Study Guide
Department of Statistics
Faculty of Natural and Agricultural Sciences
Mathematical Statistics
WST 121
Table of Contents
Contents
1
Introduction ................................................................................................................. 1
1.1
1.2
1.3
1.3.1
2.
Welcome.......................................................................................................................... 1
Educational approach ....................................................................................................... 1
Responsibilities of the student ......................................................................................... 2
Semester planner WST121 ................................................................................................................ 4
Administrative information.......................................................................................... 5
2.1
2.2
2.3
Contact details ................................................................................................................. 5
Timetable ......................................................................................................................... 6
Study material and purchases........................................................................................... 7
3. Assessments .................................................................................................................. 8
3.1
3.2
3.3
Assessment plan .............................................................................................................. 8
Semester mark calculation ............................................................................................... 9
Assessment policy ............................................................................................................ 9
3.3.1 Procedure to be followed when a semester test cannot be written ................................................... 10
3.4
3.5
3.6
3.6.1
3.6.2
4
Plagiarism ...................................................................................................................... 11
Programme/Departmental/Module rules, requirements and guidelines ........................ 12
Code of conduct ............................................................................................................. 13
Communication via email (wst121stats@gmail.com) ...................................................................... 13
Compliments and complaints .......................................................................................................... 13
Module information................................................................................................... 14
4.1
4.2
4.3
4.4
4.5
4.6
4.7
Purpose of the module ................................................................................................... 14
Module outcomes .......................................................................................................... 14
Articulation with other modules in the programme ........................................................ 14
Module structure ........................................................................................................... 15
Learning presumed to be in place ................................................................................... 15
Credit map and notional hours ....................................................................................... 15
Units .............................................................................................................................. 16
Unit: Statistics and Sampling Distributions ...........................................................................................16
Unit: Point estimation .........................................................................................................................17
Unit: Statistical intervals based on a single sample ...............................................................................17
Unit: Tests of hypotheses based on a single sample..............................................................................18
Unit: Inferences based on two samples ................................................................................................19
Unit: Analysis of Variance ....................................................................................................................20
Unit: Regression and correlation .......................................................................................................... 21
Unit: Goodness-of-fit tests and categorical data analysis ......................................................................22
Unit: Alternative approaches to inference ............................................................................................23
5 Support services ............................................................................................................ 24
E-learning support ..................................................................................................................... 24
Other support services .............................................................................................................. 24
Annexure 1 ....................................................................................................................... 25
1
Introduction
1.1 Welcome
A hearty welcome to all students who are doing Mathematical Statistics 121 – the last stretch of first
year Mathematical Statistics. This is your second step towards the “sexiest career of the 21st
century” – whether it is called statistics, data analysis or data science. But heed some wise words:
“Don’t let what you cannot do interfere with what you can do.” – John Wooden
“There are no shortcuts to any place worth going.” – Beverly Sills
“There is no substitute for hard work.” – Thomas Edison
You’ll be in the experienced hands of lecturers:
Dr Seite Makgai
(Course coordinator)
Chapters 9, 10, 11, 12
Dr Najmeh Rad
Chapters 6, 7, 8 & 13, 14
1.2 Educational approach
Lecturers should be seen as facilitators rather than conveyors of knowledge. It is your responsibility to
engage, utilize and capitalize on all learning opportunities. Quality instruction requires students to
diligently prepare for lectures, tutorials and practicals, as this enables teaching to build actively on
common prior knowledge.
We will be functioning in a hybrid environment consisting of both online and in-person activities.
Details of each week’s activities will be posted weekly on ClickUP.
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© 2023 University of Pretoria: Adapted for NAS
ClickUP Roadmap:
•
Administrative matters, which includes Announcements, Lecturers & Tutor
information, will be posted under Admin.
•
Throughout the course of the semester, information will be regularly posted
under Announcements. These announcements will include test information
(scope, arrangements, etc.), various other updates, etc. Please check for
announcements regularly (at least twice a day).
A weekly schedule detailing what needs to be covered on a weekly basis will
be posted under the Weekly Schedule page.
