (your arm-span is the distance from the fingertips of your left hand to the fingertips of your right hand, when your arms are stretched out). Tuahu wondered if this was true. He collected measurements from 100 randomly selected year 10 boys and girls. He drew a scattergraph of the results. He added a line of best fit to the graph. The results are shown on the graph below and some statistics are listed in the table. Arm-span (cm) Height (cm) Statistics Height Arm-span mean minimum lower quartile median upper quartile maximum range inter-quartile range Mathematics and Statistics 91037, 2011

5 5 (a) (i) What is the height of the tallest person on the graph? ASSESSOR S USE ONLY (ii) What is the height of the person with the smallest arm-span? (iii) How many people have an arm-span between 120 and 135 cm? (b) Why was a scattergraph appropriate to show the data Tuahu had collected? (c) There are some points on the graph that seem to be unlikely measurements for a year 10 student. Give the height and arm-span for THREE points that seem unlikely. Explain why you think they are unlikely measurements for a year 10 student. Mathematics and Statistics 91037, 2011

6 (d) I te mutunga ka tau te whakaaro o Tuahu mai i tana kauwhata kei te tika te kōrero a tōna koroua: e ai ki te toharitenga, he ōrite anō te roa o te whanganga tangata ki tōna tāroa. Kei te whai mana te whakatau a Tuahu? Me homai e koe ngā take e RUA neke atu rānei mō tō whakautu. 6 MĀ TE KAIMĀKA ANAKE Te Pāngarau me te Tauanga 91037, 2011

7 (d) Tuahu concludes from his graph that the statement made by his grandfather is correct: on average, a person s arm-span is the same as their height. Is Tuahu s conclusion valid? You should give at least TWO reasons for your answer. 7 ASSESSOR S USE ONLY Mathematics and Statistics 91037, 2011

8 8 PĀTAI TUARUA Kei te whakahaere te whare hokomaha o Alice i tētahi whakataetae. Kei muri i ia rīhiti ko tētahi pū o te kupu ANKARA, arā, he taone nui kei Whenua Korukoru. Mēnā ka taea e Alice te kohi i ngā pū e ono e tātaki ana i te kupu Ankara, ka uru ia ki roto i te whakataetae mō tētahi hararei ki te Whenua Korukoru. (a) Ia rā o ngā rā mahi e 5 mō ngā wiki e 5 ka kite a Alice i tētahi rīhiti kua whiua i a ia e hipa ana i te toa hokomaha. Ko te raupapa o tana kohi mai, koinei ngā pū i kohia e ia: N K K R R N A K A R N N A K K N R N A A A K K R R MĀ TE KAIMĀKA ANAKE (i) Whakaotia te papatau hei whakarāpopoto i ana raraunga: Pū A K Auautanga N 6 R (ii) Mā te whakamahi i ana raraunga, he aha te tūponotanga ka whiwhi i a Alice tētahi K i tana rīhiti o muri mai? (iii) He pēhea te whaimana o tēnei tūponotanga? Homai kia 2 neke atu rānei ngā pūtake tauanga mō tō whakautu. Te Pāngarau me te Tauanga 91037, 2011

9 9 QUESTION TWO ASSESSOR S USE ONLY Alice s local supermarket is running a competition. On the back of each docket is printed one of the letters of the word ANKARA, a city in Turkey. If Alice can collect the six letters needed to spell Ankara, she will go in the draw for a holiday to Turkey. (a) On each of the 5 weekdays for 5 weeks Alice finds a discarded docket as she passes the supermarket. In the order that she collects them, the letters collected are: N K K R R N A K A R N N A K K N R N A A A K K R R (i) Complete the table to summarise her data: Letter A K Frequency N 6 R (ii) Using her data, what is the probability of Alice getting a K on the next docket? (iii) How valid is this probability? Give at least 2 statistical reasons for your answer. Mathematics and Statistics 91037, 2011

