SAMPLE ASSESSMENT TASKS
MATHEMATICS APPLICATIONS
ATAR YEAR 12
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2015/25819v4
1
Sample assessment task
Mathematics Applications – ATAR Year 12
Test 3 – Unit 3
Assessment type: Response
Conditions:
Time for the task: Up to 50 minutes, in class, under test conditions
Materials required:
Section One: Calculator-free
Section Two: Calculator-assumed
Standard writing equipment
Calculator (to be provided by the student)
Other materials allowed:
Drawing templates, one page of notes in Section Two
Marks available:
Section One: Calculator-free
Section Two: Calculator-assumed
52 marks
(28 marks)
(24 marks)
Task weighting:
8% for the pair of units
_________________________________________________________________________________
Sample assessment tasks | Mathematics Applications | ATAR Year 12
2
Section One: Calculator-free
(28 marks)
Suggested time: 20 minutes
Question 1 (3.3.1)
(4 marks)
(a) Define the set of vertices and a list of edges for Graph 1.
(b) Is Graph 1 planar? Explain
(2 marks)
(2 marks)
Question 2 (3.3.2)
(9 marks)
A Friday night darts competition has five competitors: Alex (A), Bob (B), Colin (C), Dave (D) and
Evan (E), who are due to play each other in a round-robin competition over a number of weeks.
On the first Friday, rounds 1 and 2 were played where:
Round 1 – Alex played Bob; Colin played Dave; and Evan had a bye
Round 2 – Alex played Evan; Colin played Bob; and Dave had a bye.
(a) Draw a graph to represent the games played on the first Friday night.
(b) Draw the complement to the graph in (a) and explain what it represents.
(c) Draw a graph to represent all the games to be played in the whole competition.
Sample assessment tasks | Mathematics Applications | ATAR Year 12
(3 marks)
(3 marks)
(3 marks)
3
Question 3 (3.2.9)
(6 marks)
The set of numbers 1, 3, 6, 10 is named triangular because the units (objects) can be arranged to
form triangles, as in the diagram below.
(a) Draw the next pattern in the diagram below and give its value.
(2 marks)
(b) Given T10=55 evaluate T11.
(1 mark)
(c) Give the first order recurrence for the sequence.
(3 marks)
Question 4 (3.2.4, 3.2.3)
(9 marks)
A wedding photographer is quoting the following price for producing a wedding album for the
newlyweds:
A fixed minimum cost of $150, with 80 photos in a hard-backed album. Further photos may also be
added in lots of 10 photos at $0.70 per photo, up to a maximum of 200 photos.
He wants to set up a table below, showing:

the type of album where T1 is the basic album, with 80 photos at a cost of $150

the number of photos in each of the possible album sizes

the cost in dollars of each of the different albums.
(a) Complete each of the blank cells of the table.
Type
Number of
pictures
$ cost of
album
T1
80
T2
T3
(3 marks)
T4
T5
Tn
200
$150
Sample assessment tasks | Mathematics Applications | ATAR Year 12
4
(b) Write a rule that will calculate the number of pictures in album type = Tn.
(3 marks)
(c) Write a rule that will calculate the cost of album type = Cn.
(3 marks)
Sample assessment tasks | Mathematics Applications | ATAR Year 12
5
Section Two: Calculator-assumed
(24 marks)
Suggested time: 30 minutes
Question 5 (3.2.9)
(3 marks)
(a) Complete the table below for the first five terms of the sequence, defined by
an1  an   n  1 , a1  1 .
n
1
(2 marks)
2
3
4
5
an
(b) Evaluate a50 .
(1 mark)
Question 6 (3.2.10, 3.2.11)
(12 marks)
A fish farm operates a fish breeding pond in which the population of a particular fish increases by 3%
per month.
(a) Give the recurrence formula to calculate the fish population at the end of each month, assuming
the rate does not change and the initial population is 1000 fish.
(2 marks)
(b) Calculate the population numbers at the end of the first six months of its operation, given the
initial population is 1000 fish.
