Exemplar 91586
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Exemplar 91586. 2013. Question One. 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.

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Exemplar 91586

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Exemplar 91586

Exemplar 91586

2013


Question one

Question One

  • 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.

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


What proportion of guavas will weigh either less than 200 g or 220 g and over

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

The proportion of guavas that will weigh either less than 200 g or 220 g and over is 0.4803

200

215

220


Exemplar 91586

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


Exemplar 91586

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

The probability that a guava selected at random is over 205 g, given that it is under 220 g in weight=

205

215

220


Exemplar 91586

A grower has a contract to supply the factory with 5 000 pineapples. The pineapples are normally distributed with a mean weight of 1.3 kg and a standard deviation of 0.234 kg.

The canning factory pays $2.70 per kilogram of pineapple. The grower is hoping to earn $17 500 from his contract with the factory.

By considering the variation of the pineapple weights, determine if the grower’s expectations for income from this contract are reasonable. Support your answer with calculations.


Exemplar 91586

A grower has a contract to supply the factory with 5 000 pineapples. The pineapples are normally distributed with a mean weight of 1.3 kg and a standard deviation of 0.234 kg.


Exemplar 91586

The canning factory pays $2.70 per kilogram of pineapple. The grower is hoping to earn $17 500 from his contract with the factory.


Exemplar 91586

By considering the variation of the pineapple weights, determine if the grower’s expectations for income from this contract are reasonable. Support your answer with calculations.

Since the expected value for the income if $17 550, with a standard deviation of only approximately $45, it is reasonable that the grower could gain an income close enough to $17 500 (the coefficient of variation is only 0.3% (45/17550).


Alternate answer

Alternate Answer

17500

17550


Alternate answer1

Alternate Answer

95% of income is expected to lie within 2 standard deviations from the mean i.e.

Between $17460 and $17640

So it is reasonable to expect an income of

$17 500 as this amount lies within the range for the central 95% of incomes.


Question two

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.


C alculate the expected number of queue jumpers in any given 15 minute interval

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


C alculate the expected number of queue jumpers in any given 15 minute interval1

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


Do not round to 1 because the question says expected number

Do not round to ‘1’ because the question says “expected number”


Exemplar 91586

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.

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.


Highlighting

Highlighting

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.

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.


Question states a rate poisson distribution

Question states a ‘rate’ – Poisson distribution

no more than one queue jumper per half an hour.


Question states a rate poisson distribution1

Question states a ‘rate’ – Poisson distribution

In your answer, you should justify your choice of distribution, identify the parameters of this distribution,

Applying this distribution because:

  • discrete (queue jumpers) within a continuous interval (time)

  • it cannot occur simultaneously (only one queue jumper at a time)

  • queue jumping is a random event with no pattern to it

  • for a small interval (eg half an hour) the mean number of occurrences (queue jumpers) is proportional to the size of the interval.


Exemplar 91586

and state any assumptions you make

Assumption is that each queue jumper is independent of other queue jumpers.


Question states a rate poisson distribution2

Question states a ‘rate’ – Poisson distribution

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.

P(X ≤ 1) = 0.3084


Exemplar 91586

  • 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%.

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


Exemplar 91586

  • 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%.


No more than one queue jumper per half an hour

“no more than one queue jumper per half an hour”

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

The probability of there being more than one queue jumper per half an hour is less than 5%, so this could be evidence of the store being ‘queue-jump free’.


Question three

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.


Exemplar 91586

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


Exemplar 91586

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


Binomial distribution

Binomial Distribution

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

Applying this distribution because:

  • fixed number of trials (four seasons/years)

  • fixed probability (67% success rate for pollination)

  • two outcomes (pollinated, not pollinated)

  • independence of events (a plant being pollinated or not does not affect chances of the same plant being pollinated or not in another season/year).


Binomial distribution1

Binomial Distribution

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

Parameters:

n = 4, π = 0.67


Binomial distribution2

Binomial Distribution

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

Assumption is that the bees visit all the plants.


Exemplar 91586

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

  • P(X ≥ 3) = 1 – P(X ≤ 2) = 1 – 0.4015 = 0.5985


Exemplar 91586

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.


Exemplar 91586

Binomial distribution

n = 12, p = 0.91

P(X = 11) = 0.3827

P(11 plants produce fruit for both seasons/years) = 0.38272 = 0.1465


Independence

Independence

Assuming whether a plant is pollinated or not in one season does not affect the chances of the same plant being pollinated or not in another season/year, and assuming whether a plant is pollinated or not in one season/year does not affect the chances of another plant being pollinated or not in the same season/year.


Exemplar 91586

A commercial grower of this particular fruit plant would like to use a combination of both bee pollination and hand pollination, as bee pollination is much cheaper than hand pollination.

The grower has 150 plants, of which they are contracted to provide 100 pollinated flowers to a local plant shop. The grower is considering using bee pollination for 90 of the plants, and hand pollination for 60 of the plants.

The graph below shows the distributions of number of plants pollinated using each method.


Exemplar 91586

Comment on how likely it is that the grower will fulfil his contract of providing 100 pollinated flowers using this combination of pollination methods. Support your answer with appropriate calculations and assumptions.


Exemplar 91586

  • Using parameters given earlier


Exemplar 91586

  • Parameters for Normal distribution


Using the normal distribution

Using the Normal distribution


Exemplar 91586

Using 2 standard deviations (10)

from the mean (115)

Range for central 95%, 105 < T < 124

So it is very likely that the grower will fulfil his contract using this combination of methods.


Exemplar 91586

  • You could have used estimates from the graph but this would have been more difficult.


Note the marking per question

Note the marking per question


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