91586
Marlborough Girls College
Level 3 Mathematics and Statistics
(Statistics)
AS91586: Apply probability distributions in solving problems
Credits: Four
Answer ALL questions in the spaces provided in this booklet.
Show ALL working for ALL questions.
Check that this booklet has pages 2 – 8 in the correct order and that none of these pages is blank.
YOU MUST HAND THIS BOOKLET TO THE SUPERVISOR AT THE END OF THE EXAMINATION.
For Assessor’s use only
Achievement Criteria
Achievement
Apply probability
distributions in solving
problems.
Achievement with Merit
Apply probability
distributions, using
relational thinking, in
solving problems.
Overall level of Performance
Achievement with Excellence
Apply probability
distributions, using extended
abstract thinking, in solving
problems.
Page 2 of 8
You are advised to spend 60 minutes answering the questions in this booklet.
The Beach Resort
Question One
(a) The amount of money spent by a tourist in a beach resort is normally distributed with a mean of $782
with a standard deviation of $158. For marketing purposes, the local tourism board identifies three
categories of tourists according to their spending: less than $600, at least $600 but less than $850, and
$850 and over.
(i)
What proportion of tourists spends either less than $600 or $850 and over?
(ii)
Calculate the probability that a tourist selected at random spends more than $600, given that she
spend less than $820.
Page 3 of 8
(b) In order to further develop tourism in the beach resort, the council plans to build an artificial reef which
could attract more surfers.
In order to finance the artificial reef, the council plans to collect a levy from all operators of
accommodations based on their income according to the following table:
Tourist spending
Levy
Up to $450
$5
More than $450
but less than $600
$8
$600 and more
$12
The plans for the reef will only go ahead if the council will be able to collect at least $100 000 in levies in
the next season.
It is expected that 12 000 tourists will spend their holiday in the resort in the next season and their
spending on accommodation is expected to be normally distributed with a mean of $502 and a standard
deviation of $83.
By calculating the total expected levies, determine if the plan for the reef should go ahead.
Page 4 of 8
Question Two
The number of goals scored by the Glasgow Rangers Football Team over a large number of matches was
recorded and is shown below.
Goals per match
(a) Calculate the expected number of goals in any given match.
(b) A 2012 Home Game for Glasgow Celtic Football team had mean stadium occupancy of 92.33%
with a standard deviation of 1.65%.
(i)
Use a distribution model to estimate the probability that a randomly selected match had
greater than 95% occupancy.
Page 5 of 8
(ii)
State any assumptions that you made in (i). Comment on the validity of these
assumptions.
(c) Calculate an estimate for the probability that 4 consecutive games all have less than 90%
occupancy. State any assumptions made in your estimate.
Page 6 of 8
Question Three
(a) A hotel is fully booked with 22 guests every night during summer. The manager was asked about how
many people he expected to eat in the restaurant each night this coming year. He said that in the past
there were rarely less than 5 people eating and usually about 12.
Using an appropriate distribution, calculate the probability that at least 15 hotel guests eat at the hotel
restaurant on an evening chosen at random.
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(b) For 89 days in summer this year, the restaurant kept records of the number of guests who ate in the hotel
restaurant each evening (graphed below).
(i)
Calculate the probability of more than 15 guests eating in the restaurant, as shown in the graph
above.
Page 7 of 8
(c) Discuss the differences between the model you used in part (a) and the distribution of actual guest
numbers as shown by the graph above.
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(d)
Discuss how True Probability, Experimental Probability and Theoretical Probability are evident in the
context of this problem.
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Page 8 of 8
(d) Show how you would change your distribution from part (a) in light of this new information in part (b).
Justify your choices and re-calculate the probability of at least 15 guests eating at the restaurant.