A Mathematical Skills Fundamental for the Pulp and Paper Industry
STATISTICS AND PROBABILITIES 4
Assessor Guide
NQF Level 4
Credits: 5
Unit Standard 9015
Compiled by:
Amanda Gilfillan
Johan Els
for
FIETA
Sparrow Research and Industrial Consultants © July 2005
ASSESSOR GUIDE
US 9015: Apply knowledge of statistics and probability to critically interrogate and effectively
communicate findings on life related problems
Candidate’s name
Candidate’s current job title
Candidate’s contact number
(Work)
(Cell)
Assessor’s name
Assessor’s current job title
Assessor’s contact number
(Work)
(Cell)
Date
Title of Unit Standard
Apply knowledge of statistics and probability to critically interrogate
and effectively communicate findings on life related problems
Unit Standard Number:
9015
NQF Level
4
Credits
5
RECOGNITION OF PRIOR LEARNING
Use the assignments indicated by the  icon for RPL
ASSESSMENT SCHEDULE
Assessment
Method
Theoretical
Written
Theoretical
Oral
Practical
Feedback
session
Date
Time
Location
Resources
required
Candidate’s signature
Assessor’s signature
Manager’s signature
Assessor Guide
US 9015 Statistics & Probabilities
2
Sparrow Research and Industrial Consultants © July 2005
US: 9015: ASSESSMENT PLAN
Unit Standard Number: 9015
Unit Standard Title: Apply knowledge of statistics and probability to critically interrogate and effectively communicate
findings on life related problems
NQF Level: 4
Credits: 5
No
Specific
outcomes
T1
Use theoretical and experimental probabilities to develop models
1
SO2 AC1-4
2
Learning Outcomes
Assess
Time
Study Resources
Required by
Learner
Study
Time
Assessment
Instruments
Calculate probabilities
Theoretical Written
Learning Materials
Memorandum
SO2 AC1
Draw a venn diagram and
calculate probabilities
Theoretical –
Written /
diagram
Learning Materials
Memorandum
3
SO2 AC1
Calculate probabilities
Theoretical Written
Learning Materials
Memorandum
4
SO2 AC1
Draw a probability tree
Theoretical Written
Learning Materials
Memorandum
T2
Critically interrogate and use probability and statistical models
5
SO3 AC1-5
Learning Materials
Memorandum
6
SO3 AC1-5
Learning Materials
Memorandum
T3
Critique and use techniques for collecting, organising and representing data
Assessor Guide
The learner will have a
basic understanding of
probability
Assessment
Methods
Assessment
Critically interrogate and
effectively communicate
results
Find the size of a sample
Theoretical Written
Draw a graph
Theoretical –
Written / graph
US 9015 Statistics & Probabilities 4

3
Sparrow Research and Industrial Consultants © July 2005
No
Specific
outcomes
7
SO1 AC1-9
Look at samples in terms
of size and
representativeness
Interpret a graph
Theoretical Written
8
SO1 AC1-9
Understand what a
normal distribution is
Calculate averages and
draw a graph
Theoretical –
Written / graph
Assessor Guide
Learning Outcomes
Assessment
Methods
Assessment
Assess
Time
US 9015 Statistics & Probabilities 4
Study Resources
Required by
Learner
Study
Time
Assessment
Instruments
Learning Materials
Memorandum
Learning Materials
Memorandum

4
Sparrow Research and Industrial Consultants © July 2005
ASSESSMENT MEMORANDUM - US: 9013
Unit standard: Apply knowledge of statistics and probability to critically interrogate and
effectively communicate findings on life related problems
US: 9015
TEST MEMORANDUM
Question 1
NQF level: 4
Credits: 5
Time
Answer
Complete the following questions
1
An ordinary die is thrown. What is the probability that the number thrown is:
a
less than 5
less than 5 is 1, 2, 3 and 4 so
b
more than 4
more than 4 is 5 and 6 so
c
2 1

