NCEA Level 3 Mathematics and Statistics 91584 (3.12) — page 1 of 6
SAMPLE ASSESSMENT SCHEDULE
Mathematics and Statistics 91584 (3.12): Evaluate statistically based reports
Assessment Criteria
Achievement
Achievement with Merit
Achievement with Excellence
Evaluate statistically based reports involves identifying
and commenting on key features in reports relevant to
any conclusions made in those reports.
Evaluate statistically based reports, with justification
involves supporting the comments made by referring to
statistical evidence and processes used in reports,
relevant to conclusions made in those reports.
Evaluate statistically based reports, with statistical
insight involves integrating statistical and contextual
information to assess the quality of reports with respect
to conclusions made in those reports.
Evidence Statement
One
Expected Coverage
The margin of error is 1/√200 (using the rule of thumb 1/√n)
= 7.1%
The 95% confidence interval for a survey percentage of 53% is
[45.9% , 60.1%].
(a) (i)
and
(a) (ii)
Merit
The margin of error is
calculated and used to
calculate a confidence
interval.
The confidence interval
is calculated and
interpreted as part of the
explanation that the
claim is supported.
As this confidence interval includes percentages less than 50%, the claim is
not supported.
Excellence
The margin of error is
calculated and used to
determine the lowest
survey percentage to
allow a claim to be
made.
The survey percentage would need to be at least 57.1% to have evidence to
support the claim that most (the majority) of New Zealand mothers have
accessed their children’s Facebook page, as this would provide a 95%
confidence interval of [50% , 64.2%].
Note: The formula
(b) (i)
Achievement
could also be used to calculate the margin of error.
The rule of thumb (1/√n) should only be used as an estimate of the margin of
error with survey percentages between approximately 30% and 70% – which
excludes (USA and) Japan. For Japan, the survey percentage is well outside
this range. The margin of error for the survey percentage for Japan would be
much smaller than the ‘rule of thumb’ margin of error of 1/√400.
Survey percentages for
which the ‘rule of
thumb’ should not be
used are identified.
At least one country is
identified being
unsuitable to use the
‘rule of thumb’ and a
partial explanation is
given regarding the size
of the associated margin
of error.
At least one country is
identified being
unsuitable to use the
‘rule of thumb’ and a
full explanation is
given regarding the
size of the associated
margin of error.
NCEA Level 3 Mathematics and Statistics 91584 (3.12) — page 2 of 6
(b) (ii)
A possible explanation, for example, for Japan’s relatively low survey
percentage is that a very small percentage of Japanese parents are Facebook
users and hence it is only possible for a very small percentage of Japanese
parents to be Facebook ‘friends’ with their children.
The study was funded by AVG – a company that sells products concerned with
internet security. This survey is about concerns about using technology, in
particular internet-based activities, eg downloading illegal music, and
educating teenagers about using the internet responsibly. A desire to
encourage parents to buy their products could motivate AVG to produce
findings that suggest the internet is unsafe.
(c)
Not Achieved
The link is identified
between the funding of
the research by an
internet-related
company and the
survey being about
internet-related activity.
NØ
No response; no relevant evidence.
N1
Candidate gives a partial response to ONE part of the question.
N2
Candidate gives partial responses to TWO parts of the question.
A3
Candidate gives ONE opportunity from the Achievement criteria.
A4
Candidate gives TWO opportunities from the Achievement criteria.
M5
Candidate gives ONE opportunity from the Merit criteria.
M6
Candidate gives TWO opportunities from the Merit criteria.
E7
Candidate gives ONE opportunity from the Excellence criteria.
E8
Candidate gives TWO opportunities from the Excellence criteria.
Achievement
Merit
Excellence
Two
Partial explanation is
given about how the
survey percentages
could be affected by the
percentage of Facebook
users within a country.
Expected Coverage
Achievement
Full explanation is
given about how the
survey percentages
could be affected by
the percentage of
Facebook users within
a country.
