Download Demonstrate understanding of chance and data Example Task A

NOTE
This assessment task has been written at Level 6 of the curriculum. When
using this task, you may want to adapt it for your students. Suggestions
include:
 Question One: Narrowing McGregor’s short-list to only two cars, or
taking out the old cars. This will reduce the number of calculations
and comparisons needed.
 Question Two: A reasonably high level of literacy is required to read
the media report. You may need to adapt the language in the report
as appropriate.
 Question Three: The complexity of GAME 2 can be reduced by playing
this game with only two dice.
Any adaptations to the assessment will need to be reflected in changes to the
assessment schedule.
NZAMT 91038 (1.12):
Demonstrate understanding of chance and data Example Task A 2011 – page 1 of 14
New Zealand
Association of
Mathematics
Teachers
1
Level 1 Mathematics and Statistics
91037 (1.12): Demonstrate understanding of chance
and data
Example Task A 2011
Credits: Four
Check that you have been supplied with the resource booklet for Mathematics and Statistics 91037 (1.12).
You should answer ALL parts of ALL questions in this booklet.
You should show ALL working.
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 2–6 in the correct order and that none of these pages is blank.
YOU MUST HAND THIS BOOKLET TO YOUR TEACHER AT THE END OF THE ALLOTTED TIME.
For Assessor’s
use only
Achievement
Demonstrate understanding of
chance and data.
Achievement Criteria
Achievement
with Merit
Demonstrate understanding of
chance and data, justifying
statements and findings.
Achievement
with Excellence
Demonstrate understanding of
chance and data, showing
statistical insight.
Overall Level of Performance
NZAMT 91038 (1.12):
Demonstrate understanding of chance and data Example Task A 2011 – page 2 of 14
You are advised to spend 60 minutes answering the questions in this booklet.
Question One
McGregor has recently bought a copy of the Cat and Orange Guide to help him buy his first car. He has
narrowed his choices down to the three used cars presented in the table. He is particularly interested in
the potential cost of repairs for his new car, as he is on a limited budget.
The following table is reproduced from the book showing owner responses of popular second-hand 10 – 15
year old cars.
Number of cars where the total cost of repairs in the
first year of ownership was…
Car
Awesome
Car
Better
Car
Age of car
Total number of cars
surveyed
No repairs, or
minor repairs up
to $100
Between $100
and $1000
$1000 or more
10 – 15 years old
530
208
112
850
Over 15 years old
245
190
65
500
10 – 15 years old
50
12
18
80
Over 15 years old
17
0
8
25
10 – 15 years old
152
20
53
225
Over 15 years old
255
95
100
450
Cool Car
Your task is to use this information and probability ideas to recommend which car McGregor should
purchase.
Justify your recommendation clearly showing all calculations.
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NZAMT 91038 (1.12):
Demonstrate understanding of chance and data Example Task A 2011 – page 3 of 14
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NZAMT 91038 (1.12):
Demonstrate understanding of chance and data Example Task A 2011 – page 4 of 14
Question Two
Refer to Question Two RESOURCE SHEET to answer this question.
Evaluate the report about Narcissism (self importance) in Facebook users.
You may consider discussing:

What is the purpose of the article?

What are the variables of interest?

How were these measured?

Are the claims made in the article valid and sensible?

How widely can these be applied?
Question Three
(Idea sourced from nrich: The Better Bet
http://nrich.maths.org/4334)
Two games each cost $1 to play.

Game 1:
Toss four fair coins. If you get two heads and two tails you win $3.