6 3
a factor of 4
factors of 4 are 1, 2, and 4 so
Assessor Guide
4 2

6 3
2
2
3 1

6 3
US 9015 Statistics & Probabilities 4
2
5
Sparrow Research and Industrial Consultants © July 2005
TEST MEMORANDUM
Question 2
2
Time
Answer
A group of 60 people were asked if they had watched the cricket game or the news during the past week on TV. Thirty-five said they watched cricket, 20
said they hade watched the news and 14 said they had watched neither.
2.1
Draw a Venn-diagram
2.2
What is the probability that a person chosen at random watched:
a
both
b
soccer but not the news
c
either soccer or the news?
Assessor Guide
4
P(C ∩ N ) =
9
3

60 20
3
P(C U N’) 
26 13

60 30
3
26  9  11 46 23


60
60 30
3
P(C U N) =
US 9015 Statistics & Probabilities 4
6
Sparrow Research and Industrial Consultants © July 2005
TEST MEMORANDUM
Question 3
Time
Answer
3
If E and F are two events such that P(E) = ½; P(F) = ⅛, and P(E ∩ F) =
a
P(E U F)
1 ,
16
find
P(E U F) = P(E)+P(F) – P(E ∩ F)
4
1 1 1
 
2 8 16
8  2 1

16
9

16

b
P(E U F)’
P(E U F)’
 1

Assessor Guide
2
9
16
7
16
US 9015 Statistics & Probabilities 4
7
Sparrow Research and Industrial Consultants © July 2005
TEST MEMORANDUM
Question 4

