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Question and response book
Mathematical Methods SEE
SEE 1
Time allowed
• Planning time — 15 minutes
• Working time — 180 minutes
General instructions
• Answer all questions in this question and response book.
• Write using black or blue pen.
• QCAA-approved calculator permitted.
• QCAA formula book provided.
• Planning paper will not be marked.
Section 1 (50 marks)
• 9 short response questions
External assessment 2022
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Book
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Section 1
Instructions
• Questions worth more than one mark require mathematical reasoning and/or working to be shown to support answers.
• If you need more space for a response, use the additional pages at the back of this book.
– On the additional pages, write the question number you are responding to.
– Cancel any incorrect response by ruling a single diagonal line through your work.
– Write the page number of your alternative/additional response, i.e. See page … – If you do not do this, your original response will be marked.
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QUESTION 1 (10 marks)
The Pareto distribution is a power-law distribution used in economics to model wealth distribution in a society. It is defined as
( )
= aka+a1 for ≥ , >0, >0f x x k a k
x , and a and k are constants in a given society,
where x is the independent variable (e.g. weekly income), a is the shape parameter and k is the scale parameter.
a) Show that ln
(
f x( ) )
= − +(
a 1 ln) ( )
x +ln( )
aka . [2 marks]Do not write outside this box.
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b) When ln
(
f x( ) )
is plotted against ln( )
x , a linear relationship is produced.Use the data from Stimulus 1 in the stimulus book to determine the value of a. [2 marks]
c) Determine the value of k. [2 marks]
Do not write outside this box.
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d) Evaluate the reasonableness of the Pareto distribution as a model for the data in
Stimulus 1 by considering the coefficient of determination and the residual plot. [4 marks]
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Note: If you make a mistake in the residual plot, cancel it by ruling a single diagonal line through your work and use the additional response space at the back of this question and response book.
Do not write outside this box.
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QUESTION 2 (3 marks)
The cumulative distribution F of households that earn less than weekly income x (in dollars) for a Pareto distribution is given by
1 , where , 0, 0
=
∫
kx a+a ≥ > >F ak dt x k a k
t , and a and k are constants in a given society.
Show that F is given by = − 1
k a
F x .
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QUESTION 3 (3 marks)
Given that 1 = − k a
F x , show that for the Pareto distribution, weekly income x (in dollars) written in terms of the cumulative distribution F is given by x kx kx kx kx k=====
( ( ( ( (
11111−−−−−FFFFF) ) ) ) )
−−−−−1a1a1a1a1a.Do not write outside this box.
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QUESTION 4 (6 marks)
Income distribution can be plotted using cumulative share in income earned (L) against cumulative share of households, from lowest to highest income (F). The curve that represents this is called the Lorenz curve.
Line of equality
0.00.0 0.2 0.4 0.6 0.8 1.0
0.2 0.4
Cumulative share of households, lowest to highest income (F)
Cumulative share in income earned (L)
0.6 0.8 1.0
Lorenz curve
In a society where everyone has equal income (i.e. perfect equality), the Lorenz curve is a straight line from the origin to (1,1), known as the line of equality. The equation of the line of equality is L = F.
The Lorenz curve for the Pareto distribution is defined as
( )
( )
1
01 1
0
1 1
−
−
= −
−
∫
∫
F a
a
k T dT L
k T dT
( )
( )
1
01 1
0
1 1
−
−
= −
−
∫
∫
F a
a
k T dT L
k T dT
( )
( )
1
01 1
0
1 1
−
−
= −
−
∫
∫
F a
a
k T dT L
k T dT
( )
( )
1
01 1
0
1 1
−
−
= −
−
∫
∫
F a
a
k T dT L
k T dT
( )
( )
1
01 1
0
1 1
−
−
= −
−
∫
∫
F a
a
k T dT L
k
(
T)
dT( )
1
01 1
0
1 1
−
−
= −
−
∫
∫
F a
a
k T dT L
k T dT
( )
( )
1
01 1
0
1 1
−
−
= −
−
∫
∫
F a
a
k T dT L
k T dT
( )
( )
1
01 1
0
1 1
−
−
= −
−
∫
∫
F a
a
k T dT L
k T dT
( )
( )
1
01 1
0
1 1
−
−
= −
−
∫
∫
F a
a
k T dT L
k T dT
( )
( )
1
01 1
0
1 1
−
−
= −
−
∫
∫
F a
a
k T dT L
k T dT .
