Download NZAMT 91038 (1.12): Demonstrate understanding of chance and

New Zealand
Association of
Mathematics
Teachers
1
Level 1 Mathematics and Statistics
91037 (1.12): Demonstrate understanding of chance
and data
Example Task B 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 2–7 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 B 2011 – page 1 of 12
You are advised to spend 60 minutes answering the questions in this booklet.
QUESTION ONE
Invest in
Easy-to-win LOTTO
Easy-to-win LOTTO has only four balls, painted with the numbers 1 to 4.
To enter, pay $2 and choose two numbers.
You win if we draw your two numbers (in any order) from our lotto bag.
WIN $10
It’s so easy to win – half the numbers are winners - just like tossing a coin!
Invest $2 and get 5 times that amount back!
OR
Try BIG-WIN LOTTO: 5 balls numbered 1 to 5
Pay $2 to enter, choose two balls, and win $20 if they are correct
Evaluate the claims made in the advertisement for “Easy-to-win LOTTO”.
In your answer you may find it helpful to comment on:
 What is the purpose of the advertisement?
 Are the claims made valid and sensible? Or can alternative/or additional claims be made?
 Would you recommend playing these games?
NZAMT 91038 (1.12):
Demonstrate understanding of chance and data Example Task B 2011 – page 2 of 12
NZAMT 91038 (1.12):
Demonstrate understanding of chance and data Example Task B 2011 – page 3 of 12
QUESTION TWO
Each school day you and your three friends have a treat from the canteen.
Each time someone buys the treats for all four of you.
Who is going to buy the four treats is decided by the following method:
Everyone starts with a coin.



(a)
Each person tosses their coin.
If everyone has the same (all heads, all tails) or if there are two heads and two tails, then everyone tosses
again.
If there is three heads and one tail, or if there is three tails and one head, then the person whose coin is
different from everyone else’s is the “odd one out” and must they buy the treats for everyone that day.
What is the probability that an “odd one out” will be found on the first coin toss?
(b) What is the probability that you will have to buy the treats after the first coin toss?
NZAMT 91038 (1.12):
Demonstrate understanding of chance and data Example Task B 2011 – page 4 of 12
(d)
One person leaves the group. You decide to change the game for 3 players, but you want it to last longer.
OPTION 1 (3 coins)
You still have a single coin each, but toss it twice. If there is a player who is “odd one out” in BOTH tosses, then
they must buy the treats. If there is no one who is “odd one out” in the two consecutive throws the coins are
tossed again.
OPTION 2 (6 coins)
You could have two coins each, toss them all at once.
If there are 5 heads and 1 tail, or if there are 5 tails and 1 head, then the person with the “odd one out” is the one
who buys the treats. If there is no “odd one out” all the coins are tossed again.
Which game would be likely to last longer? Show all your calculations to justify your answer?
NZAMT 91038 (1.12):
Demonstrate understanding of chance and data Example Task B 2011 – page 5 of 12
QUESTION THREE
Alcohol Heath Watch Director Rebecca Williams says that while New Zealand’s alcohol marketing in the past was
predominantly focused on beer and aimed at men, it has switched to mixes and now young women are being
targeted.
The number of girls going to the Emergency Department as a result of binge drinking has increased 30% over the
last four years. Rebecca suspects the problem is now as bad for females as it is for males.
For the past two years Rebecca has been collecting data. Every month she records the number who come to the
Emergency Department with problems associated with binge drinking. She calculated the mean number of girls
presenting each month as 19.3 and the mean number of males was 21.9. A graph showing the data is given
below.
Monthly Number of arrivals at the Emergency Department as a result of Binge Drinking
Rebecca concludes that males are still having more binge drinking problems than females.
E valuate Rebecca’s conclusion. Make at least FOUR evaluative statements. You may comment on
 the data that Rebecca gathered
 the way she has analyzed it
 what can be concluded from her investigation
 what she could do next to further her investigation.
NZAMT 91038 (1.12):
Demonstrate understanding of chance and data Example Task B 2011 – page 6 of 12
NZAMT 91038 (1.12):
Demonstrate understanding of chance and data Example Task B 2011 – page 7 of 12
Assessment Schedule
Mathematics and Statistics 91037 (1.12): 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
NO Purpose of article not identified.
Claims not examined
N1 Article is to sell Lotto game .