(2 marks)
Sample assessment tasks | Mathematics Applications | ATAR Year 12
6
(c) After the initial six months, 40 fish per month are removed at the end of each month. Assuming
the population growth is maintained at 3%, how many fish are expected to be in the tank at the
end of 12 months?
(3 marks)
(d) Describe what is happening to the population.
(1 mark)
(e) Estimate, to the nearest whole number, the maximum number of fish that may be removed
from the tank per month without the numbers of fish decreasing.
(4 marks)
Question 7 (3.2.7)
(9 marks)
For the recurrence relation an 1  an  0.6 and a0  3.1
(a) Deduce the rule for the nth term of the relation.
(3 marks)
(b) Check the truth of the following proposition a10  2  a9 - a8 .
(2 marks)
(c) Prove the above proposition can be generalised to an2  2  an1- an .
(4 marks)
Sample assessment tasks | Mathematics Applications | ATAR Year 12
7
Marking key for Test 3 – Unit 3
Section One: Calculator-free
(28 marks)
Question 1
(a) Define the set of vertices and a list of edges for Graph 1.
Solution
 
V G1   A, B, C , D, E, F  and E  G1   AB, AD, AF , BF , BC , CD, DE , EF
Behaviours
Defines the vertices of Graph 1
Defines the edges of Graph 1
Marks
1
1
Item*
(S/C)
simple
simple
(b) Is Graph 1 planar? Explain
Solution
Yes the graph is planar since it can be drawn such that none of the edges intersect other than at the vertices.
Behaviours
States that Graph 1 is planar
States that Graph 1 can be drawn where no edges intersect.
Marks
1
1
Item*
(S/C)
complex
complex
Sample assessment tasks | Mathematics Applications | ATAR Year 12
8
Question 2
A Friday night darts competition has five competitors: Alex (A), Bob (B), Colin (C), Dave (D) and
Evan (E) who are due to play each other in a round-robin competition over a number of weeks.
On the first Friday, rounds 1 and 2 were played where:
Round 1 – Alex played Bob; Colin played Dave; and Evan had a bye
Round 2 – Alex played Evan; Colin played Bob; and Dave had a bye.
(a) Draw a graph to represent the games played on the first Friday night.
Solution
Behaviours
Draws and labels all five vertices
Draws only the four correct edges required
Marks
1
2
Rating
simple
simple
Marks
1
1
1
Rating
simple
simple
simple
(b) Draw the complement to the graph in (a) and explain what it represents.
Solution
a)
These represent the games still to be played.
Behaviours
Includes all five vertices in the graph
Draws only the six correct edges required
Gives the correct interpretation of the complement
Sample assessment tasks | Mathematics Applications | ATAR Year 12
9
(c) Draw a graph to represent all the games that will be played in the whole competition.
Solution
Behaviours
Marks
Includes all five vertices in the graph
Draws all 10 correct edges
1
2
Item*
(S/C)
simple
simple
Question 3
The set of numbers 1, 3, 6, 10 is named triangular because the units (objects) can be arranged to
form triangles, as in the diagram below.
(a) Draw the next pattern in the diagram above and give its value.
Solution
Behaviours
Marks
Draws the correct pattern
Correctly counts the units of the fifth term
1
1
Item*
(S/C)
simple
simple
(b) Given T10=55 evaluate T11.
Solution
T10  55  T11  55  11  66
Behaviours
Calculates the correct term T11  value
Marks
Item*
(S/C)
1
simple
Sample assessment tasks | Mathematics Applications | ATAR Year 12
10
(c) Give the first order recurrence for the sequence.
Solution
T1  1
T2  1  2  3
 T2  T1  2
T3  3  3  6
 T3  T2  3
T4  6  4  10  T4  T3  4
 Tn  Tn 1  n, given T1  1
Behaviours
Marks
1
1
1
States correctly the first term T1
Shows the recursive operation was addition
Indicates the term to be added equals the term number n
Item*
(S/C)
simple
complex
complex
Question 4
The photographer wants to set up a table below showing:

the type of album where T1 is the basic album with 80 photos at a cost of $150

the number of photos in each of the possible album sizes

the cost in dollars of each of the different albums.
(a) Complete each of the blank cells of the table.
