Download Solutions and marking key for Investigation 4 – Sampling

SAMPLE ASSESSMENT TASKS
MATHEMATICS SPECIALIST
ATAR YEAR 12
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2015/3534v4
1
School name
Mathematics Specialist
Unit 3 and Unit 4
Investigation 4
Sampling means
Student name: _______________________ Teacher name: _______________________
Class:
________
Time allowed for this task:
One week in class and at home
Materials required:
Standard writing equipment
Calculator and/or appropriate statistics software
Other materials allowed:
Drawing templates
Marks available:
37 marks
Task weighting:
1% (+ 4% for validation)
Note to teachers: The teacher should introduce the investigation by expanding on the introduction page below. It is a
good idea also to allow students to use actual dice for at least the first three activities. The teacher should also decide on
the type of technology to use, such as Excel spreadsheets or CAS calculators, and how the students should present their
results, either as an electronic file or hard copy. The set solutions provided are used as a guide on the use of some of the
technologies available. The marking key is about whether each individual student has completed the set tasks.
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
2
Introduction
If everyone was the same, there would be no need for statistics or statisticians; you could find out everything
you needed to know from one person (or one event or one result). Statistics involves the study of variability so
that estimates and predictions can be made in complex situations where there is no certain answer. The
quality and usefulness of these predictions depend entirely on the quality of the data upon which they are
based.
Sampling distribution of means and the Central Limit Theorem
The following investigation reflects the content of the dot points contained in the Statistical inference section
of the Unit 4 Mathematics Specialist syllabus.
Sample means
4.3.1 Examine the concept of the sample mean X as a random variable whose value varies between
samples where X is a random variable with mean  and the standard deviation 
4.3.2 Simulate repeated random sampling, from a variety of distributions and a range of sample sizes, to
illustrate properties of the distribution of X across samples of a fixed size n , including its mean μ,

its standard deviation
(where μ and  are the mean and standard deviation of X ), and its
n
approximate normality if n is large
Overview of the investigation
Consider the roll of a standard six-face die which, when rolled, will come to rest with a number
X = 1, 2, 3, 4, 5, or 6 on the upper face. Assuming the die is a fair one, we say that X is a random variable.
Each time we roll the die, we are sampling the outcome from an infinite population. In this investigation, we
are going to generate a set of averages when we roll a single die four times.
x
x1  x2  x3  x4
3156
for example
 3.75
4
4
Each time you roll a single die 4 times, or roll 4 dice simultaneously, you are sampling the mean X and the
sample size is n = 4.
The distribution of an infinite number of sample means is called the sampling distribution of the mean. This
notion leads into the Central Limit Theorem which states that, when a number of successive random samples
is taken from a population, the distribution of sample means (sampling distribution of the mean) approaches
normality as the sample size increases (irrespective of the shape of the population distribution). With this in
mind, confidence intervals for the population mean can then be considered.
This investigation will require students to:
 demonstrate successive sampling of sample size n = 4 from a population
 calculate the mean and standard deviation of the sampling distribution of X
 compare the above statistics to the population mean μ and standard deviation  .
Note:
 The Central Limit Theorem only applies to the mean and not to other statistics.
 All samples must have the same sample size.
