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Enter School name here NAME: _______________________________________ 2 Teacher: __________ Level 2 Mathematics and Statistics 91267 (2.12): Apply probability methods in solving problems Credits: Four Achievement Apply probability methods in solving problems Achievement with Merit Apply probability methods, using relational thinking, in solving problems. Achievement with Excellence Apply probability methods, using extended abstract thinking, in solving problems You should answer ALL parts of ALL questions in this booklet. You should show ALL your working. If you need more space for any answer, use your own paper and clearly number the question. Check that this booklet has pages 2–9 in the correct order and that none of these pages is blank. YOU MUST HAND THIS BOOKLET TO YOUR TEACHER AT THE END OF THE ALLOTTED TIME. TOTAL AS91267 Kohia 2014 GRADE You are advised to spend 60 minutes answering the questions in this booklet. QUESTION ONE (a) The Eastpaddock Mall has two types of food outlets: Cafés and Fast food outlets. 70% of visitors to the Mall stay at least 60 minutes. Of the visitors to the Mall that stay at least 60 minutes, 45% of them use a food outlet. Only a quarter of visitors to the Mall that stay less than 60 minutes use a food outlet. Some of the information is shown on the probability tree below. 0.7 Stay at least 60 minutes Stay less than 60 minutes Use a food provider Don’t use a food provider Use a food provider Don’t use a food provider (i) Calculate the probability that a visitor to the Mall will stay at least 60 minutes and will use a food provider. (ii) Calculate the percentage of visitors to the Mall that don’t use a food provider. (iii) 60% of food providers are Fast food outlets. If a visitor to the Mall stays at least 60 minutes, calculate the probability that they will use a café. AS 91267 2.12 PROBABILITY KOHIA Page 2 Assessor’s e only (b) At the Eastpaddock Mall there is a large store called The HaveOne. 65% of visitors to the Mall are female. 81% of females visit The HaveOne store. Assessor’s e only Some of the information is shown on the probability tree below. Female Visited The HaveOne Don’t visit The HaveOne 0.65 Male (i) What proportion of visitors to the Mall were female that visited The HaveOne? (ii) There were 4500 visitors to the Mall last week. How many would you have expected to be female that didn’t visit the The HaveOne store? (iii) Two thirds of the visitors to the Mall visit The HaveOne store. If a male is selected, calculate the probability they visited The HaveOne store. (iv) If a visitor to the Mall visited The HaveOne store, what is the probability that they were female? AS 91267 2.12 PROBABILITY KOHIA Page 3 QUESTION TWO (a) Assessor’s e only The Eastpaddock Mall collected the following information about visitors spending in a survey at the Mall last week. The table below gives the number of ‘groups’ in each category. Spent at least $100 Spent less than $100 Totals Individuals Couples Families Totals 1560 1250 310 3120 480 220 680 1380 2040 1470 990 4500 (i) What proportion of ‘groups’ in the survey were individuals at the Mall? (ii) Of those groups that spent less than $100 at the Mall, what proportion were individual ‘groups’? (iii) Last year 60 000 ‘groups’ visited the Mall. Using the survey results, how many couple ‘groups’ would you expect to spend at least $100 at the Mall? (iv) Show that for the ‘groups’ surveyed, the risk of spending at least $100 is about 7/10. AS 91267 2.12 PROBABILITY KOHIA Page 4 (v) A newspaper headline on a report summarising the results of the survey stated “Survey shows individuals are two and half times as likely to spend at least $100 on their visit to the Mall than a family ‘group’.” Show whether or not you agree with this headline, stating full reasons and calculations. (b) The survey also recorded the age group of the individual ‘groups’ that visited the Mall. This showed: There were 580 individual ‘groups’ aged at least 50 years old that spent at least $100 at the Mall There were 1170 individual ‘groups’ aged less than 50 years old. There were 480 individual ‘groups’ who spent less than $100. Individuals at least 50 Individuals less than 50 Totals Spent at least $100 Spent less than $100 2040 Totals AS 91267 2.12 PROBABILITY KOHIA Page 5 Assessor’s e only (i) What proportion of individual ‘groups’ visiting the Mall who spent at least $100 were aged less than 50? (ii) A researcher claimed that individual ‘groups’ aged less than 50 are 25% more likely to spend at least $100 at the Mall than those aged at least 50. Comment on this claim using the survey data. Use suitable reasons and calculations to support your choice. AS 91267 2.12 PROBABILITY KOHIA Page 6 Assessor’s e only QUESTION THREE (a) Assessor’s e only Visitors to the Eastpaddock Mall walk a variety of distances inside the Mall. The distance walked by visitors in the Mall is normally distributed, with a mean of 1250 metres and standard deviation of 310 metres. (i) What is the probability that a visitor to the Mall will walk between 1000 and 1500 metres? (ii) What percentage of visitors to the Mall, walk less than 500 metres? (iii) 5% of visitors to the Mall are classified as ‘brief visitors’. Calculate the maximum distance (to the nearest metre) walked in the Mall by a ‘brief visitor’. (b) Mall officials claim that older visitors to the Mall generally don’t walk as far as younger visitors. It was found that only 10% of older visitors to the Mall walk more than 1000 metres. Calculate the mean distance walked by older visitors to the Mall. Assume that a normal distribution can be used to model the distances walked by older visitors in the Mall and that it has the same standard deviation (310 metres). AS 91267 2.12 PROBABILITY KOHIA Page 7 (c) A survey of the distances walked by male and female visitors to the Mall last week produced the following results. Distance walked by Males Distance walked by Females AS 91267 2.12 PROBABILITY KOHIA Page 8 Assessor’s e only (i) Give a possible reason why the distances walked by males is not independent of the distances walked by females. (ii) Compare and contrast the distributions and males and for females. You should discuss shape, centre and spread in relation to the context. AS 91267 2.12 PROBABILITY KOHIA Page 9 Assessor’s e only AS 91267 2.12 PROBABILITY KOHIA Page 10