Solution
Type
Number of
pictures
$ cost of
album
T1
T2
T3
T4
T5
Tn
80
90
100
110
120
200
$150
157
164
171
178
234
Behaviours
Completes the numbers of pictures in each cell correctly
Completes the cost of each type correctly
Completes Tn correctly
Sample assessment tasks | Mathematics Applications | ATAR Year 12
Marks
1
1
1
Item*
(S/C)
complex
complex
complex
11
(b) Write a rule that will calculate the number of pictures in album type = Tn.
Solution
Tn  a   n  1 d
given a  80 and T2  90
 T2  80  d  90
 d  10
 Tn  80   n  110
 Tn  70  10n
Behaviours
Marks
Applies the linear rule with correct common difference d
Calculates a from the table of values
States the rule in terms of n
1
1
1
Item*
(S/C)
simple
simple
simple
(c) Write a rule that will calculate the cost of album type = Cn.
Solution
Cn  a   n  1 d
given a  150 and C 2  157
 C2  150  d  157
d 7
 Cn  150   n  1 7
 Cn  143  7 n
Behaviours
Applies the linear rule with correct common difference d
Calculates a from the table of values
States the rule in terms of n
Marks
1
1
1
Item*
(S/C)
simple
simple
simple
Sample assessment tasks | Mathematics Applications | ATAR Year 12
12
Section Two: Calculator-assumed section
(24 marks)
Question 5
(a) Complete the table below for the first five terms of the sequence, defined by
an1  an   n  1 , a1  1 .
Solution
n
an
1
1
2
3
3
6
Behaviours
4
10
5
15
Marks
Calculates at least two terms correctly
Calculates all terms correctly
1
1
Item*
(S/C)
simple
simple
(b) Evaluate a50 .
Solution
a50  1275
Behaviours
Evaluates a50  1275
Sample assessment tasks | Mathematics Applications | ATAR Year 12
Marks
1
Item*
(S/C)
simple
13
Question 6
A fish farm operates a fish breeding pond in which the population of a particular fish increases by 3%
per month.
(a) Give the recurrence formula to calculate the fish population at the end of each month, assuming
the rate does not change and the initial population is 1000 fish.
Solution
Pn 1  1.03  Pn where P0  1000
Behaviours
Marks
Defines the recurrence relationship
1
1
States P0  1000
Item*
(S/C)
simple
simple
(b) Calculate the population numbers at the end of the first six months of its operation, given the
initial population is 1000 fish.
Solution
n
Pn
0
1000
1
1030
2
1060
Behaviours
Starts the sequence at P1 = 1030
Completes the sequence at P6=1194
3
1092
4
1125
5
1159
Marks
1
1
6
1194
Item*
(S/C)
simple
simple
Sample assessment tasks | Mathematics Applications | ATAR Year 12
14
(c) After the initial six months, 40 fish per month are removed at the end of each month. Assuming
the population growth is maintained at 3%, how many fish are expected to be in the tank at the
end of 12 months?
Solution
Given Qn  Q0  1.03  40, and  Q0  1194
 Q6  1167
Behaviours
Marks
States the new recurrence relation
States the initial condition Q0
Calculates the correct population
1
1
1
Item*
(S/C)
complex
complex
complex
(d) Describe what is happening to the population.
Solution
The population of fish is declining slowly.
Behaviours
States the correct change in population
Sample assessment tasks | Mathematics Applications | ATAR Year 12
Marks
1
Item*
(S/C)
simple
15
(e) Estimate, to the nearest whole number, the maximum number of fish that may be removed
from the tank per month without the numbers of fish decreasing.
Solution
Given Qn  Q0  1.03  k , where k is the number of fish taken from the pond
Q0  1194 and k  36
Or
Q0  1194 and k  35
Q5  Q0  1.03  36 ,
Q5  Q0  1.03  35 ,
 Q5  1193
 Q5  1198
 Q is decreasing
 Q is increasing
Hence
maximum
number
to be per
taken
perwithout
month depleting the population.
Hence kk=3535is is
thethe
maximum
number
of fishoftofish
be taken
month
without depleting the population.