 Samples must be random.
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
3
Definitions
Population:
A population is the set of all elements of interest for a particular study. Quantities such as
the population mean μ are known as population parameters.
Sample:
A sample is a subset of the population and is selected to represent the whole population.
Quantities such as the sample mean X are known as sample statistics and are estimates
of the corresponding population parameters.
Random sample: A random sample is a sample in which each member of the population has an equal chance
of being selected. Random samples generate unbiased estimates of the population mean,
whereas non-random samples may not be unbiased. Also, the variability within random
samples can be mathematically predicted.
Activity 1
(5 marks)
Consider the roll of a standard six-face die which, when rolled, will come to rest with a number X = 1, 2, 3, 4, 5,
or 6 on the upper face. Assuming the die is a fair one, we say that X is a random variable. Each time we roll the
die, we are sampling the outcome from an infinite population X. In probability theory, each number on the
face of the die has an equal chance of one in six of occurring.
(a) Complete the probability distribution of X in Table 1 below.
(1 mark)
Table 1: Single role of a die
X
1
2
3
4
5
6
P X  x
(b) Draw the probability distribution associated with Table 1 (single role of the die).
(2 marks)
(c) The probability distribution of X is a uniform distribution.
(2 marks)
Use your calculator to calculate the mean of the population = μ and the standard deviation =  to two
decimal places.
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
4
Activity 2 (sample size is constant at n = 4)
(9 marks)
Take a sample size n = 4 from this population by rolling a single die four times or rolling four dice
simultaneously.
(a)
Record the mean of the four dice for each trial x 
x1  x2  x3  x4
3156
for example
 3.75
4
4
Repeat this six times (i.e. six samples, each of sample size four) for Table 2.
(3 marks)
Table 2: six samples, each of sample size four
(b)
Sample #
1
Mean x
3.75
2
3
4
Sort your values of x into Table 3, below.
5
6
(2 marks)
Table 3
x
frequency
 x
 x
 x
 x
 x
 x
(c)
Plot the frequency distribution of the means of these six samples.
(2 marks)
(d)
Calculate the mean and the standard deviation of these six sample means.
(2 marks)
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
5
Activity 3 (sample size is constant at n = 4)
(9 marks)
(a) Keep the sample size n = 4 from the population by rolling a single die four times, or rolling four dice
simultaneously, and increase the number of samples to 10 and complete Table 3.
(3 marks)
Table 3: 10 samples, each of sample size four
Sample#
1
2
3
4
5
6
7
8
9
10
Mean x
(b) Sort your values of x into Table 4, below.
(2 marks)
Table 4
x
frequency
 x
 x
 x
 x
 x
 x
(c)
Plot the frequency distribution of the means of these 10 samples.
(2 marks)
(d)
Calculate the mean and the standard deviation of these 10 sample means.
(2 marks)
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
6
Activity 4 (sample size is constant at n = 4)
(14 marks)
(a) Keep the sample size n = 4 and increase the number of samples to 40 and complete Table 5.
(3 marks)
(See Appendix 1 on how to use a spreadsheet to simulate the data)
Table 5
Sample #
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
mean
Sample #
mean
Sample #
mean
Sample #
mean
(b) You now have 40 sample means from samples of size n = 4. Sort your values of X into Table 6, below.
(2 marks)
Table 6
x
frequency
 x
 x
 x
 x
 x
 x
(c) Plot the frequency distribution of the means of these 40 samples.
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
(2 marks)
7
(d) Calculate the mean and standard deviation of the distribution of sample means.
(2 marks)
(e) How does the shape of this distribution compare with the sampling distributions when you did six samples
and when you did 10 samples?
(2 marks)
(f)