10 10 E whakaaroaro ana a Alice kia hia te tau toharite o ngā rīhiti me kohi ia kia tātakina te kupu ANKARA. MĀ TE KAIMĀKA ANAKE (b) Mā te whakamahi i ana kohinga rīhiti ki (a), e hia ngā rīhiti i kohia e Alice i mua i tana riro katoa mai o te kupu ANKARA? (c) E mārama ana a Alice he roa rawa te wā ka kitea tētahi whakautu mēnā ia ka kohikohi rīhiti. Whāia, ka tīkina e ia tētahi mataono me te tuhi i nga pū e ono o A, N, K, A, R, A ki runga. Kei te hiahia a Alice kia rapua te nui toharite o ngā wā me whiu ia i te mataono hei tātaki i te kupu ANKARA. Ka whiua e ia te mataono, ā, ko te pū ka puta ki runga, ka whakaaro noa ia koia ko te pū tērā i kitea e ia ki muri i te rīhiti. Te rironga mai o ngā pū katoa hei tātaki i te kupu Ankara, ka tīmata anō ia. Ka mutu tana whakamātautau ina oti i a ia te tātaki i te kupu Ankara mō ngā wā 10. R, K, N, A, A, K, K, K, A A, A, N, A, R, A, K A, N, N, K, A, N, R, A K, A, K, K, R, A, A, A, N R, A, N, A, A, R, A, K N, A, R, R, A, A, A, K A, A, A, A, R, A, R, A, N, R, A, A, A, A, R, A, A, N, R, N, A, K N, K, A, A, N, A, A, N, A, K, N, A, R N, K, A, N, N, A, N, A, A, R A, A, K, A, K, A, N, N, A, R Kātahi a Alice ka whakamahi i ana hua kia kitea e hia ngā rīhiti me riro i a ia hei tātaki i te kupu katoa o ANKARA. Anei ana hua: 9, 7, 8, 9, 8, 8, 22, 13, 10, 10 (i) Mā te whakamahi i ngā raraunga a Alice, homai he whakautu mō tana pātai: E hia te tau toharite o ngā rīhiti hei kohi māna hei tātaki i te kupu ANKARA? Homai kia RUA neke atu ngā toharite. Te Pāngarau me te Tauanga 91037, 2011

11 (ii) 11 Whakamāramahia ko tēhea te toharite pai hei whakamahi mā Alice, ā, he aha ai? MĀ TE KAIMĀKA ANAKE (iii) E kī ana te toa hokomaha he ōrite te tūpono ki ngā pū o A, N, K me R. Whakamāramahia he aha e kore ai e whaimana te whakamātau a Alice. Te Pāngarau me te Tauanga 91037, 2011

12 12 Alice wonders how many dockets she would have to collect, on average, to be able to spell the word ANKARA. ASSESSOR S USE ONLY (b) Using her collection of dockets in (a), how many dockets did Alice collect before she had the whole word of ANKARA? (c) Alice realises that it will take too long to find an answer by collecting actual dockets. Instead, she takes a dice and puts the six letters of A, N, K, A, R, A on it. Alice wants to find out, on average, how many times she must roll the dice to spell the word ANKARA. She rolls the dice and whatever letter is on top, she imagines is the letter she has found on the back of a docket. Once she has all the letters she needs to spell Ankara, she begins again. She stops her experiment when she has spelt the word Ankara 10 times. R, K, N, A, A, K, K, K, A A, A, N, A, R, A, K A, N, N, K, A, N, R, A K, A, K, K, R, A, A, A, N R, A, N, A, A, R, A, K N, A, R, R, A, A, A, K A, A, A, A, R, A, R, A, N, R, A, A, A, A, R, A, A, N, R, N, A, K N, K, A, A, N, A, A, N, A, K, N, A, R N, K, A, N, N, A, N, A, A, R A, A, K, A, K, A, N, N, A, R Alice then uses her results to find out how many dockets she needed to spell the whole word of ANKARA. Her results are: 9, 7, 8, 9, 8, 8, 22, 13, 10, 10 (i) Using Alice s data, give Alice an answer to her question: How many dockets would she have to collect, on average, to be able to spell the word ANKARA? Give at least TWO averages. Mathematics and Statistics 91037, 2011

13 13 (ii) Explain which average you would suggest Alice uses and why. ASSESSOR S USE ONLY (iii) The supermarket says that each letter, A, N, K and R, is equally likely to be found. Explain why Alice s experiment is not valid. Mathematics and Statistics 91037, 2011

14 14 NGĀ KAUWHATA MŌ TE PĀTAI TUATORU Puna raraunga: 30 Ngā pāmahana i Nairopi Ngā toharite ā-marama Mōrahi Mōkito 25 Pāmahana (waeine C) Poutūterangi 08 Kohitātea 08 Whiringa-ā-rangi 07 Mahuru 07 Hōngongoi 07 Haratua 07 Poutūterangi 07 Kohitātea 07 Mahuru 09 Hōngongoi 09 Haratua 09 Poutūterangi 09 Kohitātea 09 Whiringa-ā-rangi 08 Mahuru 08 Hōngongoi 08 Haratua 08 Whiringa-ā-rangi 09 Marama 30 Ngā pāmahana i Roma Ngā toharite ā-marama Mōrahi Mōkito 25 Pāmahana (waeine C) Poutūterangi 08 Kohitātea 08 Whiringa-ā-rangi 07 Mahuru 07 Hōngongoi 07 Haratua 07 Poutūterangi 07 Kohitātea 07 Haratua 09 Poutūterangi 09 Kohitātea 09 Whiringa-ā-rangi 08 Mahuru 08 Hōngongoi 08 Haratua 08 Hōngongoi 09 Whiringa-ā-rangi 09 Mahuru 09 Marama Te Pāngarau me te Tauanga 91037, 2011