Under the Study Material division, you will find:
o The study guide is included under this division. Please read this study guide
thoroughly.
o Information regarding the textbook and the instructions to access it
via the library will also be placed under the Study Material division.
o Lecture material (Some additional material-if necessary will be posted
under this page. Please note that not all Chapters 6-14 will have
additional material.)
o Assessment Material (memorandums of semester tests, formula
sheets and tables) will be also be posted under the Study Material
division.
•
•
•
•
•
Under the Tutorials & Practicals division you will find:
o Tutorials (Question worksheets, tutorial assignment submission links and memos)
o Practicals (Question worksheets, practical assignment submission links and memos)
Your marks for tests, tutorial and practical assignments, as well as any other
assessments can be found under My Grades.
Under the Online lectures & interaction division you will find:
o Lecturers and tutors consultation hours (these times will be announced).
o Discussion board forums to post any/all of your content related questions.
o Class collaborate for any live online lectures. The recordings, if
available, can also be found under the Class Collaborate page.
Content will be added to these sections during the course of the semester.
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1.3 Responsibilities of the student
⇒
Start your studies with the determination to make a success of it. Tertiary studies require a lot
of sacrifice, perseverance and hard work and all this still does not mean you cannot fail. But remember
where there is a will there is a way.
⇒
The tempo of the semester is extremely fast, so falling behind could be detrimental to your
performance. Construct a study schedule, and keep to it! See to it that you revise all your subjects at
least once a week and that you understand all the work. If you fall behind, try to catch up within one
week. Guard against the mistake of concentrating on one subject during test periods.
⇒
Work through all the lectures as indicated by the weekly plan posted. Discipline yourself to
keep to the schedule as posted every week. Concentrate on the explanations and use the terminology
and notation of the subject. You need to learn the statistical language in order to communicate the
concepts!
⇒
Take immediate steps if you see that you are not making progress with your studies or if you
are losing interest. If a problem arises, deal with it as soon as possible. Talk to someone who can help
you, and remember no one can help you if they do not know about your problem. The lecturers and
tutors and student advisors are available online to see to your needs.
⇒
The subject Mathematical Statistics, as the name indicates, is more mathematical in nature.
All the new terminology is based on the old, which has to be known. Do your best to understand the
work that is done each day. Post on the discussion board if something is not clear to you.
⇒
Mathematical Statistics is a study subject that cannot be mastered within a day or two. During
the preparations for any test, it is important that you write out the definitions, concepts, propositions
and proofs related to the scope of the test. In this way, you improve your concentration and thus will
know your work sooner.
⇒
See to it that you understand the subject in its entirety. Schematic representations and tables
of summations can help you to achieve this. This takes a lot of time but is always worth the effort
when it comes to revising the work.
⇒
Always be proud of your work. Keep it systematic and neat. If something does not make any
sense, do it over and do it correctly. Do not settle for anything less than the best.
⇒
Don't be an academic wreck! Vary you study time by doing sport or any other recreational
activity. But do not over indulge in the last two. Remember you enrolled at university to study and
get your degree.
To end with: successful studies depend on you being MOTIVATED. If a course in this department is
included in your curriculum, you can accept that there is a good reason why this is so.
On the next page find a calendar which gives an indication of weekly activities. On ClickUP an updated
list of activities is published weekly and stored afterwards in case you missed something or need a
reminder of what was done during a particular week.