, where   1.71 is the population standard deviation, with the standard deviation of the
n
samples of size n = 4 .
(3 marks)
Compare
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
8
Appendix 1
To set up the simulation for the roll of a die, we use the Excel spreadsheet.

Choose ‘Formulas’ on the toolbar ‘Calculation options’‘Manual’ which will stop the random
number generator recalculating while you are setting up your table,

Use the random number generator and the integer function to get the formula = INT(RAND()*6)+1.

Copy this formula across four cells in rows A3 to E3 and this will generate four random integers
between one and six, one in each cell.

In cell F3 in this row, calculate the average of the four integers using = AVERAGE(B3:E3).

Now highlight these five cells B3 to E3 and copy them down for 40 rows in Activity 3 as this is the
number of samples required.

Each row represents a sample of size = 4.

Now go to ‘Formulas’ ‘calculate now’ to complete the table.

Copy and sort the mean/average column as this will help you do a frequency count.
1
2
3
4
5
6
7
8
9
A
B
C
D
E
Dice #
Sample
#
1
2
3
4
5
6
7
1
2
3
4
F
mean
.=INT(RAND()*6)+1
5
4
4
5
1
4
6
3
1
1
1
3
1
4
6
6
3
2
1
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
1
4
2
2
5
6
6
=AVERAGE(B3:E3)
4G.75
3.75
3.25
3.5
2.5
3.5
Sorted
list
1.75
2
2.25
2.5
2.5
2.75
2.75
9
Dice #
1
2
3
4
Sample #
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
mean
6
5
4
4
5
1
4
5
4
6
3
5
4
3
3
3
3
1
4
1
6
4
4
6
6
4
4
3
1
1
2
4
3
4
3
2
1
3
3
1
3
6
3
1
1
1
3
4
2
2
5
6
5
5
1
2
3
3
1
5
5
5
1
3
2
4
2
3
2
5
5
6
5
4
3
1
2
5
3
6
1
4
6
6
3
2
1
4
4
4
3
3
2
1
2
3
4
2
3
6
3
6
2
4
1
2
3
5
6
1
5
2
4
2
3
2
3
1
5
4
1
4
2
2
5
6
6
6
4
4
6
2
6
2
5
2
1
2
3
1
3
1
2
4
5
2
4
6
3
6
2
3
2
6
4
6
1
3
2
1
2.75
4.75
3.75
3.25
3.5
2.5
3.5
4.75
3.5
4
4.25
4
4.25
2.75
2.75
2.5
2.75
2
2.75
3.25
4.25
4
2.25
4.25
3.5
3
3.25
4.25
3
3.25
3.5
3.75
3.5
4
3.25
2.75
1.75
3
3.25
3
Sorted
list
1.75
2
2.25
2.5
2.5
2.75
2.75
2.75
2.75
2.75
2.75
3
3
3
3
3.25
3.25
3.25
3.25
3.25
3.25
3.5
3.5
3.5
3.5
3.5
3.5
3.75
3.75
4
4
4
4
4.25
4.25
4.25
4.25
4.25
4.75
4.75
Frequency table
1<=x<2
1
2<=x<3
10
3<=x<4
18
4<=x<5
11
5<=x<6
0
6<=x<7
0
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
10
Solutions and marking key for Investigation 4 – Sampling means
Activity 1
(5 marks)
Consider the roll of a standard six-face die which, when rolled, will come to rest with number
X = 1, 2, 3, 4, 5, or 6 on the upper face. Assuming the die is a fair one, we say that X is a random variable.
Each time we roll the die, we are sampling the outcome from an infinite population X. In probability theory,
each number on the face of the die has an equal chance of one in six of occurring.
(a) Complete the probability distribution of X in Table 1 below:
(1 mark)
Table 1: Single role of a die
X
1
2
1
6
Behaviours
Correctly completes the probability table
P X  x
1
6
3
4
5
6
1
6
1
6
1
6
Marks
1
1
6
Item
simple
(2 marks)
Probability
(b) Draw the probability distribution associated with Table 1 (single role of a die).
Face value of die
Behaviours
Draws the class limits correctly
Draws the frequency values correctly
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
Marks
1
1
Item
simple
simple
11
(c) The probability distribution of X is the uniform distribution.
Use your calculator to calculate the mean of the population = μ and the standard deviation
=  to two decimal places.
(2 marks)