The link between the
funding of the research
by an internet-related
company and the survey
being about internetrelated activity is
explained in terms of the
possible motivation for
results.
Merit
Excellence
NCEA Level 3 Mathematics and Statistics 91584 (3.12) — page 3 of 6
(a)
The target population for this survey is New Zealand cat and dog owners. In
order for inferences to be made from this sample, it should be a random
sample of all New Zealand cat and dog owners and / or it should be
representative of all New Zealand cat and dog owners. The sample should
contain cat and dog owners from both rural and urban areas of New Zealand,
proportional to the number of pet owners in these areas. The ratio of dog and
cat owners should be around 5 : 3 in the sample.
The target population
is identified.
One characteristic of
the sample is
identified, eg it is
randomly selected or it
is representative.
The target population is
identified and at least
one characteristic of the
sample is described in
context and linked to
representativeness.
Note: Other characteristics of the sample are possible as long as they are
clearly linked to the target population. It is acceptable to refer to New Zealand
cat owners as one population and New Zealand dog owners as another
population.
(b)
A potential difficulty could be about how respondents of the survey interpret
the words ‘keeping to yourself’ – a clear definition of what ‘keeping to yourself’
means would need to be provided for people completing the survey so that
respondents were interpreting these words in a consistent way.
A potential difficulty
with the question is
identified.
Note: Other potential difficulties are possible.
The 400 respondents could be split into 250 cat owners and 150 dog owners.
Margin of error for cat owners = 1/√250 = 6.32%
Margin of error for dog owners = 1/√150 = 8.16%
Average margin of error = 7.2%
Comparing two subgroups so margin of error for comparison = 1.5 x 7.2% =
10.8%
(c)
The difference between the two survey percentages is 61% - 53% = 8
percentage points.
The 95% confidence interval for the difference between the two survey
percentages is [-2.8%, 18.8%].
With 95% confidence, I estimate that the percentage of New Zealand cat
owners who prefer to ‘keep to themselves’ is somewhere between 2.8% lower
and 18.8% higher than the corresponding percentage of New Zealand dog
owners. This means the evidence presented in the report could not be used to
support the claim that a higher percentage of New Zealand cat owners prefer
to ‘keep to themselves’ than New Zealand dog owners.
Margin of errors of
each group (New
Zealand cat owners
and New Zealand dog
owners) correctly
calculated.
The margin of errors of
each group are
calculated and used to
construct a confidence
interval for the
difference between the
proportion of New
Zealand cat owners and
New Zealand dog
owners who prefer to
‘keep to themselves’.
The confidence
interval for the
difference between
the proportion of New
Zealand cat owners
and New Zealand dog
owners who prefer to
‘keep to themselves’ is
constructed by
calculating and using
the margin of errors of
each group and is
interpreted as part of
the explanation as to
whether the evidence
could support the claim.
NCEA Level 3 Mathematics and Statistics 91584 (3.12) — page 4 of 6
Note: The formula
1.96
p1 q1 p 2 q 2
could also be used to calculate the

n1
n2
margin of error for the confidence interval.
There are likely to be pet owners who own both cats and dogs, as well as
other pets. People may also own more than one cat and / or dog. We do not
know whether the question used had a ‘both’ option or if there were any other
options about owning other pets or owning more than one cat or dog (question
effect). This might be a problem as it could bias survey responses and it might
be that the number of pets owned is what explains your relationship status
(this might be the explanation for a higher proportion of cat owners being
single).
(d)
Two potential nonsampling errors,
biases, or issues are
identified.
There may be a potential issue if only Purina customers were used for the
survey (selection bias), as the people who buy Purina cat or dog food might
not be representative of all New Zealand pet owners. For example, if Purina
customers are mostly older people, they may own cats as companions as they
are easier to look after than dogs, rather than using them ‘as a substitute for a
significant other’ as stated in the report. Additionally, any inferences made
could only be applied to Purina customers, not necessarily all New Zealand
pet owners.