Game 2:
Roll three fair six-sided dice. You win $2 for every six that you roll.
Which game are you best to play? Explain your answer clearly, using probability ideas.
NZAMT 91038 (1.12):
Demonstrate understanding of chance and data Example Task A 2011 – page 5 of 14
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NZAMT 91038 (1.12):
Demonstrate understanding of chance and data Example Task A 2011 – page 6 of 14
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Assessment Schedule
NZAMT 2011 task B 91037
Demonstrate understanding of chance and data
Assessment Criteria
Achievement
Demonstrate understanding of chance
and data will involve using a range of
appropriate concepts and terms to
demonstrate an understanding of
statistical literacy and probability.
Merit
Demonstrate understanding of
chance and data, justifying
statements and findings will involve:
 providing supporting evidence
such as summary statistics,
probabilities, data values,
trends or features of visual
displays
 reference to the context and the
population.
Excellence
Demonstrate understanding of
chance and data, showing
statistical insight will involve:
 integrating the statistical and
contextual information and
knowledge to show a deeper
understanding
 critical reflection on the validity
of the processes and
conclusions given in contexts
involving probability or
statistics.
Question One
Analysis
N1
Recommendations
Lack of understanding of information in table leading
to meaningless probabilities calculated
EG creating totals around table and basing
probabilities on these
AND Some attempt at recommendation needed
based on analysis
OR
N2
Reading table correctly, but not calculating
probabilities
Some correct probabilities calculated
A3
Probabilities correctly calculated for either
 P(no repairs or minor repairs only) OR
 P(repair needed) OR
 P(expensive repair needed)
A4
Probabilities correctly calculated for either
 P(no repairs or minor repairs only) OR
 P(repair needed) OR
 P(expensive repair needed)
M5
Further calculation of second level probabilities as
NZAMT 91038 (1.12):
AND Recommendation based on counts only, not
probabilities
AND No recommendation given
OR
Recommendation does not follow from
probabilities calculated
AND Basic recommendation provided based on
probabilities calculated, but too much
emphasis given to minor differences in
probabilities
EG Best to buy either old Car B because its got
the lowest probability of repairs needed
(32.0% compared to 32.4%)
AND Basic recommendation provided based on
probabilities calculated
EG Best to buy either old Car B or new Car C as
both have about a 32% of needing a repair
OR Best to buy new Car C as similar likelihood
of repair as old Car B but car is newer
AND A4 - level recommendation
Demonstrate understanding of chance and data Example Task A 2011 – page 7 of 14
M6
E7
E8
well as A4 calculations
EG P(expensive repair needed) as well as P(repair
needed)
Probabilities correctly calculated for
 P(no repairs or minor repairs only) OR
 P(repair needed)
AND
 P(expensive repair needed)
to allow comparison of at least the short-listed
candidates
AND Consideration of other factors when making
recommendation
EG major repair costs vs minor repair costs
OR Best to buy new Car C as similar likelihood
of needing a repair as old Car B but new Car C
is less likely to have a repair costing more
than $1000.
OR
AND As above
Similar further analysis to help make a better
recommendation
Analysis to M6 level.
Analysis to M6 level. Candidates may included
extended relevant probabilities calculated
EG Conditional probabilities
OR ALL of each type of car regardless of age
AND Some discussion of other factors without
developed detail and understandings
presented in E8
AND Detailed consideration of other factors when
making recommendation
EG major repair costs vs minor repair costs
Sample size issues
Disregarding age of car
Probabilities
Probability of…
any repairs
Probability of…
repairs costing over
$1000
Probability of…
repairs costing over
$1000 given that
repair is needed
850 62.35%
37.65%
13.18%
35.00%
500 49.00%
51.00%
13.00%
25.49%
80
62.50%
37.50%
22.50%
60.00%
25
68.00%
32.00%
32.00%
100.00%
225 67.56%
32.44%
23.56%
72.60%
450 56.67%
43.33%
22.22%
51.28%
n
10 – 15
Awesome years old
Car
Over 15
years old
10 – 15
years old
Better
Car
Over 15
years old
10 – 15
years old
Cool Car
Over 15
years old
NZAMT 91038 (1.12):
Probability of…
No repairs or
minor repairs
only
Demonstrate understanding of chance and data Example Task A 2011 – page 8 of 14
Question Two
Idea
N
A
M
E
Purpose of
article
Purpose not clearly
identified