 pxi
 3.5
 pi

 pxi 2
 x
 pi

2
 1.71
Behaviours
Marks
2
Correctly calculates each statistic
Activity 2 (sample size is constant at n = 4)
Item
simple
(9 marks)
Take a sample size n = 4 from this population by rolling a single die four times or rolling four dice
simultaneously.
(a) Record the mean of the four dice for each trial x 
x1  x2  x3  x4
3156
for example
 3.75 .
4
4
Repeat this six times (i.e. six samples, each of sample size four) for Table 2.
(3 marks)
Table 2: six samples, each of sample size four
Dice #
1
2
3
4
Sample #
mean
1
3
1
5
6
3.75
2
5
6
4
4
4.75
3
4
3
3
2
3.00
4
4
3
6
4
4.25
5
5
1
3
5
3.5
6
5.5
6
4
6
6
Behaviours
Shows evidence of how the random numbers were generated
Marks
2
Sample #
1
2
3
4
5
6
Mean x
3.75
4.75
3.00
4.25
3.50
5.50
Behaviours
Completes the table of means
Marks
1
Item
simple
Item
simple
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
12
(b) Sort your values of x into Table 4, below.
(2 marks)
x
frequency
1 x  2
0
2 x 3
0
3 x 4
3
4 x 5
2
5 x 6
1
Behaviours
Calculates the class width correctly
Allocates the frequency values correctly
Marks
1
1
(c) Plot the frequency distribution of the means of these six samples.
Behaviours
Draws the class limits correctly
Draws the frequency values correctly
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
Item
simple
simple
(2 marks)
Marks
1
1
Item
simple
simple
13
(d) Calculate the mean and the standard deviation of these six sample means.
(2 marks)
Mean = 4.125
Standard deviation
=0.9048
Behaviours
Marks
2
Correctly calculates each statistic
Item
simple
Activity 3 (sample size is constant at n = 4)
(9 marks)
(a) Keep the sample size n = 4 from the population by rolling a single die four times, or rolling four dice
simultaneously, and increase the number of samples to 10 and complete Table 3.
(3 marks)
Dice #
Sample #
1
2
3
4
5
6
7
8
9
10
1
2
3
4
mean
6
2
4
4
5
1
4
3
6
1
3
6
3
3
4
1
3
2
2
2
1
2
6
6
3
2
1
2
4
1
2
1
2
2
4
6
3
5
4
1
Behaviours
Shows evidence of how the random numbers were generated
3.00
2.75
3.75
3.75
4.00
2.50
2.75
3.00
4.00
1.25
Item
simple
Marks
2
Table 3: 10 samples, each of sample size four
Sample#
1
2
3
4
5
6
7
8
9
10
Mean x
3.00
2.75
3.75
3.75
4.00
2.50
2.75
3.00
4.00
1.25
Behaviours
Completes the table of means
Marks
1
Item
simple
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
14
(b) Sort your values of x into Table 4, below.
(2 marks)
Table 4
x
frequency
1 x  2
1
2 x 3
3
3 x 4
4
4 x 5
2
5 x 6
0
Behaviours
Calculates the class width correctly
Allocates the frequency values correctly
Marks
1
1
(c) Plot the frequency distribution of the means of these 10 samples.
Behaviours
Draws the class limits correctly
Draws the frequency values correctly
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
Item
simple
simple
(2 marks)
Marks
1
1
Item
simple
simple
15
(d) Calculate the mean and the standard deviation of these 10 sample means.
(2 marks)
Mean = 3.78
Standard deviation = 0.97
Note: The frequency graph using the class
mid-point has been included here as an
alternative to that shown in part (b).
Behaviours
Marks
2
Correctly calculates each statistic
Item
simple
Activity 4 (sample size is constant at n = 4)
(14 marks)
(a) Keep the sample size n = 4 and increase the number of samples to 40 and complete Table 5.
(3 marks)
See Appendix 1 for the simulated data.
Table 5
Sample #
1
2
3
4
5
6
7
8
9
10
2.75
4.75
3.75
3.25
3.5
2.5
3.5
4.75
3.5
4
11
12
13
14
15
16
17
18
19
20
4.25
4
4.25
2.75
2.75
2.5
2.75
2
2.75
3.25
21
22
23
24
25
26
27
28
29
30
4.25
4
2.25
4.25
3.5
3
3.25
4.25
3
3.25
Sample #
31
32
33
34
35
36
37
38
39
40
mean
3.5
3.75
3.5
4
3.25
2.75
1.75
3
3.25
mean
Sample #
mean
Sample #
mean
Behaviours
Shows evidence of how the random numbers were generated
Completes the table of means
Marks
2
1
3
Item
simple
simple
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
16
(b) You now have 40 sample means from samples of size n = 4. Sort your values of x into Table 6, below.
(2 marks)
Table 6
x
frequency
1 x  2
1
2 x 3
10
3 x 4
18
4 x 5
11
5 x 6
0
Behaviours
Defines appropriate class intervals
Assigns correct frequencies
Marks
1
1
(c) Plot the frequency distribution of the means of these 40 samples.
Behaviours
Draws the class limits correctly
Draws the frequency values correctly
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
Item
simple
simple
(2 marks)
Marks
1
1
Item
simple
simple
17
(d) Calculate the mean and standard deviation of the distribution of sample means.
(2 marks)
Mean = 3.475
Standard deviation = 0.800
Note: The frequency graph has
been included here as an
alternative to that shown in part
(b).
Behaviours
Marks
2
Correctly calculates each statistic
Item
simple
(e) How does the shape of this distribution compare with the sampling distributions when you did six samples
and when you did 10 samples?
(2 marks)
The distribution for n = 40 has higher frequencies for the middle values of the mean, giving it a roughly normal
distribution or symmetric shape.
Behaviours
Marks
Item
States that the central values have higher frequencies
1
simple
States the mean is near the centre of the range of values
1
simple
(f)

, where   1.71 is the population standard deviation, with the standard deviation of the
n
samples of size n = 4 .
(3 marks)
Compare
The mean of the sampling distribution, 3.4754 approximates the population mean (   3.5 ) and the standard

, the population standard deviation divided by the square root of the sample
n
 1.71

 0.85 .
size. Hence, sx  0.800 
n
4
deviation that approximates
Behaviours
States the mean of the sample means is close to the population mean
Calculates