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© 2023 University of Pretoria: Adapted for NAS
Semester planner WST121
Week
(lectures
per week)
Date*
General**
Content
1 (4)
24-28 July
6.1-6.2
2 (4)
31 Jul-4 Aug
6.3-6.4
3 (3)
4 (4)
7-11 Aug
(9=PH)
14-18 Aug
5 (4)
21-25 Aug
6 (4)
Tutorial
Practical
Tutorial
Test
(During the
Tutorial
Session)
Lecturer in
charge***
0
N
1
1
N
7.1
1
1
8.1-8.2
2
2
N
8.3-8.4
3
3
N
28 Aug-1 Sep
9.1-9.2
3
3
S
7 (4)
4-8 Sep
9.3-9.4
4
4
S
8 (2)
11-15 Sep
(Mon=WedT,
Tue=MonT,
13=SD)
18-22 Sep
9 (0)
ST1
(Mon 21 Aug)
Tut test 2
moved
to this week
1
N
9.4
Recess (14-23 Sept)
Tut & Prac 4 due on
18th Sept
10 (3)
25-29 Sep
10.1-10.2
S
(25=PH)
11 (4)
2-6 Oct
12 (4)
9-13 Oct
13 (4)
16-20 Oct
14 (4)
15 (4)
16 (4)
ST2
(Wed 4 Oct)
5
5
S
11.1, 11.3,
11.4
5
5
S
6
6
*Trial Practical
12.1-12.3
Test/Assignment
Practical Test
(Fri 27 Nov)
30 Oct-3 Nov
ST3
(Mon 30 Oct)
23-27 Oct
6-10 Nov
10.3-10.5
Sick Practical Test
(Wed 1 Nov)
No tests this weekonly Tutorial 8
(optional)
3
S
12.4-12.6
7
S
13.1-13.3
7
N
14.1-14.2
8
N
15 Nov (Preliminary date)
Exam 15h00-18h00
* PH=Public Holiday, SD=Spring Day, MonT= Monday Timetable, WedT= Wednesday Timetable
** All dates are provisional and subject to change.
*** Lecturer in charge: N=Dr Najmeh Rad, S=Dr Seite Makgai
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2.
Administrative information
2.1
Contact details
Lecturers and Tutors
Name
Office
Contact
Module coordinator
& Lecturer
Dr Seite Makgai
IT 6-12
- wst121stats@gmail.com
Lecturer
Dr Najmeh Rad
IT 5-27
Ms Ramadimetje Leshilo
Tutors
Mr Lizo Sanqela
Graduate
Centre 1-66
- During Tutorial and Practical
sessions and via the Discussion
boards.
Other Support
Building and
room number
Name
Faculty Student
Advisor*
Mpho Mmadi
Subject
librarian
Katlego Aphane
Maths
Building,
Room 1-29
Merensky
library, level
5, office 5-4
Office
Telephone
number
Email address
012 420 6740
mpho.mmadi@up.ac.za
012 420 4791
katlego.aphane@up.ac.za
* Your Faculty Student Advisor can advise you on goal-setting, adjustment to
university life, time management, study methods, stress management and career
exploration. Book an individual consultation or attend a workshop. For other
support services see Section 5. Click HERE for contact details for other NAS FSAs.
For more info on the Department of Statistics, visit us on the internet:
https://www.up.ac.za/statistics
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2.2
Timetable
Scheduled session
Day
Time
Venue
Lecture 1
Monday
14:30 -15:20
Thuto 1-2
Lecture 2
Wednesday
07:30-8:20
Pre-recorded video i.e. no live
lecture
Lecture 3
Thursday
14:30-15:20
Thuto 1-2
Lecture 4
Friday
10:30-11:20
Thuto 1-2
Tutorial group 1
Monday
10:30-12:20
Te Water hall
Tutorial group 2
Wednesday
14:30-16:20
IT 2-23
Practicals
Practicals
Monday
Wednesday
12:30-13:20
16:30-17:20
Informatorium
Fill the labs in this order:
1. Orange
2. Blue 3
3. Blue 2
4. Blue 1
Informatorium
Fill the labs in this order:
1. Red
2. Grey
3. Purple
4. Blue 1
5. Blue 2
The course consists of:
●
FOUR lecture periods per week (3 in-person, and 1 pre-recorded lecture on Wednesdays).
The lectures will be marked as Lecture 1, Lecture 2, Lecture 3 and Lecture 4 in the weekly
schedule. The lecturer will indicate the material that will be covered in this session in the
Weekly Work Scheduled.
●
ONE tutorial session per week (in-person). A worksheet with individual assignments will be
published on ClickUP weekly. Your completed assignment and answers must be submitted on
ClickUP. Memos will be published on ClickUP. You will have one or two weeks to complete
your tutorial assignment and submit your answers on ClickUP. The submission link opens on
Monday at 7am of the first week of the assignment and closes at 7am on Monday of the week
when the next assignment is released. The purpose of these assignments is to give you better
insight into the subject-matter treated in class and a better understanding of applications of
the subject in order to improve self-tuition.
