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Draft — For School Planning Purposes Only
Level 3 Mathematics and Statistics
(Statistics)
3.14: Apply probability distributions in solving problems
Credits: Four
Check that you have completed ALL parts of the box at the top of this page.
You should answer ALL parts of ALL questions in this booklet.
If you need more room for any answer, use the space provided at the back of this booklet.
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Check that this booklet has pages 2–11 in the correct order and that none of these pages is blank.
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YOU MUST HAND THIS BOOKLET TO YOUR TEACHER AT THE END OF THE ALLOTTED TIME.
OVERALL LEVEL OF PERFORMANCE
© New Zealand Qualifications Authority, 2012
All rights reserved. No part of this publication may be reproduced by any means without the prior permission of the New Zealand Qualifications Authority.
2
You are advised to spend 60 minutes answering the questions in this booklet.
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QUESTION ONE
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Plants generally produce several flowers each season that need to be pollinated in order for fruit to
be produced.
A particular fruit plant produces one flower each season. This flower must be pollinated in order for
a single fruit to be produced. This plant can be pollinated by:
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• bees carrying pollen from flower to flower between plants of the same type OR
• hand, with a person transferring pollen from flower to flower of the same type.
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Each of these methods has a different success rate and different cost structures for the grower.
• Bee pollination has a 62% successful pollination rate.
• Hand pollination has a 91% successful pollination rate.
A grower of this particular fruit has 150 plants, of which they are contracted to provide 100
pollinated flowers to a local plant shop. The grower receives $12.50 for each pollinated plant.
Importing hives of bees costs $500 per year. Hand pollination costs $1 000 per season.
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Make a recommendation to the grower as to which pollination method they should use with
reference to the highest income and the best guarantee of fulfilling their contract.
Mathematics and Statistics 3.13
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Mathematics and Statistics 3.13
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QUESTION TWO
Another grower produces guava for the local canning factory. The mean number of guava per
tree is 250.
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(a)
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Source: WikiCommons, 2006, Sakurai Midori, digital image, http://commons.wikimedia.org/wiki/File:Guava_ID.jpg
Calculate the probability that the grower gets 260 guava per tree.
(b)
The weight of individual guavas is normally distributed with a mean of 215 g with a standard
deviation of 13.2 g.
The canning factory accepts guava for processing in two categories: less than 210 g, and 210 g
and over.
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The quality controller at the factory will select a guava at random from the canning line to
ensure the grading process is working satisfactorily. Calculate the probability that a guava
selected at random from the canning line is over 205 g given that it is under 210 g in weight.
Mathematics and Statistics 3.13
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The grower also has a contract to supply the factory with 3 000 pineapples for canning. The
pineapples are normally distributed with a mean weight of 1.3 kg and a standard deviation of
0.234 kg.
The grower is hoping to earn $52 500 from the fruit of his 200 guava trees and the 3 000
pineapples that the canning factory buys this year. The canning factory pays $3.90 per
kilogram of guava and $2.70 per kilogram of pineapple.
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(c)
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Comment on whether the grower’s expectations are likely to occur.
Mathematics and Statistics 3.13
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Mathematics and Statistics 3.13
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QUESTION THREE
The following probability function was the result of the observation of the number of people
who queue jumped (pushed ahead of others waiting to be served) at the express checkout line
at a local supermarket in 15 minute intervals.
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Probability
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Queue jumpers
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(a)
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Jumpers per 15 mins
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Using an appropriate distribution, calculate the expected number of queue jumpers in any
given hour. In your answer, you should justify your choice of distribution and state any
assumptions you make.
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(b)
The store manager makes a statement to staff that he wants the store to be known as ‘queue–
jump free’. This is defined by the manager as the probability of no queue jumpers per 10
minutes being 0.9.
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(c)
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State the effect that reducing the probability of no queue jumpers per 10 minutes to 0.9 will
have on the expected number of queue jumpers in any given hour. You should provide
statistical evidence to support your answer.
Observation of the express checkout also found that 43% of customers required the checkout
operator to ask for supervisor assistance. This meant that the mean queuing time for
customers was 5.6 minutes with a standard deviation of 1.9 minutes.
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Calculate the probability that three of the next 10 customers require supervisor assistance and
have to wait longer than 7 minutes in the queue. State any assumptions you make.
Mathematics and Statistics 3.13
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The store manager thought that customer service could be improved if specific express
checkouts were introduced for those customers who required supervisor approval to make
their purchases.
81.4% of those that required supervisor assistance moved to the new checkouts. This reduced
the probability of a customer queuing for longer than 5 minutes to 0.23.
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Calculate the new mean queuing time. State any assumptions you make.
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(d)
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Extra space if required.
Clearly number the question (if required).
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Extra space if required.
Clearly number the question (if required).
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Question
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Mathematics and Statistics 3.13