n
States s x is approximately equal to

n
Marks
1
Item
simple
1
simple
1
simple
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
18
School name
Mathematics Specialist
Unit 3 and Unit 4
Investigation 4
Validation – Sampling means
Student name: _______________________ Teacher name: _______________________
Class:
________
Time allowed for this task:
50 minutes in class
Materials required:
Standard writing equipment
Calculator (to be provided by the student)
Other materials allowed:
Drawing templates
Marks available:
15 marks
Task weighting:
4% (+ 1% for extended part)
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
19
Question 1
(3 marks)
For this activity, we have set up a spreadsheet to randomly generate the integers one to eight such that each
number has an equally likely chance of coming up. This can generate an infinite set of integers from one to
eight. This set is called the population of the variable X which has a mean  and standard deviation  x .
(a) Complete the probability table below
X
1
(1 mark)
2
3
4
5
6
7
8
P X  x
(b) Hence, calculate the mean and standard deviation for the theoretical population of numbers generated in
this way.
(2 marks)
Question 2
(12 marks)
In this activity, we have sampled the population described above. The results are shown in Appendix 2. The
sample size was n = 10 and we took 40 samples.
(a) You now have 40 sample means from samples of size n = 10. Sort your values of X into the frequency
table below.
(2 marks)
x
frequency
 x
 x
 x
 x
 x
 x
 x
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
20
(b) On the axes below, plot the frequency distribution of the means of these 40 samples.
(3 marks)
(c) Use a grouped data frequency table to calculate the mean and standard deviation for the sample
distribution.
(4 marks)
(d) Compare the mean and standard deviation of the sample distribution to the population mean μ and
standard deviation  .
(3 marks)
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
21
Solutions and marking key for Investigation 4 – Validation – Sampling means
Question 1
(3 marks)
For this next activity, we have set up a spreadsheet to randomly generate the integers one to eight, such
that each number has an equally likely chance of coming up. This can generate an infinite set of integers
from one to eight. This set is called the population of the variable X which has a mean  and  x .
(a) Complete the probability table below.
(1 mark)
Solution
X
1
2
3
4
5
6
7
8
P X  x
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
Behaviours
Correctly completes each the probability values
Marks
1
Item
simple
(b) Hence, calculate the mean and standard deviation for the theoretical population of numbers generated
in this way.
(2 marks)
Solution
Mean = 4.5
Standard deviation = 2.291
Behaviours
Accurately calculates the mean
Accurately calculates the standard deviation
Marks
1
1
Item
simple
simple
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
22
Question 2
(12 marks)
In this activity, we have sampled the population described above. The results are shown in Appendix 2.
The sample size was n = 10 and we took 40 samples.
(a) You now have 40 sample means from samples of size n = 10. Sort your values of X into the frequency
table below.
(2 marks)
Solution
x
frequency
1 x  2
0
2 x 3
0
3 x 4
9
4 x 5
19
5 x 6
12
6 x 7
0
7 x 8
0
Behaviours
Defines appropriate class intervals
Assigns correct frequencies
Marks
1
1
(b) On the axes below, plot the frequency distribution of the means of these 40 samples.
Item
simple
simple
(3 marks)
Solution
Behaviours
Marks the axes correctly
Uses an appropriate frequency scale
Draws an accurate frequency distribution
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
Marks
1
1
1
Item
simple
simple
simple
23
(c) Use a grouped data frequency table to calculate the mean and standard deviation for the sample
distribution.
(4 marks)
Solution
Mean = 4.575
Standard deviation = 0.7299
Note: The frequency
distribution has been
included here as an
alternative to that shown in
part (b).
Behaviours
Sets up the table using the mid-points of the intervals
Uses the correct frequency values in the table
Calculates the mean of the distribution correctly
Calculates the standard deviation of the distribution correctly
Marks
1
1
1
1
Item
simple
simple
simple
simple
(d) Compare the mean and standard deviation of the sample distribution to the population mean μ and
standard deviation  .
(3 marks)
Solution
The mean of the sample means X = 4.575 approximates the population mean  = 4.5 and the standard
deviation s x  0.7299 approximates the population standard deviation divided by the square root of the
2.2913
  
 0.7445 .
 . Since n  10  sx 
10
 n
sample size 
Behaviours
States the mean of the sample means is close to the population mean
Calculates