Marks
Item*
(S/C)
Uses recurrence formula Qn  Q0  1.03  k to check values of k
1
complex
Shows that k  36 causes a decrease in Q
Shows that k  35 causes an increase in Q
States k=35 is the maximum value for which Q increases
1
1
1
complex
complex
complex
Behaviours
Sample assessment tasks | Mathematics Applications | ATAR Year 12
16
Question 7
For the recurrence relation an1  an  0.6 and a0  31
.
(a) Deduce the rule for the nth term of the relation.
Solution
 Tn  a   n  1 d
an 1  an  0.6 and a0  3.1
 Tn  3.1  n  1 0.6
Tn  0.6n  2.5
Marks
Item*
(S/C)
Uses a linear relation such as Tn  a   n  1 d
1
simple
Defines a and d correctly
Simplifies the expression
1
1
simple
simple
Behaviours
(b) Check the truth of the following proposition a10  2  a9 - a8 .
Solution
a10  2  a9 - a8  8.5  2  7.9- 7.3
 8.5  15.8- 7.3 True
Behaviours
Marks
Substitutes the correct values into the equation
Demonstrates the LHS of the equation = RHS
1
1
Item*
(S/C)
simple
simple
(c) Prove the above proposition can be generalised to an  2  2  an 1 - an .
Solution
a

n 1
 a  0.60  ........ 1
n
a
a
0.6...... 2  subtracting equations
n  2 n 1
a
a
a
a ......... 2 1 rearranging
n 2 n 1 n 1 n

a
n2
 2 a
n 1
- a .....,  3 
n
Behaviours
Sets up equations (1) and (2)
Eliminates the constant by subtraction
Rearranges the equation to give the result
Sample assessment tasks | Mathematics Applications | ATAR Year 12
Marks
2
1
1
Item*
(S/C)
complex
complex
complex
17
Sample assessment task
Mathematics Applications – ATAR Year 12
Investigation 1 – Unit 3
Assessment type: Investigation
This investigation reviews the statistical investigation process: identify a problem; pose a statistical
question; collect or obtain data; analyse data; interpret and communicate results.
Notes for teachers
 It is assumed the class has the appropriate access to technology for data handling, such as CAS
calculators or spreadsheets.

The teacher may decide the best type of technology suitable or available for the purpose of the
investigation, and modify the scope of the questions to suit.

This investigation is set to coordinate with the teaching of the Topic 3.1 Bivariate data analysis.

Students should have been taught the concepts and skills required for each section of the
Investigation, especially the use of appropriate software.
Conditions:
Time for the task: to take home/or in class over one week
Materials required:
Internet access
Calculator and/or computer with appropriate software
Other materials allowed:
Tape measures
Marks available:
23 marks
Task weighting:
1% (+ 4% for validation) for the pair of units
__________________________________________________________________________
Sample assessment tasks | Mathematics Applications | ATAR Year 12
18
INTRODUCTION
CensusAtSchool was a nation-wide project that collected real data from students by way of an online
questionnaire. It provided a snapshot of the characteristics, attitudes and opinions of those students
who completed questionnaires from 2006 to 2014. It can be located online:
http://www.abs.gov.au/websitedbs/CaSHome.nsf/Home/Home
While the database is very large, you can sample the results of the survey to allow you to answer
statistical questions posed.
For example, archaeologists/forensic experts often unearth skeletons which are not complete. The
proposition is that we can predict the height of a skeleton using the arm span.
Task 1
(3 marks)
(a) Go to CensusAtSchool, using the link above, and download a random sample with n~50 of the
complete list of variables for the 2014 Questionnaire. This is the database you will use when
completing the various tasks and questions.
(b) State the survey questions (variables) to help you consider the proposition: It is possible to use
the length of an arm span of a body to estimate the height of that individual.
(c) State the explanatory variable and the response variable.
Task 2
(9 marks)
(a) Document how you will go about selecting and presenting the data from your downloaded
database.
(b) Calculate the statistical measures you will use to decide whether the proposition is reasonable
or not.
Task 3
(5 marks)
Use the statistical results from Task 2 to decide whether the proposition is reasonable or not and, if
so, how you would use these measures.
