Section 1: Descriptive Statistics & Basic Probability Theory

Section 1: Descriptive Statistics & Basic Probability Theory

Part A. Short answer questions (5 questions worth 1 mark each for a total of 5 marks)
1. If a descriptive statistic is resistant to outliers, what does this mean? Explain your
answer using a suitable example.
2. If two events A and B are independent and P(A)=0.2, P(B)=0.3, calculate P(A or B).
3. A media article claimed that there was a correlation between the colour of a car and
the likelihood of a car being in an accident. What is wrong with this statement?
4. What do you understand by a “misleading graph”? Briefly explain your answer with a
suitable example of a variable on a labelled diagram.
5. If Event A = New product sold and Event B = Discount on new product, P (A and B)
would be simply calculated as P (A)*P(B). True or False? Explain your answer.
Part B. Data Interpretation and Problem Solving

Question 1 (9 marks)

The sensors measure the human traffic passing a restaurant for a week. The results
are in the table below. The figures are number of people passing the restaurant.

(a) For each time slot, calculate the mean human traffic for the weekdays, MondayFriday.
(b) Plot a Histogram of the human traffic for Saturday.
(c) Discuss your findings from the histograms, noting any patterns in the data.
(d) Explain how the restaurant might use this data to target sales better for Saturday.

Question 2 (4 marks)

The following table shows the annual incomes (in thousands of dollars) and
weekly rent payments (dollars) of a sample of 10 individuals in country Victoria:
(a) Draw a scatter diagram (using Excel) for these data with rent payment on the vertical
axis.
(b) Describe the relationship between rent payment and income.

Question 3 (6 Marks)

A disease has a 6% prevalence in the population. You need to choose between two tests.
If the disease is present, the first test will give a positive result with probability 0.90. If the
disease is not present, it will give a positive result with probability 0.07. If the disease is
present, the second test will give a positive result with probability 0.96. If the disease is not
present, it will give a positive result with probability 0.10.

(a) For each of the tests, what is the probability that someone has the disease given a
positive test result?
(b) The disease is highly contagious, and so those with the disease need to be
quarantined. Which test would you recommend? Explain your reasoning.

Question 4 (6 Marks)

Income 45 51 40 44 48 35 51 55 49 47
Rent 230 280 190 220 270 200 300 310 260 250
An investment firm has classified its clients according to their gender and the
composition of their investment portfolios (primarily bonds, primarily stocks, or a balanced
mix of bonds and stocks). The proportions of clients falling into the various categories are
shown in the following table:

Portfolio Composition
One client is selected at random, and two events A and B are defined as follows:
A: The client selected is male.
B: The client selected has a balanced portfolio.
Find the following probabilities.
(a) P(A).
(b) P(B).
(c) P( ).
Section 2: Discrete Probability Distributions

Question 5 (7 Marks)

You wish to form an investment portfolio, and have three assets to choose from, a high-risk
asset A, and medium-risk asset B, and a zero-risk asset C.
The returns per $1,000 invested for each of these assets in each of three possible economic
conditions is give in the table below:
(a) What is the expected return and variance of returns for each asset?
(b) What is the expected return and variance of returns for a portfolio that invests $600
in A, $300 in B, and $100 in C?
(c) If the economic forecast changes such that there is now a 35% chance of recession,
a 50% chance of a normal economy and only a 15% chance of expansion, determine
the new expected return for the same investments as in Part 2.

Question 6 (5 Marks)

A telephone receptionist takes an average of six calls per hour. Calculate the probability that
the receptionist takes:
(a) exactly five calls in the next 45 minutes.
(b) two or more calls in the next 30 minutes.
(c) from eight to twelve calls inclusive in the next two hours.
Gender Bonds Stocks Balanced
Male 0.18 0.20 0.25
Female 0.12 0.10 0.15
A
Asset A Asset B Asset C
Recession (P=0.2) -$170 -$40 $30
Normal (P=0.5) $10 $90 $30
Expansion (P=0.3) $210 $140 $30

Question 7 (6 Marks)

An insurance company has evidence to suggest that 24% of cars are not insured. Calculate
the following probabilities:
(a) That of the next seven cars involved in accidents, exactly two are not insured.
(b) That of the next twelve cars involved in accidents, fewer than three are not insured.
(c) That of the fifteen cars involved in accidents, more than twelve are insured.
Section 3: Continuous Probability Distributions

Question 8 (6 Marks)

A trucking company has worked out that on average its trucks drive 115,000 kilometres a
year, with a standard deviation of 22,000 kilometres. The distances driven are normally
distributed.
(a) What percentage of the trucks will drive between 100,000 and 120,000 kilometres a
year?
(b) What percentage of trucks will drive less than 60,000 or more than 140,000
kilometres per year?
(c) What minimum distance will be driven by at least 75% of the trucks?
Section 4: Sampling Distributions

Question 9 (6 Marks)

In the first quarter of 2014, the rental cost of a three-bedroom house in a regional town was
$300 with a standard deviation of $30. Assume that the rental costs are normally distributed.
If you select a random sample of ten rental properties, what is the probability that the sample
will have a mean rental cost of:
(a) Less than $275?
(b) Between $280 and $300?
(c) Greater than $310?                                            

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