Time
Answer
4
The probability that Jeffrey is late for college on any one day is 18% and is independent of whether he was late on the previous day.
4.1
Draw a probability tree
8
Outcome
Probability
Late-Late
0,18 x 0,18
0,0324
Late-On time
0,18 x 0,82
0,1476
On time-Late
0,82 x 0,18
0,1476
On time-On time
0,82 x 0,82
0,6724
4.2
Find the probability that Jeffrey is
a
late Monday and on- time on Tuesday
From the table, P(late – on time) = 0,1476 = 14,76%
2
b
arrives on time on at least one of these
days
From the table, option 2, 3 and 4 indicates on time at least one of the days, P(on time at least one of
the days) = 0,1476 + 0,1476 + 0,6724 = 0,9676 (96,76%)
2
Assessor Guide
US 9015 Statistics & Probabilities 4
8
Sparrow Research and Industrial Consultants © July 2005
TEST MEMORANDUM
Time
Question 5
5
Answer
Find the size of sample required at the 90% confidence limit for the following distribution and intervals.  = 11 cm , interval 8 cm
Normal distribution Table
z
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
1.6
0.94520
0.94630
0.94738
0.94845
0.94950
0.95053
0.95154
0.95254
0.95352
0.95449
1.7
0.95543
0.95637
0.95728
0.95813
0.95907
0.95994
0.96080
0.96164
0.96246
0.96327
1.8
0.96407
0.96485
0.96562
0.96638
0.96712
0.96784
0.96856
0.96926
0.96995
0.97062
1.9
0.97128
0.97193
0.97257
0.97320
0.97381
0.97441
0.97500
0.97558
0.97615
0.97670
1,65 11
8
n
1,65 11
 n
8
n  2,26875
n  5,14
Assessor Guide
US 9015 Statistics & Probabilities 4
n  6
9
Sparrow Research and Industrial Consultants © July 2005
TEST MEMORANDUM
Time
Question 6
6
Answer
The table below shows that 83% of all HIV diagnoses have been in people aged 20-45 years old. On average, women and girls have been diagnosed at a
slightly younger age than men and boys – 47% of females were under 30 years old compared to only 34% of males
South African HIV / AIDS diagnosis up to end March 2005
Age
Group
(years)
Male
HIV
No
Female
AIDS
%
No
HIV
%
No
Total
AIDS
%
No
HIV*
%
No
AIDS
%
No
%
0-4
471
1
224
1
432
2
223
6
905
1
447
2
5-9
262
1
73
0
157
1
52
1
421
1
125
1
10-14
232
0
49
0
76
0
26
1
309
0
75
0
15-19
949
2
71
0
647
3
34
1
1598
2
105
0
20-24
5152
10
550
3
2798
14
230
6
7955
11
780
4
25-29
10102
20
2416
14
4966
26
789
21
15071
21
3205
15
30-34
11479
22
3876
22
4608
24
941
25
16088
23
4817
23
35-39
9241
18
3620
21
2818
15
630
17
12060
17
4250
20
40-44
5726
11
2689
15
1367
7
363
10
7094
10
3052
14
45-49
3308
6
1779
10
676
3
178
5
3984
6
1957
9
50-54
1985
4
1046
6
386
2
92
2
2371
3
1138
5
55-59
1096
2
614
3
213
1
68
2
1309
2
682
3
60-64
634
1
333
2
101
1
44
1
735
1
377
2
65+
408
1
238
1
72
0
25
1
481
1
263
1
51045
100
17578
100
19317
100
3695
100
70381
100
21273
100
Total
Unknown
Assessor Guide
332
5
48
US 9015 Statistics & Probabilities 4
2
402
7
10
Sparrow Research and Industrial Consultants © July 2005
TEST MEMORANDUM
Time
Notes:
* Due to rounding to the nearest whole number, percentage columns may appear not to total 100%
* Includes 41 HIV infections with sex not stated on the report
6.1
Use the values in the total for Aids and
draw an appropriate graph to show the
information for the different age groups
7
AIDS Diagnoses in Different Age Groups
6000
Persons (number)
5000
4000
3000
2000
1000
0
0-4
5-9
10-14
15-19
20-24
25-29
30-34
35-39
40-44
45-49
50-54
55-59
60-64
65+
Age Group
6.2
Discuss the findings of the graph in a
paragraph
Assessor Guide
The graph has a normal distribution pattern except for the very young ages. The high number of 0 to 4
years indicates the children born with Aids. It seams that they have already died by the age of 4. The
highest number of Aids victims is from 30 to 39.
US 9015 Statistics & Probabilities 4
3
11
Sparrow Research and Industrial Consultants © July 2005
TEST MEMORANDUM
Time
Question 7
7
Answer
Study the following HIV graph for Namibia
HIV Infection in Namibia
25
22
20
18.3
17.4
HIV Infection (%)
15.4
15
10
5.4
5
4.2
0
1992
1994
1996
1998
2000
2002
7.1
What is the increase (%) between 1994
and 1996
10%
2
7.2
According to the graph do you think that
the health program is preventing the
spread of HIV if the Percentage in 2004
was 22,5 %
The percentage increase has slowed down considerably – this indicates that the health program is
starting to work
2
Assessor Guide
US 9015 Statistics & Probabilities 4
12
Sparrow Research and Industrial Consultants © July 2005
TEST MEMORANDUM
Question 8
8
Time
Answer
Farm
workers
Total
Remuneration
Eastern Cape
33 718
R 32 935 100
Free State
57 607
R 58 088 800
Gauteng
20 815
R 34 462 900
KwaZulu-Natal
75 799
R 76 343 900
Limpopo
62 635
R 52 539 000
Mpumalanga
61 603
R 59 961 700
312 177
R 314 331 400
Province
TOTAL
8.1
Calculate the average salary of the
workers in Gauteng and in Eastern Cape
Average salary Gauteng =
R 34 462 900
 R 1 655,67
20 815
4
So the average salary for a farm worker in Gauteng is R 1 655,67 per month
Average salary Eastern Cape =