Eg: Facebook
Purpose simply
identified from article
title
Eg: Facebook users are
narcissistic
Recognising that the
article summarises
someone’s research
relating Facebook users
to narcissistic users
Recognising that the
article summarises
someone’s research
relating psychological
measures to Facebook
users
Variables
of interest
Variables of interest not
clearly identified
Variables of interest
simply identified
Eg: Facebook use &
narcissism score
Number of times
facebook page was
checked per day &
psychological tests
How were
these
measured?
Are the claims valid and sensible?
Sample
size
100 students sampled
are not many of the
facebook users.
Sample was only of 1825 year olds
Number of times
facebook page was
checked per day (selfreported through
interview) &
psychological tests
Some thought on HOW
the data on number of
times facebook pages
were checked was
collected/measured
Age
Specific description of
graph with no linkage to
report.*
Students
Specific description of
graph with no linkage to
report.*
Sample was only of
students
Sample was only of
students. 28% of
Facebook users are in
Uni so this data might
not be representative of
the all Facebook users.
Nationality
Specific description of
graph with no linkage to
Sample was only of
Canadians
Sample was only of
Canadians but the
NZAMT 91038 (1.12):
Sample was of 18-25
year olds. From the
graph this age group is
the largest proportion of
Facebook users
Sample was of 18-25
year olds. From the
graph this age group is
the largest proportion of
Facebook users in the
US. It seems reasonable
to assume that the US
age distribution would
be similar across other
countries.
Sample was only of
students. 28% of
Facebook users in the
US are in Uni so this
data might not be
representative of the all
Facebook users
internationally.
Query proportion of
population of each country
Demonstrate understanding of chance and data Example Task A 2011 – page 9 of 14
Idea
N
A
report.*
Claims
widely
applied
Score
N1 – Two N statements
OR – One A statement
N2 – Two N statements
plus on A statement.
A3 – Two A statements:
one must include
purpose or variables, 2nd
must relate to claims of
report.
OR – One M statement
only
A4 – At least 3 A
statements: one must
include purpose or
variables, another must
relate to claims of
report.
*
M
E
biggest users of
Facebook are from the
United States.
Therefore the study
results should not be
widely applied
M5 – A4 plus one M
statement
OR – two M statements
only
that are Facebook users –
nationality graph given is
in thousands
M6 – A4 plus two M
statements
Media article implies
results are applicable to
all Facebook users
E7 – As E8, but has not
pulled ideas together to
make a detailed
conclusion.
Several of the examples
above may have been
presented, but not
linked.
E8 – Cohesive extended
response to report:
report objectives and
variables clearly
identified; study only of
Canadian students from
one University in one
age group, so difficult to
generalise with
confidence.
Descriptions of graphs need to be at Curriculum Level 5, not just simple graph reading and statements. Should be
quantified and in context of the graph. Eg: The United States has 21000(000) users compared to 7500(000) users in
Canada – nearly three times as many users in the US.
NZAMT 91038 (1.12):
Demonstrate understanding of chance and data Example Task A 2011 – page 10 of 14
Question Three
Grade Game 1
4 coins
N1
N2
A3
A4
OR:
M5
M6
E7
OR:
E8
Game 2
3 dice
Attempt at finding sample space
OR
Attempt at finding sample space
 Random listing of outcomes
 Random listing of outcomes
 Errors in understanding shown
 Errors in understanding shown in
in tree diagrams
tree diagrams
Attempt at finding sample space for both GAMES
 Random listing of outcomes
 Errors in understanding shown in tree diagrams
Good systematic attempt at finding
OR
Good systematic attempt at finding sample
sample space, but not necessarily
space, but not necessarily complete
complete
(student shows understanding that they
(student shows understanding that they
need to find some probabilities)
need to find some probabilities)
Good systematic attempt at finding sample spaces for both GAMES, but not necessarily
complete (student shows understanding that they need to find some probabilities)
Probability calculated correctly for
GAME 1
AND Sound attempt made to calculate
probability for GAME 2
Probability calculated correctly for
 Must included viewing GAME as
GAME 1 with some justification
rolling a 6 or NOT rolling a 6
 (eg using 1/6 and 5/6 on tree
diagram)
Probability calculated correctly for
AND Correct probabilities for GAME 2 with some