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●
ONE practical session per week (in-person). You must fill the labs according to the order
given in the study guide. In the practical sessions, using the programming language R/RStudio,
you will be given the opportunity to practically apply theoretical concepts covered during
lectures. A worksheet with a practical assignment will be published on ClickUP. Unless
otherwise stated in your assignment instructions, you will have two submissions for each
Practical. The first is the submission of your R script. You will be able to submit your script
as many times as you like and will receive immediate feedback on this. The aim of this is to
give you the opportunity to correct your code and make coding less intimidating. Once you
are satisfied that your code is correct, you can move on to the results/interpretation
submission. You will have only one opportunity for this submission and will not receive
immediate feedback. Your R script as well as results/interpretations must be uploaded to
Gradescope. To access Gradescope, please use the link provided on ClickUP → Practicals.
You will have either one or two weeks to complete your practical and submit your code and
results on ClickUP. The submission link opens on Monday at 7am of the first week of the
assignment and closes at 7am on Monday of the week when the new assignment is released.
Sick notes are NOT accepted for missed
•
•
•
tutorial/practical submissions,
ClickUP assessments or
tutorial tests
and there will not be additional opportunities or supplementary assessments to make up for them or
to improve marks. In order to accommodate cases where a student may miss an assessment, only
selected assessments are counted towards the semester mark: Only the best 6 tutorials, the best 2
tutorial tests and the best 4 practical assignments (excluding Practical 0, and including the Trial Prac
test) will count towards your semester mark. Note that this does not apply to the semester tests and
practical test. ALL semester tests and the practical test are compulsory and count towards your
semester mark. If you miss any semester test or the practical test, you must apply for the relevant sick
test. (See section 3.3.1)
2.3
Study material and purchases
Prescribed Book (e-book) (DB):
Title: Modern Mathematical Statistics with Applications
Authors: Devore, J.L.,Berk,K. N.
Edition: 2nd (2012)
The textbook is available on the Library website. Only selected parts of the textbook will be covered
in this module. See Study Themes and Units in this guide for detail.
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All students are expected to use their own calculators. Scientific calculators with facilities for
regression and correlation are recommended. No programmable calculators allowed.
R and RStudio should be downloaded and installed on your computer. Students who are unable to
install this software can make use of Google Collaboratory. Details of these can be found on ClickUP
in Course Materials → Practicals → Practical 0.
3. Assessments
3.1
Key
T
Assessment plan
Assessment type
Assessment
venue/platfor
m
About
Due date
Weight
Selected Mondays at
7am***
Selected Mondays at
7am***
21 Aug*
4 Oct*
30 Oct*
27 Oct*
37.5 or
37.5 or
37.5 or
10
Tutorials
ClickUP
P
Practicals
ClickUP
ST1
ST2
ST3
PT
Sick
PT
Semester test 1
Semester test 2
Semester test 3
Practical test
Written
Written
Written
Lab
Exercises based on each
chapter
Practical application of theory
covered each chapter
Chapter 6, 7**
Chapter 8, 9**
Chapter 10, 11**
All Practicals
SICK Practical Test
Lab
All Practicals
1 Nov*
10
TT
Tutorial Tests
Written
Tutorial test 1: 6.1-6.2**
Tutorial test 2: 8**
Tutorial test 3: 10**
During the tutorial
Classes*
5
5
5
100
* Preliminary dates, subject to change ** Subject to change
*** See Section 2.2 for details
****Weight is according to the number of semester tests missed
●
Semester Tests: Three semester tests will be written. All tests will be written on campus – all 3
tests are compulsory. The best 2 out of the 3 semester tests will be taken.
●
Practical test: A final practical test (in R) based on the practical assignments will be written at the
end of the semester and submitted to Gradescope.
●
Tutorial tests: Three tutorial tests will be written during the Monday/Wednesday tutorial sessions.
Test memos/explanations will be published on ClickUP. If you have queries about marks or the
allocation of marks first check the memo and send queries within the specified time. Please send an
email to wst121stats@gmail.com along with a clear explanation of your query. Your query will be
checked and your mark will be changed if necessary. You will have until one week after the marks are
published to send queries. After this time no marks will be adjusted.