15 15 PĀTAI TUATORU Kei te hiahia a Richard ki te neke ki tāwāhi ki tētahi tāone mahana ake. Kei te hiahia ia ki te neke ki Roma, ki Nairopi rānei. Kei te whārangi 14 e whakaatu ana te toharite ā-marama mō ngā pāmahana mōkito, mōrahi hoki ki ia tāone mō ngā tau e toru mai i te tau 2007 ki MĀ TE KAIMĀKA ANAKE Ka whakatau a Richard ki te neke ki Roma i runga i tana whakaaro: He teitei ake te pāmahana o Roma i Nairopi, nā reira he mahana ake a Roma. Ka kōtiu te mōrahi pāmahana o Roma i ia tau, ka mutu he pārekareka ake te noho i konei. He iti iho te taurangi o te pāmahana o Roma i roto i te tau, nā reira he pārekareka ake tēnei. Ko te āhua nei kei te piki haere te pāmahana o Roma, nā reira ā tōna wā ka mahana haere ake. Kāre e tino rerekē ake te pāmahana mōkito me te pāmahana mōrahi o Roma, nā reira he āhuru ake. Whakamahia ngā kauwhata kei te whārangi 14 hei whakautu i ia pātai (a) ki (f). Hei aha te whakamārama he aha e pā mai ai ēnei āhuatanga āhuarangi. (a) Kei te tika a Richard ki te whakapono he kaha ake te pāmahana o Roma i Nairopi, nā reira he mahana ake a Roma? Parahautia tō whakautu mā te whakamahi i ngā kauwhata. (b) Kōrero mō te rerekētanga o te pāmahana mōrahi me te pāmahana mōkito o ēnei tāone e rua i roto i te tau. Te Pāngarau me te Tauanga 91037, 2011

16 16 GRAPHS FOR QUESTION THREE Data source: 30 Temperatures in Nairobi Monthly averages Max Min 25 Temperature (degrees C) Mar 08 Jan 08 Nov 07 Sep 07 Jul 07 May 07 Mar 07 Jan 07 May 09 Mar 09 Jan 09 Nov 08 Sep 08 Jul 08 May 08 Jul 09 Nov 09 Sep 09 Month 30 Temperatures in Rome Monthly averages Max Min 25 Temperature (degrees C) Mar 08 Jan 08 Nov 07 Sep 07 Jul 07 May 07 Mar 07 Jan 07 Sep 09 Jul 09 May 09 Mar 09 Jan 09 Nov 08 Sep 08 Jul 08 May 08 Nov 09 Month Mathematics and Statistics 91037, 2011

17 17 QUESTION THREE ASSESSOR S USE ONLY Richard wants to move overseas to a warmer city. He would like to move to either Rome or Nairobi. The graphs on page 15 show the monthly average minimum and maximum temperatures in each city for three years from 2007 to Richard decides to move to Rome because he thinks: The temperature in Rome is higher than the temperature in Nairobi, so Rome is warmer. The maximum temperature in Rome peaks every year, which is more pleasant to live in. The temperature in Rome is less variable over a year, so this is more pleasant. The temperature appears to be rising in Rome, so it will get warmer in future. There is less difference between the maximum and minimum temperature in Rome, so it will be more comfortable. Use the graphs given on page 16 to answer each of questions (a) to (f). You do not need to explain why the climate features happen. (a) Is Richard correct to believe that the temperature in Rome is higher than the temperature in Nairobi, so Rome is warmer? Justify your answer using the graphs. (b) Comment on how the maximum and minimum temperatures in the two cities vary over a year. Mathematics and Statistics 91037, 2011

