Download Level 3 Statistics AS91586 Apply probability distributions in solving

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Level 3 Statistics
AS91586 Apply probability distributions in
solving problems
3
Credits: Four
Achievement
 Apply probability
distributions in solving
problems.
Achievement with Merit
 Apply probability
distributions, using relational
thinking, in solving
problems.
Achievement with Excellence
 Apply probability
distributions, using
extended abstract thinking,
in solving problems.
BURGER PRINCE
You should attempt ALL the questions in this booklet.
Show ALL working.
Make sure that you have the Formulae and Tables Booklet L3--STATF.
If you need more space for any answer, use the page(s) provided at the back of this booklet and
clearly number the question.
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.
TOTAL
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GRADE
1
You are advised to spend 60 minutes answering the questions in this booklet.
BURGER PRINCE
QUESTION ONE: CLEANING STANDARDS
At a local fast food shop called Burger Prince there are strict cleaning controls in place
from head office. The controls must be followed or the chain may lose its licence to use
the name Burger Prince. The manager of Burger Prince must take a daily swab from the
surface of the refrigerator and test the amount of bacteria in the sample. Unacceptable
levels are above 15 bacteria colonies per cm2 and will cause the restaurant to fail its
hygiene requirements. The table below shows the results of last 90 samples round to the
nearest 10 colonies/cm2.
Bacteria
levels
Frequency
0 (or
negligible)
4
10
20
30
40
Over 45
72
13
0
1
0
a) Identify an appropriate distribution to model the distribution and justify your choice.
Identify the parameters of the distribution and state any assumptions you make.
b) Use your model to predict if the restaurant will fail its hygiene requirements.
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c) New hygiene standards have been introduced by head office. They now want the
unacceptable bacteria colony count to be less than 8 colonies per cm2.
What is the probability over the following week the colony count will be below 8
colonies/cm2 for at least 6 out of 7 days.
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QUESTION TWO: MARKETING CAMPAIGN
The amount spent by a typical single customer is normally distributed with a mean of
$15.67 and a standard deviation of $7.25
The chain of outlets is considering changing its marketing attract higher paying customers.
It has segmented the current market into four categories;
Low
Low to Medium
Medium
High
Less than $5.25
Between $5.26 and $10.03
Between $10.04 and $20.45
$20.45 and above
a) What percentage of customers are either High spenders or Low spenders?
The marketing department is considering a new segment of higher paying customers.
They want the top 10% of customers to be called “Super High”.
b) What typical spend per customer would this new category start from?
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b) i) The head office decides to implement the new marketing campaign. They introduce
free-range meat and eggs and adopt a more ‘café-style’ approach to their décor.
The new campaign was given 3 weeks to see if it is a success, it will be considered a
success if the mean spend has risen to $22. The marketing departing randomly selected
300 customers and recorded their amount spent at the restaurant.
Preliminary findings can be seen in the histogram below.
Explain whether a normal distribution would be an appropriate model for the distribution of
the expected spend per customer.
Does the graph provide evidence the target has been met? Justify your answer.
Burger Prince spend per customer after 3
weeks of new campaign
80
70
frequency
60
50
40
30
20
10
0
5
7.5
10
12.5
15
17.5
20
22.5
25
27.5
Spend per customer ($)
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5
30
32.5
35
37.5
40
ii) If the customers are categorised in the same proportions as previously, how much
would a Medium customer now spend?
Support your answer with calculations.
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QUESTION THREE: MYSTERY SHOPPER
a) i) The restaurant stays open 24 hours a day, seven days a week. Head office comes to
inspect the premises unexpectedly at random times during the week.
Name, draw and label the probability distribution graph which models this situation for a
week.
Space below for sketching and labelling graph:
a) ii) The inspectors can arrive at any time in the week. What is the probability the
inspector arrives between 2pm and 5pm on Tuesday?
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b) There are three ID cards at head office which the inspectors use to gain access to the
restaurant kitchens.
Each card is kept in its own drawer in the office. A new employee replaces the cards into
the drawers at random. What is the expected value and standard deviation of the number
of ID cards that are put back into their original drawers?
c) A new inspector is hired and head office want to find the average of the two newest
inspector’s rankings of restaurants.
The first inspector’s average ranking is 4 and his standard deviation is 1.2. The new
inspector’s average ranking is 3.5 and her standard deviation is 2.3. What is the mean
and standard deviation of the average of the two inspector’s rankings?
State any assumptions made.
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