Please note that due to capacity and time of the assessments, we will no longer have a sick semester
test, but we will consider the best 2 out of the 3 semester tests. If a student misses any one of the
semester tests, he/she must write the other semester tests to increase their chance of getting a good
semester mark.
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© 2023 University of Pretoria: Adapted for NAS
3.2
Semester mark calculation
Semester marks are calculated in the following manner (refer to keys given in the above table in column
2):
1. Convert all your marks to percentages.
2. Select the best 2 tutorial tests, the best 6 tutorials and the best 4 practical assignments
(excluding Practical 0, but including the Trial Prac test/assignment) will count towards
your semester mark.
3. Your semester mark is then calculated according to the following formula:
Semester mark=
(0.05 x T)+(0.05 x P)+(0.375 x ST1)+(0.375 x ST2 or 0.375 x ST3)+(0.1 x PT)+(0.05 x
TT)
Example:
1. Suppose your tutorial marks (in %) were 45, 75, 65, 60, 55, 70, 84, 90 then only the best 6
highlighted marks will count towards your semester mark. Therefore, T = (75 + 65 + 60 + 70 +
84 + 90)/6 = 74
2. Repeat step 2 for best 4 Practicals (P) and best 2 Tutorial Tests (TT) and best 2 Semester tests (ST).
3. Final calculation example:
Example:
Tuts: 75%
Pracs: 77%
ST1:60%
ST2: 75%
ST3:80%
PT:65%
TT: 90%
Semester mark =
(0.05 x 75)+(0.05 x 77)+(0.375 x 75)+(0.375 x 8.)+(0.10 x 65)+ (0.05 × 90) ≈ 76.725%
Note: A student must obtain a semester mark of at least 40% in WST121 to be allowed to
write the final examination (exam entrance). No exceptions will be made.
Semester marks are displayed on ClickUP shortly before the official closing of lectures at the end of
the semester, i.e. 10 November 2023.
3.3
Assessment policy
A student must obtain a semester mark of at least 40% in WST121 to be allowed to write
the examination.
Pre-requisite: A student needs to obtain a final mark of at least 50% in WST111 for admission
to WST121. As of the year 2023, GS does not apply to this module, i.e. your final WST 111 mark MUST
BE at least 50% (not 40%) to qualify for WST 121.
The final mark (FM) is compiled using the semester mark (SM) and the examination mark (EM). The
SM and EM either count 50% each, or the SM counts 40% and the EM 60%, depending on which set
of weights is most beneficial to the student. The 50/50 and 40/60 weightings will be calculated for
each student and the combination which gives the highest mark will be the student’s final mark.
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© 2023 University of Pretoria: Adapted for NAS
Example 1: (50 SM & 50 EM)
SM: 90%
EM: 70%
Final mark = 0.5  SM + 0.5  EM = 0.5  90 + 0.5  70 = 80%
Example 2: (40 SM & 60 EM)
SM: 70%
EM: 90%
Final mark = 0.4  SM + 0.6  EM = 0.4  70 + 0.6  90 = 82%
A student must achieve at least 40% in the exam- even if you have a very good semester mark (for
subminimum purposes). A supplementary exam must be written if your FM is above 50% but your EM
is below 40%. Students with a final mark of 40% to 49% qualify for a supplementary examination.
3.3.1 Procedure to be followed when a semester test cannot be written
●
●
●
●
●
●
In terms of the regulations of the University of Pretoria, if there is a valid reason for not being
able to write a semester test, the student must notify the lecturer beforehand or within three
(3) working days of the date of the test that was not written.
A medical certificate cannot be submitted after a student has written a test.
Note, once a test has been started it is considered to have been written.
In all cases, the application form in Annexure 1 (at the end of the study guide) must be
submitted along with supporting documentation. Clearly indicate the course (WST121), your
student number, surname and initials as well as a contact number. Documentation must be
submitted via email to the wst121stats@gmail.com address.