Show that the Lorenz curve for the Pareto distribution is given by LLLLL= − −= − −= − −= − −= − −1 11 11 11 11 1
( ( ( ( (
FFFFF) ) ) ) )
11111−−−−−1a1a1a1a1a.Do not write outside this box.
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Do not write outside this box.
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QUESTION 5 (11 marks)
In 1912, the Italian statistician Corrado Gini proposed a coefficient that could be used as a measure of wealth inequality. The Gini coefficient, G, is given by dividing the area between the line of equality and the Lorenz curve by the total area under the line of equality. Consequently, 0≤ ≤G 1, where 0 represents perfect equality and 1 represents perfect inequality (i.e. where one person has all the wealth).
a) Use mathematical reasoning to explain why perfect equality in income is represented
by G = 0 and perfect inequality in income is represented by G = 1. [3 marks]
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b) Given that the Lorenz curve for the Pareto distribution is given by LLLLL= − −= − −= − −= − −= − −1 11 11 11 11 1
( ( ( ( (
FFFFF) ) ) ) )
11111−−−−−1a1a1a1a1a, show that 1= 2 1 G −
a for a Pareto model of income distribution. [6 marks]
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12 of 21 c) Given that 1
= 2 1 G −
a , determine the restrictions on the possible values of a. [2 marks]
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QUESTION 6 (2 marks)
Use Stimulus 2 in the stimulus book to draw a conclusion about the reasonableness of the Pareto distribution for modelling gross Australian household income. Include numerical evidence from the stimulus to support your conclusion.
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QUESTION 7 (8 marks)
a) Calculate the Gini coefficient for a Pareto model of income distribution using 1
= 2 1 G −
and your value of a from Question 1b). a [1 mark]
b) Use the trapezoidal rule and Stimulus 3 in the stimulus book to approximate the value
of the Gini coefficient for the cumulative income share based on quintile data. [5 marks]
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c) Contrast the Gini coefficients calculated in Questions 7a) and 7b), given that they are modelling the same dataset. Provide mathematical justification for any observed
difference. [2 marks]
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QUESTION 8 (4 marks)
Referring to Stimulus 2, document a strength and a limitation of the Pareto model of income distribution in Australia.
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QUESTION 9 (3 marks)
The Lorenz curve for the Pareto distribution, LLLLL= − −= − −= − −= − −= − −1 11 11 11 11 1
( ( ( ( (
FFFFF) ) ) ) )
1111−−−−−1aa1a1a1a, says that if F = 0.8 (80% of the population) then L = 0.2 (20% of the wealth). Determine the value of a for which this is true.END OF PAPER
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ADDITIONAL PAGE FOR STUDENT RESPONSES
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ADDITIONAL PAGE FOR STUDENT RESPONSES
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ADDITIONAL PAGE FOR STUDENT RESPONSES
Write the question number you are responding to.Do not write outside this box.
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ADDITIONAL PAGE FOR STUDENT RESPONSES
Write the question number you are responding to.© State of Queensland (QCAA) 2022
Licence: https://creativecommons.org/licenses/by/4.0 | Copyright notice: www.qcaa.qld.edu.au/copyright — lists the full terms and conditions, which specify certain exceptions to the licence. | Attribution: © State of Queensland (QCAA) 2022
Clear zone — margin trimmed off after completion of assessment
