Mathematics and Statistics 91267 (2.12) Exam

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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
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(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.
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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
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QUESTION TWO
(a)
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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
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(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
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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.
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QUESTION THREE
(a)
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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).
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(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
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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
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e only
AS 91267 2.12 PROBABILITY
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