Task 4
(6 marks)
Test your results from Task 2 by collecting data from your fellow students and demonstrate how you
would use the second data set to support or challenge your answer in Task 3.
Sample assessment tasks | Mathematics Applications | ATAR Year 12
19
Marking key for Investigation 1 – Unit 3
Task 1
(a) Go to CensusAtSchool, using the link above, and download a random sample with n=50 of the
complete list of variables for the 2014 Questionnaire. This is the database you will use when
completing the various tasks and questions.
(b) State the survey questions (variables) to help you consider the proposition: It is possible to use
the length of an arm span of a body to estimate the height of that individual.
(c) State the explanatory variable and the response variable.
Solution
(a)
See Appendix 1 for the full list of questions
(b)
Question 9 – How tall are you without your shoes on? (nearest cm)
Question 11 – What is your arm span? (nearest cm)
(c)
Arm span is the explanatory variable and Height is the response variable.
Behaviours
Supplies the list of questions used in the 2014 survey
Chooses the required Questions 9 and 11 from the survey
Indicates Arm span as the explanatory variable and Height as the response
variable
1
1
Item*
(S/C)
simple
simple
1
simple
Marks
Task 2
(a) Document how you will go about selecting and presenting the data from your downloaded
database.
Step 1
Solution
Copy bivariate data for Height and Arm span with sample size n~ 50 into a separate
spreadsheet. (See Appendix 2)
Step 2: Use the software to draw a scatter graph with Height and Arm span.
Step 3: Delete any outliers using the scatter graph to choose them n~ 47.
Behaviours
Selects the two variables to set up a scatter plot
Draws the scatter plot with the appropriate software
Deletes appropriate outliers from scatter plot
Marks
1
1
1
Item*
(S/C)
simple
simple
simple
Sample assessment tasks | Mathematics Applications | ATAR Year 12
20
(b) Calculate the statistical measures you will use to decide whether the proposition is reasonable
or not.
Solution
Step 4: Calculate the linear regression equation with edited graph y = 0.8171x+31.955
(See Appendix 3)
Step 5: Calculate the correlation coefficient (r=0.896) and the coefficient of determination (r2=0.801)
(See Appendix 3)
Step 6: Plot a graph of residuals to check the appropriateness of the linear model
(See Appendix 4)
Marks
Item*
(S/C)
Accurately calculates the linear regression line y = 0.862 x+ 23.6
2
simple
Accurately calculates correlation coefficient (r) and the coefficient of
determination (r2)
Calculates the residuals of the data points
Refers to the random pattern of residual plot
2
1
1
simple
simple
simple
Behaviours
Task 3
Use the statistical results from Task 2 to decide whether the proposition is reasonable or not and, if
so, how you would use these measures.
Solution
r=0.896 indicates a strong positive relationship
Hence, any increase in Arm span indicates an increase in Height.
The residual plot indicates that the linear regression fit is appropriate. (See Appendix 4)
r2=0.802 also suggests that the regression line y = 0.8171x+31.955 is a good predictor of Height given
Arm span.
Therefore, the proposition is reasonable.
We can use the equation y = 0.8171x+31.955 to predict height (y) given an arm span length (x)
e.g. x=145 then y = 0.8171×145+31.955 ~ 150
Behaviours
Uses the correlation coefficient to indicate a strong positive relationship
Uses the residual to check the appropriateness of the linear model
Comments the coefficient of determination indicates the regression line is a
good predictor
States the proposition is reasonable
Gives a worked example
Sample assessment tasks | Mathematics Applications | ATAR Year 12
1
1
1
Item*
(S/C)
simple
simple
simple
1
1
simple
simple
Marks
21
Task 4
Test your results from Task 2 by collecting data from your fellow students and demonstrate how you
would use the second data set to support or challenge your answer in Task 3.
Solution
Residual
Q11. Arm span
145
160
129
157
187
172
160
162
15
10
5
0
-5 0
-10
-15
-20
Q09. Height
153
162
147
146
187
170
166
165
2
4
Residual
3
-1
10
-14
2
-3
3
1
X
1
2
3
4
5
6
7
8
y-fitted
150
163
137
160
185
173
163
164
6
8
10
X
The pattern of residuals suggests the linear model is appropriate.