Note: Other potential non-sampling errors, biases, or issues are possible.
Not Achieved
NØ
No response; no relevant evidence.
N1
Candidate gives a partial response to ONE part of the question.
N2
Candidate gives partial responses to TWO parts of the question.
A3
Candidate gives ONE opportunity from the Achievement criteria.
A4
Candidate gives TWO opportunities from the Achievement criteria.
M5
Candidate gives ONE opportunity from the Merit criteria.
M6
Candidate gives TWO opportunities from the Merit criteria.
E7
Candidate gives ONE opportunity from the Excellence criteria.
E8
Candidate gives TWO opportunities from the Excellence criteria.
Achievement
Merit
Excellence
Two potential nonsampling errors, biases,
or issues are identified
and described by
relating to the evidence
or processes presented
in the report.
Two potential nonsampling errors,
biases, or issues are
identified and
discussed, supported
with statistical and
contextual reasoning.
NCEA Level 3 Mathematics and Statistics 91584 (3.12) — page 5 of 6
Three
Expected Coverage
The explanatory variable is hours of sleep and the response variable is the
body mass index.
(a)
Achievement
Both variables are
identified.
Hours of sleep is measured by the time the child was in bed. This would not
be the same as hours of sleep (time in bed would be longer than hours of
sleep). The measuring of time in bed would rely on parents logging the time
in bed themselves or being asked to recall it at a later date by the researcher,
which means there may be inaccuracy in this measurement. To measure
hours of sleep accurately would require watching the child throughout the
night, which is unpractical.
Merit
Both variables are
identified
Both variables are
identified
AND
AND
one potential issue with
the measurement of the
explanatory variable is
stated.
one potential issue
with the measurement
of the explanatory
variable is discussed,
supported with
statistical and
contextual reasoning.
Note: Either number of hours sleep or time in bed is acceptable for the
explanatory variable.
(b)
(c)
(d)
This is an observational study as the number of hours of sleep was not
controlled by the researchers – it would not be possible for the researcher to
dictate how many hours of sleep each child had per night. The children were
observed as a group over a period of time. Because this is an observational
study, causal relationship claims should not be made, or should at least be
made with extreme care.
Identification that this
study is an
observational study.
The word ‘equals’ in the statement is misleading as it implies a causal
relationship, eg the word ‘equals’ could be replaced with the words ‘leads to’
or ‘means’.
Identification of a causal
claim in statement.
The amount of exercise that the child had during childhood could also be
associated with the adult body mass index, eg more hours of exercise during
childhood could cause lower adult body mass index.
Two related lifestyle
factors are identified.
The amount of food eaten by the participants as adults could also be
associated with the adult body mass index, eg larger amounts of food eaten
could increase adult body mass index.
Not Achieved
NØ
No response; no relevant evidence.
Excellence
Explanation given of
why this is an
observational study
(which links to the
context) and the
limitation this has on
making a causal
relationship claim.
Two related lifestyle
factors are identified and
partial explanations are
given as to how these
might affect the adult
body mass index.
Two related lifestyle
factors are identified
and full explanations
are given as to how
these might affect the
adult body mass
index.
NCEA Level 3 Mathematics and Statistics 91584 (3.12) — page 6 of 6
N1
Candidate gives a partial response to ONE part of the question.
N2
Candidate gives partial responses to TWO parts of the question.
A3
Candidate gives ONE opportunity from the Achievement criteria.
A4
Candidate gives TWO opportunities from the Achievement criteria.
M5
Candidate gives ONE opportunity from the Merit criteria.
M6
Candidate gives TWO opportunities from the Merit criteria.
E7
Candidate gives ONE opportunity from the Excellence criteria.
E8
Candidate gives TWO opportunities from the Excellence criteria.
Achievement
Merit
Excellence
Note: At any stage, for any question, a candidate may have offered other information that is statistically valid and can be used to replace points raised in the schedule.