18 NGĀ KAUWHATA (i tāia anō mai i te whārangi 14) 18 Puna raraunga: 30 Ngā pāmahana i Nairopi Ngā toharite ā-marama Mōrahi Mōkito 25 Pāmahana (waeine C) Poutūterangi 08 Kohitātea 08 Whiringa-ā-rangi 07 Mahuru 07 Hōngongoi 07 Haratua 07 Poutūterangi 07 Kohitātea 07 Kohitātea 09 Whiringa-ā-rangi 08 Mahuru 08 Hōngongoi 08 Haratua 08 Whiringa-ā-rangi 09 Mahuru 09 Hōngongoi 09 Haratua 09 Poutūterangi 09 Marama 30 Ngā pāmahana i Roma Ngā toharite ā-marama Mōrahi Mōkito 25 Pāmahana (waeine C) Mahuru 08 Hōngongoi 08 Haratua 08 Poutūterangi 08 Kohitātea 08 Whiringa-ā-rangi 07 Mahuru 07 Hōngongoi 07 Haratua 07 Poutūterangi 07 Kohitātea 07 Whiringa-ā-rangi 09 Mahuru 09 Hōngongoi 09 Haratua 09 Poutūterangi 09 Kohitātea 09 Whiringa-ā-rangi 08 Te Pāngarau me te Tauanga 91037, 2011

19 (c) 19 Kua kīia atu a Richard kāre e tino rerekē ake te pāmahana mōrahi me te pāmahana mōkito ki Roma tēnā ki Nairopi. E tautoko ana ngā kauwhata i tēnei? Parahautia tō whakautu. MĀ TE KAIMĀKA ANAKE (d) E whakaaro ana a Richard e piki haere ana te pāmahana i Roma, nā reira ka mahana ake ā tōna wā. Kei te whakaae koe? Parahautia tō whakautu mā te whakamahi i ngā kauwhata. (e) Kei te tika a Richard ki te kī he iti iho te rerekētanga o ngā pāmahana mōrahi me te mōkito i Roma? Parahautia tō whakautu mā te whakamahi i ngā kauwhata. Te Pāngarau me te Tauanga 91037, 2011

20 20 GRAPHS (reprinted from page 16) Data source: 30 Temperatures in Nairobi Monthly averages Max Min 25 Temperature (degrees C) Mar 08 Jan 08 Nov 07 Sep 07 Jul 07 May 07 Mar 07 Jan 07 May 09 Mar 09 Jan 09 Nov 08 Sep 08 Jul 08 May 08 Jul 09 Nov 09 Sep 09 Month 30 Temperatures in Rome Monthly averages Max Min 25 Temperature (degrees C) Mar 08 Jan 08 Nov 07 Sep 07 Jul 07 May 07 Mar 07 Jan 07 Sep 09 Jul 09 May 09 Mar 09 Jan 09 Nov 08 Sep 08 Jul 08 May 08 Nov 09 Month Mathematics and Statistics 91037, 2011

21 (c) Richard has been told that both the maximum and minimum temperatures vary less in Rome than they do in Nairobi. Do the graphs support this? Justify your answer. 21 ASSESSOR S USE ONLY (d) Richard thinks that the temperature appears to be rising in Rome, so it will get warmer in the future. Do you agree? Justify your answer, using the graphs. (e) Is Richard correct to say that the difference between the maximum and minimum temperature is less in Rome? Justify your answer using the graphs. Mathematics and Statistics 91037, 2011

22 (f) Kei te hiahia a Richard ki te noho ki tētahi wāhi mahana. Me tohu ia i a Roma, i a Nairopi rānei? 22 Parahautia tō whakautu mā te hoki ki ngā kauwhata. Matapakitia ngā whakawhāititanga i roto i ngā raraunga, ngā rangahau rānei e tae atu ai koe ki tētahi whakataunga whai mana. MĀ TE KAIMĀKA ANAKE Te Pāngarau me te Tauanga 91037, 2011

23 23 (f) Richard wants to live somewhere warm. Should he choose Rome or Nairobi? Justify your answer by referring to the graphs. Discuss any limitations in the data, or any other research you would need to do before you could make a valid decision. ASSESSOR S USE ONLY Mathematics and Statistics 91037, 2011

24 24 TAU PĀTAI He puka anō mēnā ka hiahiatia. Tuhia te (ngā) tau pātai mēnā ka whai wāhi ki konei. MĀ TE KAIMĀKA ANAKE Te Pāngarau me te Tauanga 91037, 2011

25 25 QUESTION NUMBER Extra paper if required. Write the question number(s) if applicable. ASSESSOR S USE ONLY Mathematics and Statistics 91037, 2011

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28 English translation of the wording on the front cover Level 1 Mathematics and Statistics, Demonstrate understanding of chance and data 9.30 am Monday 14 November 2011 Credits: Four 91037M Achievement Achievement with Merit Achievement with Excellence Demonstrate understanding of chance and data. Demonstrate understanding of chance and data, justifying statements and findings. Demonstrate understanding of chance and data, showing statistical insight. 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. 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 25 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. New Zealand Qualifications Authority, All rights reserved. No part of this publication may be reproduced by any means without the prior permission of the New Zealand Qualifications Authority.