n
States s x is approximately equal to

n
Marks
1
Item
complex
1
complex
1
complex
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
24
Appendix 2
Random integers I 1, 2, 3, 4, 5,.6, 7, 8
Random
#
Sample
#
1
2
3
4
5
6
7
8
9
10
3
2
7
8
3
3
7
3
3
1
4
Sorted
list
3.2
5
1
7
7
8
6
1
7
6
4
5.2
3.2
8
4
1
5
4
4
2
4
1
3
3.6
3.4
7
1
1
5
3
7
2
6
4
7
4.3
3.6
6
7
3
6
4
3
1
7
8
7
5.2
3.6
8
8
6
1
5
1
1
1
1
6
3.8
3.8
2
1
8
3
2
3
5
4
1
3
3.2
3.8
6
7
4
4
7
7
6
4
5
7
5.7
3.8
8
2
4
4
6
1
5
5
5
6
4.6
3.8
7
8
3
1
1
7
6
8
4
2
4.7
4
7
6
1
6
2
7
1
7
2
6
4.5
4
6
5
3
1
1
8
1
4
4
3
3.6
4
3
6
5
7
8
7
5
3
8
6
5.8
4.1
4
4
3
8
3
6
7
8
4
4
5.1
4.1
4
3
7
8
6
7
2
2
1
4
4.4
4.2
3
3
6
5
5
1
7
2
6
2
4
4.2
1
8
5
4
1
8
3
3
5
8
4.6
4.3
6
3
2
3
4
2
5
6
6
1
3.8
4.3
6
1
4
1
1
8
6
3
1
3
3.4
4.4
5
8
4
8
7
4
5
1
1
5
4.8
4.4
2
1
4
2
2
3
2
8
5
3
3.2
4.5
7
1
3
3
8
8
1
7
7
5
5
4.6
7
5
4
5
6
5
7
8
1
5
5.3
4.6
2
7
6
1
4
7
1
7
4
3
4.2
4.7
4
1
1
1
7
7
6
1
3
7
3.8
4.7
7
8
1
5
4
5
2
7
8
4
5.1
4.8
1
7
3
8
4
7
4
6
5
2
4.7
4.8
6
4
6
4
4
6
4
1
2
4
4.1
4.9
2
2
3
2
5
6
1
6
7
6
4
5
4
4
4
6
6
8
3
8
3
8
5.4
5
1
3
6
6
7
5
2
5
2
5
4.2
5.1
8
5
6
6
5
7
6
1
4
2
5
5.1
1
4
6
8
6
6
7
5
2
6
5.1
5.1
8
3
1
2
8
4
4
3
8
7
4.8
5.2
6
3
4
6
6
7
3
7
6
4
5.2
5.2
6
5
3
8
6
3
1
4
2
6
4.4
5.2
8
5
2
1
1
5
2
8
1
8
4.1
5.3
1
4
7
3
2
6
4
7
4
5
4.3
5.4
7
7
2
6
7
5
3
2
2
8
4.9
5.7
4
3
5
4
1
7
4
5
2
3
3.8
5.8
Mean
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
25
School name
Mathematics Specialist
Unit 3
Test 1
Student name: _______________________ Teacher name: _______________________
Class:
________
Time allowed for this task:
55 minutes, in class, under test conditions
Section One – calculator-free section – 35 minutes
Section Two – calculator-assumed section – 20 minutes
(30 marks)
(20 marks)
Materials required:
Calculator with CAS capability (to be provided by the student)
Standard items:
Pens (blue/black preferred), pencils (including coloured), sharpener,
correction fluid/tape, eraser, ruler, highlights
Special items:
Drawing instruments, templates, notes on two unfolded sheets of
A4 paper, and up to three calculators approved for use in
WACE examinations
Marks available:
50 marks
Task weighting:
5%
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
26
Section One – calculator-free section
Question 1
(3.1.6, 3.1.15)
(30 marks)
(8 marks)
(a) Given z  3  i evaluate z6 giving the answer in Cartesian form.
(2 marks)
(b)
(4 marks)


Given Z1  cis   and Z2  cis   evaluate the following in exact Cartesian form:
3
4
  
(i)
Z1
(ii )
iZ2
(iii )
cis  
 12 
(c)
Solve x 2  6 x  13  0 for x  Im in exact form.
Question 2
(3.1.10)
(a) Sketch the set of points defined by z  2  3i   13 .
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
(2 marks)
(6 marks)
27
Question 3
(3.1.11, 3.1.12)
(8 marks)
Determine and locate all solutions in the Argand plane to the equation z 5  1 .
Question 4
(3.1.13, 3.1.15)
(8 marks)
Given H  z   z5  2z 4  5z3  10z2  4z  8
(a)
Evaluate H  i  , H  i  and H  2 .
(3 marks)
(b)
Hence, find all roots of the equation z5  2z 4  5z3  10z2  4z  8  0 .
(5 marks)
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
28
Section Two – calculator-assumed section
Question 5
(20 marks)
(3.1.7)
(10 marks)
(a)
Expand and simplify the expression F      cos   i sin   .
(2 marks)
(b)
Hence, express the Re  F  in terms of cos .
(3 marks)
(c)
Use Re  F  to solve the equation 16 x 5 -20x 3  5x -1  0 and express the solutions in trigonometric
5
form.
Question 6
(5 marks)
(3.1.7)
(10 marks)
Given z  cos   i sin  :
(a)
1

z z 
 in trigonometric form.
Express 
1

iz  
z

(b)
Show z 2 +
1
=2 cos2 and hence prove cos2  2cos2   1 .
z2
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
(4 marks)
(6 marks)
29
Solutions and marking key for Test 1 for concurrent Unit 3 and Unit 4 program
Section One – calculator-free section
Question 1
(a)
(30 marks)
(3.1.6, 3.1.15)
(8 marks)
Given z  3  i evaluate z6 giving the answer in Cartesian form.
(2 marks)
Given z  3  i evaluate z 6
z6 