Article implies you will always win
N2 “toss a coin”, half the numbers
are winners implies that you win
½ the time
A3 Claim suggest better odds than
there are. Recognition that there
is less than a 50% chance of
winning, partially complete list of
possibilities made for either Easyto-Win or Big Win
A4 List of all possible draws for Easyto-win (either with
order:12,13,14,21,23,24,31,32,34,
41,42,43) (or without order:
12,13,14,23,24,34)
M5 Calculation of probability of
winning as 1/6 for Easy-to-win
AND list of all possible draws for
NZAMT 91038 (1.12):
OR
List of all possible draws for Big Win
(either with
order:12,13,14,15,21,23,24,25,31,32,34,35,41,
42,43,45,51,52,53,54,) (or without order:
12,13,14,15,23,24,25,34,35,45)
Calculation of probability of winning as 1/10
for Big-win AND list of all possible draws for
Easy Win
Recommendation
made without
reference to
probability
Recommendation
made without
reference to
probability
Recommendation
made without
reference to
probability or with
incorrect
reference to
probability
Recommendation
made with
reference to
probability
calculations
(possibly with
minor flaws)
Recommendation
made with
reference to the
probability
Recommendation
made with
reference to the
Demonstrate understanding of chance and data Example Task B 2011 – page 8 of 12
Big Win
M6 Correct probability for both Easyto-win(1/6) and Big Win (1/10)
E7 E(Easy-to-win)= $1.67 or some
other equivalent calculation (eg if
got back 6x investment it would
be fair but you only get back 5x
investment)
OR E(Big) win calculated
E8
Identification of conflicting claims
within article (1/2 chance implied
then “invest and get 5x amount”)
Correct probability for both Easy-to-win(1/6)
and Big Win (1/10)
E(Big Win) = $2 or some other equivalent
calculation (eg if I get back 10x the investment
so it si fair)
OR
E(Easy-to-win) calculated
All outcomes listed and probabilities calculated
correctly for both Easy-to-Win and Big Win,
Expected values correctly calculated for both
Easy-to-Win and Big Win
Expected coverage
TWO
2(a)
Pr(odd one out) = 8/16 = ½ or 0.5
TTTT,TTTH,TTHT,TTHH,THTT,THTH,THHT,
THHH,HTTT,HTTH,HTHT,HTHH,HHTT,HHTH,HHHT
,HHHH
2(b)
Pr(you) = 2/16 = 1/8
(bold in list above)
2(c)
OPTION 1
Pr(you odd one out in first toss) = 2/8 = ¼
TTT, TTH, THT, THH,HTT,HTH,HHT,HHH
So Pr(you in 2 consecutive)= ¼ x ¼ = 1/16
Three players, so Pr(winner in 2 consecutive tosses =
3/16)
OPTION 2
Pr(one odd one out ) = 12/64 =3/16
Therefore both games would have same chance of
repeating, but Option 2 requires two consecutive
throws (which would take longer) so choose
Option 2
NZAMT 91038 (1.12):
Achieveme
nt
Demonstrate
understandin
g of chance
by:
Correctly
deducing
the
probability
in any form.
Correctly
deducing
the
probability
in any form.
Correct
calculation
of simple
probability
for one part
of one of the
games
probability
Recommendation
refers to
“fairness”or
similar concept .
Recommendation
refers to expected
return over a
long-run situation,
demonstrates
understanding of
expected value
Recommendation
refers to expected
return over a
long-run situation,
demonstrates
understanding of
expected value
Merit
Excellence
Demonstrate
understandin
g of chance,
justifying
statements
by:
Demonstrate
understandin
g of change,
showing
statistical
insight by:
Correctly
calculating
the overall
probability
for one of
the games
Correctly
identifying
probability
for one of the
games and
minor error in
the other
game, with
choice and
justification
based on
(possibly
only partially
correct)
result
Correctly
identifying
probability
for both
Demonstrate understanding of chance and data Example Task B 2011 – page 9 of 12
games
Scoring
N1 – Incomplete list or flawed probability tree for
calculations of probability in (a) and (b) and (d)
N2 – One correct answer for either (a) or (b)
NZAMT 91038 (1.12):
A3 – 2a
A4 – 3a OR
2a + m
M5 – A4 +
m OR
A3+2m
M6 – A4
+2m
Replacement
evidence for
2abc
available.
E7 – M + e
with some
valid
reasoning
(may be
incomplete
or
communicate
d clumsily)
E8 – M + e
with valicd
clear and
concise
reasoning.
Demonstrate understanding of chance and data Example Task B 2011 – page 10 of 12
Expected coverage
Achievement
Demonstrate
understanding of
chance by:
THREE
3
Although the mode is the same for both
male and female binge drinking
admissions, median is lower for females
(thereby suggesting females have “caught
up to males” with binge drinking would not
be justified).
However female mean is strongly
influenced by the two outliers of 8 and 9
presentations. As the data was collected
over a two year period and Rebecca
believes female binge drinking is
increasing, it would be worth checking to
see if the outliers were from the very start
of the two year period. If so, it would be
sensible to change the data to collection for
both male and female to being over the last
22 months, and if this was done, the claim
that now female binge drinking is as bad as
male binge drinking would be justified.
Variation in number of binge drinking
presentations is much larger for females
(larger range, IQ range, Std Dev).
This could also be due to the binge
drinking incidents changing over time for
females, so lower values are more
common in the first part of the two year
period.
If the two lowest values are excluded from
the study then the measures of spread
would be much closer.
As Rebecca suspects the current situation
is a result of a growing trend, it would have
been more useful to plot the data as a time
series (line graph). This would have more
clearly shown whether females have
“caught up” to males. Eg The problem of
binge drinking could be an increasing one
(if the highest 6 female values were all in
the most recent months)
Hospital presentations for binge drinking
would generally be a good indicator of the
incidence of binge drinking, but comparing
populations when the data is gathered over
time is problematic. Especially in this case
when Rebecca knew that there has been
an increase over time.
Binge drinking presentations are also likely
to peak during holiday periods. A line
graph would have shown this.
NZAMT 91038 (1.12):
 Using a measure
of central
tendency to
make a valid
conclusion
Merit
Excellence
Demonstrate
understanding of
chance,
justifying
statements by:
Demonstrate
understanding of
change,
showing
statistical insight
by:
 Commenting that
the mean will
have been
affected by
outliers and
adjusting
conclusion.
 Justifying the
possible
exclusion of
the outliers
based on
context
Making an
observation about
a measure of
spread
Drawing a valid
conclusion
about spread
Making a valid
comment about
the
appropriateness
of this data
display for the
question being
asked.
Making a valid
comment about
the relevance of
the data to the
question.
Demonstrate understanding of chance and data Example Task B 2011 – page 11 of 12
Scoring
N1- some aspect of graphs of summary
statistics commented on, but with only
partially correct interpretation
N2 – some graphs comment on correctly,
but interpretation is unclear or incomplete
NZAMT 91038 (1.12):
A3 – 1a
A4 – 2a
M5 – 1m OR 1e
M6 – 3m OR
1m+1e
E7- M + 1e
E8 – M+ 2e
Demonstrate understanding of chance and data Example Task B 2011 – page 12 of 12