The measurements of the eight students indicate that the fitted value is often reasonably close to the actual
measured value. The two outliers may be due to a mistake in measuring?
(Note – these measurements were taken from the sampler on a different occasion.)
Item*
Behaviours
Marks
(S/C)
Sets up a table for a separate group for Arm span versus Height
1
simple
Calculates the Fitted height of these individuals
1
simple
Calculates the residuals of the data points
1
simple
Refers to the random pattern of residual plot
1
simple
Compares the Actual height to Fitted height
1
simple
States that there is a reasonably good match
1
simple
Sample assessment tasks | Mathematics Applications | ATAR Year 12
22
Sample assessment task
Mathematics Applications – ATAR Year 12
Investigation validation 1 – Unit 3
Assessment type: In-class Investigation
This investigation reviews the statistical investigation process: identify a problem; pose a statistical
question; collect or obtain data; analyse data; interpret and communicate results.
Notes for teachers:
 It is assumed your class has the appropriate access to technology for data handling.
 This in-class investigation reflects the work carried out in the extended part.
Conditions:
Time for the task: Up to 50 minutes, in class, under test conditions
Materials required:
Standard writing equipment
Calculator (to be provided by the student)
Other materials allowed:
Drawing templates
Marks available:
18 marks
Task weighting:
4% (+1% for extended section)
__________________________________________________________________________________
Question 1
(8 marks)
The table below shows the hours spent by a sample of 20 high school students watching television
and doing homework in a week.
Hours watching TV
4
5
5
0
9
6
20
4
Hours of homework
10
5
1
2
3
7
9
4
Hours watching TV
6
2
40
5
1
8
17
2
Hours of homework
4
1
1
1
1
10
20
1
Hours watching TV
13
1
14
14
Hours of homework
14
6
0
9
Some people feel that TV is a distraction to young high school students; in fact, they would claim,
‘the more hours spent by a student watching television, the less hours the student will spend doing
homework’.
Complete the following questions and use these data to comment on the above statement.
(a) Decide which is the explanatory variable and which is the response variable in the above
comment.
(2 marks)
Sample assessment tasks | Mathematics Applications | ATAR Year 12
23
(b) Calculate the statistical measures you will use to decide whether the proposition is reasonable
or not.
(2 marks)
(c) Use your measures from (b) to comment on the proposition, ‘the more hours spent by a student
watching television, the less hours the student will spend doing homework’.
(4 marks)
Question 2
(10 marks)
The diagram below is the scatter plot for the sample in the table above.
25
20
15
10
5
0
0
10
20
30
40
(a) Label the axes and draw the regression line from 1(b) on the graph.
50
(3 marks)
(b) Calculate the equation of the linear regression line, using the variables defined here:
(4 marks)
Hours watching TV = t and Hours of Homework  w
(c) Using the line of best fit in (b), calculate the residual for the student who watched 17 hours of
TV and spent 20 hours doing homework.
(3 marks)
Sample assessment tasks | Mathematics Applications | ATAR Year 12
24
Marking key for Investigation validation 1 – Unit 3
Question 1
The table below shows the hours spent by a sample of 20 high school students watching television
and doing homework in a week.
(a) Decide which is the explanatory variable and which is the response variable in the above
comment.
Solution
Hours watching TV = Explanatory variable
Hours doing homework = Response variable
(Note – the variables must be expressed in hours)
Behaviours
Marks
Defines the explanatory variable, Watching TV, in hours
Defines the response variable, Homework, in hours
1
1
Item*
(S/C)
simple
simple
(b) Calculate the statistical measures you will use to decide whether the proposition is reasonable
or not.
Solution
Behaviours
Calculates r the correlation coefficient of linear regression
Calculates r2 the coefficient of determination
Sample assessment tasks | Mathematics Applications | ATAR Year 12
Marks
1
1
Item*
(S/C)
simple
simple
25
(c) Use your measures from (b) to comment on the proposition, ‘the more hours spent by a student
watching television, the less hours the student will spend doing homework’.
