Level 3 Mathematics and Statistics(Statistics)

91586 (3.14): Apply probability distributions in solving problems

Credits: Four

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SAMPLE PAPER

3This task has been updated (14 June 2013) to reflect the standard. Linear transformations of continuous random variables have been removed from the curriculum and so are no longer in the standard.Student work exemplars were completed before this update and have not been amended.

Assessor’s use only

You are advised to spend 60 minutes answering the questions in this booklet.

QUESTION ONE

(a) The weight of individual guavas is normally distributed with a mean of 215 g with a standard deviation of 13.2 g. A canning factory accepts guava for processing in three categories: less than 200 g, at least 200 g but less than 220 g, and 220 g and over.

(i) What proportion of guavas will weigh either less than 200 g or 220 g and over?

(ii) Calculate the probability that a guava selected at random is over 205 g, given that it is under 220 g in weight.

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(b) Andrew is a mechanic who is employed by a road service centre. The time taken for Andrew to change a flat tyre is approximately normally distributed, with a mean of 12.4 minutes and a standard deviation of 3.0 minutes.

The road service centre has recently introduced a new fee structure for changing flat tyres, as shown in the table below:

Time taken to change flat tyre

Less than 10 minutesBetween 10 and 15

minutesMore than 15

minutesFee charged ($) $35 $40 $60

Typically, Andrew changes 48 flat tyres per week. The road service centre has set a target that with this new fee structure, they will generate a total of $2500 in fees from Andrew changing 48 flat tyres per week.

By considering the variation of the total fees charged for Andrew changing 48 flat tyres per week, determine if the centre’s target for the total fees are reasonable.

Support your answer with calculations.

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QUESTION TWO

The observations of the number of people who queue jumped (pushed ahead of others waiting to be served) at the express checkout at the local supermarket in a large number of 15-minute intervals produced the following graph.

Queue jumpers

Pro

por

tion

Jumpers per 15 minutes

(a) Calculate the expected number of queue jumpers in any given 15-minute interval.

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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 no more than one queue jumper per half an hour.

(i) Using your answer to part (a) and an appropriate distribution to model this situation, calculate the probability of there being no more than one queue jumper in any given half-hour period.

In your answer, you should justify your choice of distribution, identify the parameters of this distribution, and state any assumptions you make.

(ii) During the four weeks after the store manager made this statement, it was observed that the proportion of 10-minute periods with no queue jumpers was 90%.

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Can the store manager now claim that the store is ‘queue-jump free’? You should provide statistical evidence to support your answer.

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QUESTION THREE

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 in one season per year. The flower must be pollinated in order for a single fruit to be produced.

This plant type can be pollinated by bees carrying pollen from flower to flower between plants of the same type. Bee pollination has a 67% successful pollination rate.

(a) Using an appropriate distribution, calculate the probability that one plant produces at least three fruit over four years (seasons), if bee pollination is used.

In your answer, you should justify your choice of distribution, identify the parameters of this distribution, and state any assumptions you make.

(b) This plant type can also be pollinated by hand, with a person transferring pollen from flower to flower between plants of the same type. Hand pollination has a 91% successful pollination rate.

A home grower has 12 plants of this plant type.

Calculate the probability that for both of two years (seasons), only one of the 12 plants does not produce fruit if hand pollination is used. State any assumptions you make.

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(c) A gardening store that provides supplies to commercial growers has recorded the amounts per sale (in $) of their most recent individual sales.

This information is shown in the graph below:

By applying suitable probability distribution models, determine the probability that for the next eight individual sales, at least two of these sales are for $800 or higher.

Support your answer with reasoning, including justifying models selected and stating any assumptions made.

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Extra space if required.Clearly number the question (if required).

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Assessor’s use onlyExtra space if required.

Clearly number the question (if required).

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