GAME 1 with some justification
justification
Correctly connects winnings and
AND
Correct probabilities for GAME 2 with some
probability for GAME 1
justification
Eg: expected values
Correctly connects winnings and probability
for GAME 2
Eg: expected values
Correctly connects winnings and probability for BOTH GAMES with justification AND clear
conclusion given in context.
NZAMT 91038 (1.12):
Demonstrate understanding of chance and data Example Task A 2011 – page 11 of 14
Question Two (cont…) - Calcuations
GAME 1
16 possible outcomes when tossing four coins, six of these contain 2 heads and 2 tails, therefore
P(2H&2T) = 6/16 = 3/8 = 0.375
Expected total winnings from GAME 1 = 3/8 x $3 = 9/8 = $1.125
GAME 2
216 (6x6x6) possible outcomes when roll 3 dice. Tree diagram using ‘6’ and ‘not 6’ as the events of interest
is useful for developing required sample space.
 One six: Three paths that give one ‘6’, and two ‘not 6’ (6**, *6*, **6)
P(one ‘6’, two ‘not 6’) = 3 x 1/6 x 5/6 x 5/6 = 75/216
Expected winnings from one six = $2 x 75/216 = 150/216
 Two sixes: Three paths that give two ‘6’ and one ‘not 6’ (66*, 6*6, *66)
P(two ‘6’, one ‘not 6’) = 3 x 1/6 x 1/6 x 5/6 = 15/216
Expected winnings from two sixes = $4 x 15/216 = 60/216
 Three sixes: One path that gives three ‘6’ (666)
P(three ‘6’) = 1/6 x 1/6 x 1/6 = 1/216
Expected winnings from three sixes = $6 x 1/216 = 6/216
Expected total winnings from GAME 2 = 150/216 + 60/216 + 6/216 = 216/216 = $1
Conclusion: GAME 1 is best (just)!
NZAMT 91038 (1.12):
Demonstrate understanding of chance and data Example Task A 2011 – page 12 of 14
Question Two
Resource sheet
Media report from Mail-Online
Facebook users 'are insecure, narcissistic and
have low self-esteem'
By Mail Foreign Service
http://www.dailymail.co.uk/sciencetech/article-1310230/Facebook-users-narcissistic-insecure-low-self-esteem.html
Last updated at 10:43 AM on 9th September 2010
Using Facebook is the online equivalent of staring at
yourself in the mirror, according to a study.
Those who spent more time updating their profile on
the social networking site were more likely to be
narcissists, said researchers.
Facebook provides an ideal setting for narcissists to
monitor their appearance and how many ‘friends’
they have, the study said, as it allows them to thrive
on ‘shallow’ relationships while avoiding genuine
warmth and empathy.
People who constantly check Facebook may be lacking
in self-esteem, a study found
They also tend to use the site for promoting
themselves to friends or people they would like to
meet, the study concluded.
Researcher Soraya Mehdizadeh from York University
in Canada asked 100 students, 50 male and 50 female,
aged between 18 and 25 about their Facebook habits.
They all took psychology tests to measure their levels
of narcissism, which the study defined as ‘a pervasive
pattern of grandiosity, need for admiration, and an
exaggerated sense of self-importance’.
Those who scored higher on the narcissism test
checked their Facebook pages more often each day
than those who did not.
There was also a difference between men and women
posts on their Facebook page while women tended to
carefully select the pictures in their profile.
The
findings,
published
in
the
journal
Cyberpsychology, Behaviour And Social Networking,
also suggested that those with low self-esteem also
checked their Facebook pages more regularly than
normal.
This may not be altogether surprising as it is widely
thought, however contradictory it may appear, that
narcissism is linked to a deep-rooted lack of selfesteem.
Miss Mehdizadeh admitted that not everyone would
appreciate her findings.
She said: ‘I think people get sort of defensive about it
– like, “I don’t use my Facebook for that reason” –
because it’s a label that you don’t want to be slapped
with.’
Facebook has more than 500million users worldwide
and is the world’s biggest social networking website,
but it has been involved in a number of controversies.
A study earlier this week showed that the grades of
students who use Facebook while they study, even if
it is only on in the background, are 20 per cent lower
on average than those of non-users.
NARCISSISM or SELF-LOVE
– an exceptional interest in and
admiration for yourself
– men generally promoted themselves by written
NZAMT 91038 (1.12):
Demonstrate understanding of chance and data Example Task A 2011 – page 13 of 14
Graphs source:
http://www.slideshare.net/EmergenceMedia/facebook-demographics-user-statistics-emergence-media
NZAMT 91038 (1.12):
Demonstrate understanding of chance and data Example Task A 2011 – page 14 of 14