Solution
Since r=0.1984, the relationship is weak and positive
r2 =0.0394 indicates that the Hours of television watched has a small effect on the Hours of homework done
The proposition suggests a negative relationship between Hours of TV versus Hours of homework,
but the sample here is neither negative nor strong, so does not support the proposition.
Item*
Behaviours
Marks
(S/C)
States the relationship is weak and positive
2
complex
States that the coefficient of determination indicates that TV watched has only
a small effect on homework done
1
complex
Explains that the proposition suggests a negative relationship
1
complex
States that the sample does not support the proposition
1
complex
Question 2
The diagram below is the scatter plot for the sample in the table above.
(a) Label the axes and draw the regression line from 1(b) on the graph.
Solution
Hours of homework
25
20
15
10
5
0
0
10
20
30
40
50
Hours watching TV
Behaviours
Correctly labels the axes
Uses the correct intercept on the vertical axis
Uses the correct gradient
Marks
1
1
1
Item*
(S/C)
simple
simple
simple
Sample assessment tasks | Mathematics Applications | ATAR Year 12
26
(b) Calculate the equation of the linear regression line, using the variables defined here:
Hours watching TV= t and Hours of Homework  w
Solution
Hours watching TV= t and
Hours of Homework  w
w= 0.11t+ 4.46
Behaviours
Marks
Defines w = the response variable
Calculates the gradient accurately
Calculates the intercept accurately
Writes the linear equation coherently
(c)
1
1
1
1
Item*
(S/C)
simple
simple
simple
simple
Using the line of best fit in (b), calculate the residual for the student who watched 17 hours
of TV and spent 20 hours doing homework.
Solution
w  0.11×17 +4.46  6.33
Residual  20 - 6.33  13.67 hours
Behaviours
Uses the explanatory value t=17
Calculates the response value w accurately
Calculates the residual correctly
Marks
1
1
1
Item*
(S/C)
simple
simple
simple
ACKNOWLEDGEMENTS
Data from: Australian Bureau of Statistics. (2013). Retrieved May, 2014, from (2014) CensusAtSchool
www.abs.gov.au/websitedbs/CaSHome.nsf/Home/CaSMa06+ARE+MALES+BETTER+DRIVERS#hello
Used under Creative Commons Attribution 2.5 Australia licence.
Sample assessment tasks | Mathematics Applications | ATAR Year 12
Appendix 1
Australian Bureau of Statistics. (2014).
CensusAtSchool questionnaire: Variations list 2014.
Retrieved from
www.abs.gov.au/websitedbs/CaSHome.nsf/Home/Ce
nsusAtSchool+Past+Questionnaires. Used under a
Creative Commons Attribution 2.5 Australia licence.
Sample assessment tasks | Mathematics Applications | ATAR Year 12
Appendix 2
Data points for Arm span versus Height for approximately 50 students, after outliers were removed
Q11. Arm span
164
135
169
186
169
151
177
168
145
169
172
140
185
178
144
144
162
168
134
165
154
178
159
189
142
165
161
170
Sample assessment tasks | Mathematics Applications | ATAR Year 12
Q09. Height
172
134
170
186
169
152
168
169
151
176
172
149
186
198
144
148
163
163
145
167
160
191
162
178
133
165
164
160
Q11. Arm span
157
133
149
168
154
160
181
133
141
160
163
152
157
174
178
166
123
165
186
167
152
154
140
153
Q09. Height
160
134
153
171
155
159
179
143
144
164
170
156
163
174
179
163
152
160
180
175
162
155
150
155
Appendix 3
Scatter graph for Arm span versus Height for approximately 50 students, after outliers were removed
250
y = 0.8171x + 31.955
R² = 0.8019
200
Height
150
100
50
0
0
20
40
60
80
100
120
140
160
180
200
Arm span
Sample assessment tasks | Mathematics Applications | ATAR Year 12
Appendix 4
Residual plot of Actual height – Fitted height from the ~ 50 students
10
5
0
Residual
0
10
20
30
40
50
60
Series1
-5
-10
-15
X
End of assessment
Sample assessment tasks | Mathematics Applications | ATAR Year 12