ncea level 2 mathematics and statistics (91267) 2015 — page 3 of 3 assessment schedule – 2015 mathematics and statistics: apply probability

NCEA Level 2 Mathematics and Statistics (91267) 2015 — page 3 of 3
Assessment Schedule – 2015
Mathematics and Statistics: Apply probability methods in solving
problems (91267)
Evidence Statement
One
Expected Coverage
Achievement (u)
Merit (r)
Excellence (t)
(a)(i)
P(0 < Z < 0.75) = 0.2734
Probability found.
(ii)
P(0 < Z < z) = 0.4  z = 1.2815
= 1.2815
x = 44.25
z = 1.2815
found.
x = 44.25
found.
(iii)
P(0 < Z < z) = 0.45  z = 1.645
= 1.645
 = 3.647
= 10.88
z = 1.645
found.
 = 3.647
found.
10.88 or 11
extra doctors.
(b)(i)
P(0 < X < 90)
P(-3 < Z < 1.5) = 0.932
0.932 found.
(ii)
p = 1 – = 0.75
Proportion found.
(iii)
Possible valid comparative statements that may relate to the points
listed below.
Shape:
Graph 1: Not symmetrical; skewed to the left; two peaks; bunching of
values to the right.
Graph 2: As above but one peak.
Centre:
Graph 1: Mode at 75 – 90 seconds; Median at 60 – 75 seconds; mean to
the right of centre.
Graph 2: As above.
Graph 1: Range of about105 seconds
Graph 2: Range of 2 minutes (120 seconds)
Proportions:
Proportions are similar except for 30 – 60 seconds.
One valid comment about each of two aspects of shape, centre, and
Two valid comments, at least one comparative covering each of two
aspects of shape, centre, and spread. There must be numerical support
for at least one comment.
As for Merit except at least two comparative comments covering each of
two aspects of shape, centre, and spread.
There must also be some comparisons of the class proportions,
specifically noting the 30-60 second intervals in the first graph.

N1
N2
A3
A4
M5
M6
E7
E8
No response; no relevant evidence.
A valid attempt at one question
1 of u
2 of u
3 of u
1 of r
2 of r
1 of t
2 of t
Two
Expected Coverage
Achievement(u)
Merit(r)
Excellence(t)
(a)(i)
= 0.8
Proportion found.
(ii)
= 0.11
Proportion found.
(iii)
= 0.5687
Expected No. = 52500  0.5687
= 29855
Expected no. found.
(iv)
Risk of Year 12 failing = = 0.087
Risk of Year 13 failing = = 0.238
Relative risk = = 2.74
This is not very close to 4, and hence the claim is not justified.
One risk found.
Relative risk found.
Comparison
with 4 and correct conclusion.
(b)(i)
5 subjects
6 subjects
Total
Passed
626
574
1200
Failed
192
108
300
Total
818
682
1500
= 0.3827
Proportion found.
(ii)
P(passed with 6 subjects) = = 0.8416
P(passed with 5 subjects) = = 0.7653
= 1.0997
Hence 10% more likely to pass if taking 6 subjects BUT this is
deceptive, as candidates with more ability are likely to be taking 6
subjects.
There could also be comment on the representativeness of the sample.
Both risks found.
Relative risk found.
Interpretation of risks and a realistic argument presented to support
or not support taking 6 subjects.

N1
N2
A3
A4
M5
M6
E7
E8
No response; no relevant evidence.
A valid attempt at one question
1 of u
2 of u
3 of u
1 of r
2 of r
1 of t
2 of t
Three
Expected Coverage
Achievement (u)
Merit (r)
Excellence (t)
(a)(i)
p = 0.55  0.7 = 0.385
Probability found.
(ii)
P(Female) = 0.45
p = 0.45  0.8  0.35 = 0.126
Probability found.
(iii)
p = (0.55  0.3  0.2) + (0.45  0.8  0.65)
= 0.033 + 0.234
= 0.267 or 26.7%
Either 0.033 or
0.234 calculated.
Probability found.
(iv)
N = 550  (0.55  0.3  0.2)
= 18.15
18.15.
Expected numbers do not need to be integer values, but accept 18.
(v)
P(Male and retained) must be
0.033  10 = 0.33
Hence P(Female and sold) = 0.45 – 0.33 = 0.12
0.45  p = 0.12
p = = 0.267
0.12 found.
Probability found.
(b)(i)
p = 0.3 + 0.4 + 0.15 = 0.85
Probability found.
(ii)
Proportion of black pairs and three eggs
= (0.05  0.15)
= 0.0075
Proportion of nests with three eggs
= (0.75  0.4) + (0.2  0.35) + (0.05  0.15)
= 0.3775
p = = 0.0198
which is very close to or 0.02, so the researcher’s claim is
justified.
0.0075 found.
0.3775 found.
Proportions found and compared.

N1
N2
A3
A4
M5
M6
E7
E8
No response; no relevant evidence.
A valid attempt at one question
1 of u
2 of u
3 of u
1 of r
2 of r
1 of t
2 of t
Cut Scores
----------
Not Achieved
Achievement
Achievement with Merit
Achievement with Excellence
0 – 8
9 – 14
15 – 19
20 – 24