1
QUESTION TWO
After dinner Hemi and Jonno decide to drive across town to the video shop to hire another DVD.
Frequency
DVD's Hired in Month of August
10
9
8
7
6
5
4
3
2
1
0
Hemi
Jonno
Hemi and Jonno have kept count of the number of “New Release” DVD’s and the day of the
week that they were hired from the video shop in the month of August 2011.
a)
Describe a feature of Jonno’s and a feature of Hemi’s DVD’s hiring that is shown by the
graph.
b)
Jonno and Hemi have made some claims based on the data displayed in the graph.
i)
Hemi claims that Jonno hires more DVD’s in total than he does in the month of August.
Comment on the claim.
ii)
Jonno claims that Hemi spends more on DVD’s than he does. From the information
given, comment on the claim.
2
iii)
Hemi believes that he can use this data to prove that he hires more DVD’s throughout
2011 than Jonno. Comment on the claim.
c)
Hemi and Jonno often take turns to drive across town from their flat to the video shop to
hire DVDs. They each have a different way of getting to their favourite video shop.
Both Jonno and Hemi have collected a sample of their driving times (to the nearest
minute). Their driving times are shown in the graphs and tables below.
Driving Route
Driving Time from Flat to Video Shop
Jonno
Hemi
0
5
10
15
20
25
Time (minutes)
Driving Time from Flat to shop
30
35
40
Hemi
Jonno
3
0
5
10
15
20
25
30
35
40
Time (minutes)
Raw Data:
Summary
statistics
Travelling Time
(nearest minute)
Hemi
Jonno
7
7
Mean
8
10
Median
8
10
11
12
11
12
11
11
Hemi
Jonno
13.15
14.56
13
12
Mode
11
12
Range
14
33
Minimum
7
7
12
Lower quartile
11
12
12
Upper quartile
15.5
14
12
12
Maximum
21
40
13
12
13
12
13
12
14
13
14
13
14
13
15
13
16
13
16
13
17
13
18
14
21
14
14
14
14
17
25
27
40
4
QUESTION TWO (Continued)
Jonno claims that:
“My way of getting to the video shop is quicker”.
Make at least FOUR evaluative statements about whether you think Jonno’s claim is
valid or not. In your answer, you should consider:
 the data accuracy,
 the data collection method
 the appropriateness of different data displays and summary statistics
 sampling variability
 any conclusions made
5
QUESTION THREE
a)
After watching DVD’s Hemi and Jonno play a game using two different spinners:

one spinner has four equal parts: three are labelled “4” and one is “5”

the other spinner has six equal parts labelled either “5” or “6” .
5
4
4
5
5
5
5
4
5
6
i.
What is the probability of getting a 4 on the first spinner?
ii.
What is the probability of getting a 5 on the second spinner?
iii.
What is the probability of getting a total of 9 when you spin both spinners?
b) Hemi spins both spinners. He adds the result of the two spinners together and
records the total.
He repeats this 240 times.
Complete the table below, giving the expected results of the 240 totals of the
two spinners.
Total for two
spinners
9
10
11
Expected values
________________________________________________________________
6
c)
Jonno uses two of his own spinners to repeat the experiment. One spinner
has three “4’s” and one “5” marked on the equal parts. The second spinner
has six equal regions.
The results from 240 totals of the two spinners are shown in the table below.
Total for two
spinners
Frequency of
totals
9
10
11
134
91
15
Hemi looks at these results and claims that the results show Jonno has two
sixes on his second spinner.
Investigate Hemi’s claim.
7
Mathematics and Statistics 91037: Demonstrate understanding of chance and data
Evidence Statement
Two
Expected coverage
a)
e.g.
Hemi hires movies between Thursday to
Sunday only.
Jonno hires movies on every day of the
week.