R 32 935 100
 R 976,78
33 718
So the average farm worker salary for Eastern Cape is R 976,78 per month
Assessor Guide
US 9015 Statistics & Probabilities 4
13
Sparrow Research and Industrial Consultants © July 2005
TEST MEMORANDUM
Question 8
8.2
Time
Answer
Draw a graph to show the
difference in salary per
person in the different
provinces
5
Farm Worker Salaries
1800
1600
Monthly Salary (R)
1400
1200
1000
800
600
400
200
0
Eastern Cape
Free State
Gauteng
KwaZulu-Natal
Limpopo
Mpumalanga
Province
Total
Assessor Guide
65
US 9015 Statistics & Probabilities 4
14
Sparrow Research and Industrial Consultants © July 2005
INFORMAL ASSESSMENT: CRITICAL CROSSFIELD OUTCOMES
Critical outcomes
Not yet competent
(NC)
Partially competent
(PC)
Competent
(C)
Outstandingly competent
(OC)
Identify and solve problems
Learner experiences difficulty
with problem solving activities
involving statistics and
probabilities
Learner can identify and solve
only basic problems that
require recall of knowledge.
Learner is able to recall,
interpret and apply knowledge
to solve a variety of problems
involving statistics and
probabilities
Learner is able to interpret,
apply, analyse, synthesise and
evaluate knowledge to solve
problems involving statistics
and probabilities
Communicate effectively
Learner cannot use everyday
and mathematical language to
describe properties, processes
and problem solving methods
Learner's use of everyday
language and mathematical
language to describe
properties, processes and
problem solving methods is
very limited
Learner has good command of
everyday and mathematical
language to describe
properties, processes and
problem solving methods.
Confident.
Learner communicates
confidently and clearly, using
everyday language and
mathematical language to
describe properties, processes
and problem solving methods.
Very confident.
Collect, analyse, organise and
critically evaluate information
Learner experiences problems
with processing information.
Learner is able to collect and
organise information, but
cannot critically evaluate it.
Learner can collect, analyse,
organise and evaluate
information.
Learner’s ability to collect,
analyse, organise and critically
evaluate information is
outstanding.
Use mathematics
Learner cannot use
mathematics to describe and
represent realistic and abstract
situations or to solve relevant
problems
Learner's use of mathematics
to describe and represent
realistic and abstract situations
and to solve relevant problems
is very limited.
Learner
Learner is able to use
mathematics sufficiently to
describe and represent realistic
as well as abstract situations
and to solve relevant problems.
Assessor
Learner uses mathematics
skilfully in describing and
representing realistic situations
and solving relevant problems
Comments
Assessor Guide
US 9015 Statistics & Probabilities 4
Date
15
Sparrow Research and Industrial Consultants © July 2005
ASSESSOR REPORT – US 9015
Candidate Name:
Unit Standard Number
Candidate ID:
Date:
Unit Standard Title: Apply knowledge of statistics
and probability to critically interrogate and effectively
communicate findings on life related problems
Candidate Contact Number:
Level: 4
Assessor Name:
Credits: 5
Tasks
Marks
1.a
Calculate probabilities
2
1.b
Calculate probabilities
2
1.c
Calculate probabilities
2
2.1
Draw a Venn-diagram
4
2.2a
Calculate probabilities
3
2.2b
Calculate probabilities
3
2.2c
Calculate probabilities
3
3a
Calculate P(E U F)
4
3b
Calculate P(E U F)’
2
4.1
Draw a probability tree
8
4.2a
Calculate probabilities
2
4.2b
Calculate probabilities
2
Find the size of sample required
4
6.1
Draw a graph
7
6.2
Discuss the findings of the graph
3
7.1
Increase (%) between 1994 and 1996
2
7.2
Health program
2
8.1
Calculate average salary
4
8.2
Graph
6
5
Total
Assessor Guide
9015
Competency
Comment
65
US 9015 Statistics & Probabilities
16
Sparrow Research and Industrial Consultants © July 2005
MARKS
DESCRIPTORS
0-25
0-39%
Not yet competent (NC)
26-32
40-49%
Partially competent (PC)
33-48
50-74%
Competent (C)
49-65
75-100%
Outstandingly competent
(OC)
Assessor’s
signature:
Candidate’s
signature:
Date:
Date:
I _______________________________________________________________ (assessor) declare that the
learner _______________________________________________________________________________
is not yet competent
nearly competent
competent
has outstanding
competency
on the assessment criteria of US 9015: Apply knowledge of statistics and probability to critically interrogate
and effectively communicate findings on life related problems
Action required:
I ________________________________________________ (learner) declare that I have received
feedback and been informed of my overall competence for the criteria within US 114246
Candidate’s signature
Assessors’ signatures
Moderator’s signature
Assessor Guide
US 9015 Statistics & Probabilities
17