3 i

6
6
 3  6.
 3  .i 
5
1
 15.

 3  . i 
4
2
 3  . i 
3 i
3
 20.
 27  135  45  1  54 3  60 3  6
3
 15.
 3  . i 
2
4
 6.
 3  . i 
1
5
 i 
6
 64
OR

z  2cis    z 6  64cis     64
6
Specific behaviours
Mark
Item
1
1
simple
simple
1
1
simple
simple
6
Expands the Cartesian form of z
Simplifies correctly
Or
Expresses z6 in polar form
Expresses the answer in Cartesian form
(b)
(4 marks)


Given Z1  cis   and Z2  cis   evaluate the following in exact Cartesian form:
3
4
 
(i)
Z1
(ii )
iZ2
(iii )
cis  
 12 


Given Z1  cis   and Z 2  cis   evaluate the following in exact Cartesian form:
3
4
1  3i
2
(i)
Z1 
(iii )

1  3i
2
   Z
cis    1 


2
2  2i 
 12  Z2

(ii )

Writes the Cartesian form of iZ 2 correctly
Expresses polar term for Z 3 in Cartesian form
Simplifies the Cartesian form correctly
 
 2  2i
2

6 2 i


4

2 6 
Specific behaviours
Writes the Cartesian form of Z1 correctly
iZ2 
Mark
Item
1
1
simple
simple
1
1
complex
complex
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
30
Solve x 2  6 x  13  0 for x  Im in exact form.
(c)
(2 marks)
Or
Solve
Solve

x 2  6 x  12  0
a  1, b  6 and c  12

x
x 2  6 x  12  0

x 2  6 x  9  3


 x  3  3
2
 x  3  3i 2

x 3 3i
2

Specific behaviours
Completes the square correctly
Solves the equation using the exact form
Or
Uses the quadratic formula
Simplifies the expressions to the correct exact form
6  36  48
2
6  12
x
 3  3i
2
Mark
Item
1
1
simple
simple
1
1
simple
simple
Question 2 (1.1.7)
(a)
(6 marks)
Sketch the set of points defined by z  2  3i   13 .
Specific behaviours
Draws a circle
Has the correct centre (2 + 3i)
Has the correct radius
Circumference passes through (0, 0) (4 + 0i) and (0 + 6i)
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
Mark
Item
1
1
1
3
simple
simple
simple
simple
31
Question 3
(3.1.11, 3.1.12)
Determine and locate all solutions in the Argand plane to the equation z 5  1 .
(8 marks)
2 0 

z0  1  cis  0 
 1  0i
5 

2 1 

 2 
z1  1  cis  0 
 cis  

5 

 5 
2 2 

 4 
z2  1  cis  0 
 cis 


5 

 5 
2 3 

 6 
z3  1  cis  0 
 cis  

5 

 5 
2 4 

 8 
z4  1  cis  0 
  cis  5 
5


 
Specific behaviours
Expresses the five solutions correctly
Locates the solutions accurately on a polar graph
Mark
5
3
Question 4
(3.1.13, 3.1.15)
5
Given H  z   z  2z 4  5z3  10z2  4z  8 :
(a)
Item
simple
simple
(8 marks)
Evaluate H  i  , H  i  and H  2
(3 marks)
Given H  z   z5  2z 4  5z 3  10z 2  4z  8
(a)
Evaluate H  i  , H  i  and H  2
H  i   i  2  5i  10  4i  8  0
H  i   i  2  5i  10  4i  8  0
H  2  32  32  40  40  8  8  0
Specific behaviours
Evaluates each of the three terms H  i  , H  i  and H  2
Mark
3
Item
simple
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
32
(b)
Hence, find all roots of the equation z5  2z 4  5z3  10z2  4z  8  0.
(5 marks)
Given H  z   z 5  2z 4  5z 3  10z 2  4z  8 from part (a)
H i   0
  z  i  is a factor of H  z 
H  i   0
  z  i  is a factor of H  z 
And
 z 2  1 is a factor of H  z 
H  2  0
  z  2 is a factor of H  z 
Since
and
then