Achievement
A valid comment made
about both Hemi and
Jonno’s results.
b i)
Hemi in fact has a greater total (21) than
Jonno (17)
Valid comment made on
the actual totals. Must
give correct totals
b ii)
While Hemi does have a greater total of
DVD’s the graph gives no information about
money spent.
Valid comment made
Did not accept the reason:
‘Hemi spends more
because the total is
higher’
Title says “New Releases” so that it could
be implied all DVD’s cost same amount and
so therefore Hemi spends more on DVD’s
b iii)
e.g. If August is a representative month
then the claim could be true.
However, sampling variability alone could
mean that it could be the other way around
in other months.
Whether the flatmates watched the movies
together or independently The hiring of
DVD’s might be a dependent event as the
two both watch them.
Valid comment made.
Eg Assuming the data is
representative it is true
Would need to collect
more data.
Merit
Valid comment made
regarding recognition that
they may not be the same
price and giving reason
why
Valid comment made with
justification.
Looked for an explanation
for why August is not
representative
Excellence
8
c)
Comments about data collection
e.g.
Time of day?
Valid comment made
Maybe rush hour was the reason for the
outlier.
How was rounding of times carried out
What speed did the two flat-mates travel?
Traffic lights/road works
We have no knowledge as to the timing
(experiment) was carried out except from
graph of days. Lurking variable could be
day that DVD hired. Jonno and Hemi tend
to hire on different days/
Valid comment made with
justification
Valid comment made with
insight
Valid comment made with
insight
Valid comment made with
justification
Comment regarding sampling
variability in context of
problem
Comments about data display
e.g. No obvious difference in graphs
Comments about data analysis
e.g. Mean suggests that Hemi’s route
should be used as his mean travelling time
is lowest. Measure of central tendency
needs to be more robust than simply the
mode.
IQR suggests that Jonno’s times are more
consistent (also seen in the box and
whisker graph).
Std. Deviation and range suggest that the
Hemi’s times are more consistent (also
seen in the frequency graph).
With samples of this size the amount of
overlap in the IQR’s would mean that I
would be unable to claim that there is a
difference in travel times. This could have
happened by chance.
Sufficiency
using a measure of
central tendency to make
a valid conclusion
noticing that the mean will
have been affected by the
outliers thus raising the
mean for Jonno’s mean
time.
making an observation
about a measure of
spread but not relating it
to the conclusion
using a measure of spread
to make a valid conclusion
Comment regarding
sampling variability
Two statements from any
area (e.g. central tendency,
spread or method) making a
valid evaluations
2a
Making an informal
inference based on overlap
of middle 50%
Three statements from data
analysis and collection
methods evaluations
Making an informal inference
based on overlap of middle
50%
At least three statements from
at least two areas (one must be
about the method) with valid
evaluations
All statements made must be
correct.
2m
1e
9
Three
Expected coverage
Achievement
a i)
¾ or equivalent
a ii)
5/6 or equivalent
a iii)
5/8 or equivalent
correctly deducing the
probability in any form.
correctly deducing the
probability in any form.
correctly deducing the
probability in any form.
Correctly calculating
expected number for
totals 9 and 11
b)
c)
Total
9
10
11
for two
spinner
s
Expect
ed
150 80
10
values
If the second spinner does have two 6’s you
could expect the following
Total
9
10
11
for two
spinner
s
Expect
ed
120 100
20
values
Merit
Excellence
correctly calculating all the
expected numbers
correctly deducing the
expected number or
probabilities for Hemi’s
claim
and
making a statement
regarding Hemi’s claim
agreeing with it or not.
Valid comment made with
insight – insightful answer
would require too hard to call.
Could be either 1 or 2 sixes.
Very likely that second
spinner has 2 6’s etc.
Statement should not state
that there are 2 6’s on the 2nd
spinner.
Too hard too call. Could be either situation.
Sample could have happened by chance.
Need more data to be sure.
Sufficiency
Overall Sufficiency
2a
2A
1m
1e
2M
2E