z5  2z 4  5z 3  10z 2  4z  8  z 2  1
z
5
3

 2z 2  4z  8   z  2
4
3
  z
 z
3
 2z 2  4z  8
2
4


  z  i  z  i  z  2 z  2i  z  2i 
2
z  2z  5z  10z  4z  8
Hence the roots to z  2z  5z  10z  4z  8  0 are z  i , 2i ,2
5
4
3
2
Specific behaviours
Mark
Item
Uses the factor theorem to give factors  z  i  z  i  z  2
1
simple
Determines the remaining factors  z  2i  z  2i 
2
complex
Correctly writes all the roots
2
complex
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
33
Section Two – calculator-assumed section
Question 5
(a)
(20 marks)
(3.1.7)
(10 marks)
Expand and simplify the expression F      cos   i sin   .
5
F  
(2 marks)
  cos   i sin  
5
 (cos5   10cos3 .sin2   5cos .sin 4 )  (sin5   5cos4 .sin   10cos2 .sin3 )i
Specific behaviours
Shows the real and imaginary terms correctly
(b)
Mark
2
Hence, express the Re  F  in terms of cos .
Re  F   cos   10cos .sin   5cos .sin 
5
3
2

(3 marks)
4



  5cos  1  2cos   cos  
 cos5   10cos3  1  cos2   5cos  1  cos2 
 cos5   10cos3   10cos5
Item
simple
2
2
4
 16cos5   20cos3   5cos 
Specific behaviours
Writes the real part of F   
2
2
Substitutes for sin   1  cos 
Gives the correct expression for Re  F 
Mark
1
Item
simple
1
simple
1
simple
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
34
(c)
Use Re  F  to solve the equation 16x5  20x3  5x  1  0 and express the solutions in trigonometric
form.
F  
(5 marks)
  cos   i sin  
5
 cos5  i sin5
DeMoivre
 Re  F   cos5
Re  F   16cos5   20cos3   5cos  from part (b)
Hence cos5  16cos5   20cos3   5cos .......... 1
To solve
16 x 5  20x 3  5x  1  0

16 x 5  20x 3  5x  1
5
Let x  cos 
3

16cos   20cos   5cos   1


cos5  1
5  0,2 ,...



2n
, n  0,1,2,3,4
5
 2n 
x  cos 
 , n  0,1,2,3,4
 5 
Specific behaviours
Mark
Item
Uses De Moivre to state Re  F   cos5
1
complex
Makes the substitution x  cos  in polynomial
Replaces the polynomial in cos with cos5
Solves cos5  1 in terms of 
Gives all five solutions in terms of x
1
1
1
1
complex
complex
complex
complex
Question 6
(3.1.7)
Given z  cos   i sin  :
1

z z 
 in trigonometric form.
(a)
Express 
1

iz  
z

1

z z 


1

iz  
z

=
(10 marks)
(4 marks)
 cos   i sin     cos   i sin  
i   cos   i sin     cos   i sin   
2i sin 
2i cos 
 tan 

Specific behaviours
Mark
Item
1
in trig form
z
2
simple
2
simple
Rewrites the complex numbers z and
Simplifies both numerator and denominator
Writes the correct final term
Sample assessment tasks | Mathematics Specialist | ATAR Year 12
35
(b) Show
z2 +
1
= 2 cos2 and hence prove cos2  2cos2   1
z2
(6 marks)
1
  cos2  i sin2    cos2  i sin2 
z2
 2 cos2.....................................(1)
1
2
2
z 2 + 2   cos   i sin     cos   i sin  
z
z2 +

 
 cos2   2i sin  cos   sin2   cos2   2i sin  cos   s in 2 



 2 cos2   sin2  ...................(2)

(1)  (2)  2cos2  2 cos2   sin2 

 cos2  2cos2   1
Specific behaviours
1
using double angle form 2
z2
Gathers terms and simplifies
1
Rewrites z 2 and 2 using single angle form 
z
Gathers terms and simplifies
Equates both equations
Writes correct final expression
Rewrites z 2 and
Question
Simple
Complex
1
8
0
8
2
6
0
6
3
8
0
8
4
4
4
8
5
5
5
10
Mark
Item
1
1
complex
complex
1
complex
1
1
1
complex
complex
complex
6
4
6
10
Total
35
15
50
Sample assessment tasks | Mathematics Specialist | ATAR Year 12