In those situations where a certificate from a medical practitioner is the supporting
documentation:
o Only original certificates issued by medical practitioners registered with the Council
for Health Professions and the Allied Health Professions Council of SA will be accepted.
o The certificate from the medical practitioner must be dated on or before the date of
the test. Certificates dated after this date will not be accepted.
o The certificate must clearly specify the period for which the student is booked off.
o Any certificate from a medical practitioner stating “I have been informed that. ” or
similar will not be accepted or considered.
o Furthermore, a certificate from a medical practitioner will not be accepted or
considered if it merely states that the student appeared ill or declared him/herself
unfit.
o The validity of the certificate from the medical practitioner will be verified directly
with that practitioner.
In those situations where a certificate from a medical practitioner is not the supporting
documentation:
o A sworn affidavit (original copy) must be submitted together with other original,
suitable and verifiable documentation. In the event of a funeral, a copy of the death
certificate of the deceased or other substantiating evidence is required together with
an explanation of the relationship between the student and the deceased. In the
event of load shedding, proof of the student’s address and proof that there was load
shedding taking place in the area must be supplied.
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© 2023 University of Pretoria: Adapted for NAS
o
●
●
●
●
●
●
The validity of the affidavit and the other supporting documentation will be verified
with the corresponding authorities and persons concerned.
Students not complying with these regulations do not have any right to be otherwise
accommodated or to be given an alternative opportunity to write the test.
The worn excuses of having overslept or read the timetable incorrectly will not be accepted.
So-called sick tests are not granted automatically – all relevant authorities and persons will be
consulted to establish the merit of the case.
False medical certificates or sworn affidavits will be interpreted as unethical and fraudulent.
Any unethical or fraudulent behaviour will be reported to the Faculty and disciplinary steps
will be taken.
If a test in WST121 is scheduled at the same date and time as a test in another subject, the
student must notify the lecturer at least 1 week before the test date.
Examination: In case you could not write the exam, you should consult the administration of the
Faculty of Natural and Agricultural Sciences - no medical certificates or affidavits from students may
be accepted by the departments (Statistics 1-Stop, lecturers or Head of Department) with respect to
the exam. Also, no application for a sick exam that is submitted to Faculty after 3 days of the exam will
be accepted, even if a student handed it in with the lecturer or Head of Department.
THE DISHONEST MISSING OF A TEST AS WELL AS DISHONESTY DURING THE WRITING OF TESTS WILL
NOT BE TOLERATED UNDER ANY CIRCUMSTANCES. ALL IRREGULARITIES WILL BE SEEN IN A SERIOUS
LIGHT AND WILL BE REPORTED TO THE REGISTRAR (ACADEMIC).
The University of Pretoria commits itself to produce academic work of integrity. You have affirmed,
by signing the integrity statement on ClickUP, that you are aware of and have read the Rules and
Policies of the University, more specifically the Disciplinary Procedure and the Tests and
Examinations Rules, which prohibit any unethical, dishonest or improper conduct during tests,
assignments, examinations and/or any other forms of assessment. You are aware that no student
or any other person may assist or attempt to assist another student, or obtain help, or attempt to
obtain help from another student or any other person during tests, assessments, assignments,
examinations and/or any other forms of assessment.
The University encourages students to familiarise themselves with the Disciplinary Code for
Students, contained in the UP General Rules and Regulations
3.4
Plagiarism
Plagiarism is a serious form of academic misconduct. It involves both appropriating someone else’s
work and passing it off as one’s own work afterwards. Thus, you commit plagiarism when you present
someone else's written or creative work (words, images, ideas, opinions, discoveries, artwork, music,
recordings, computer-generated work, etc.) as your own. Only hand in your own original work.
Indicate precisely and accurately when you have used information provided by someone else.
Referencing must be done in accordance with a recognised system. Indicate whether you have
downloaded information from the Internet. For more details, visit the library’s website:
http://www.library.up.ac.za/plagiarism/index.htm.
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© 2023 University of Pretoria: Adapted for NAS
3.5
Programme/Departmental/Module rules, requirements and guidelines
The WST121 module is at first year level. A WST111 FINAL MARK (combined semester and examination
mark) of at least 50% is the prerequisite for WST121. The two modules (WST111 and WST121) are
prerequisites for WST211. Only students who passed WST111 and WST121 (at least 50% in both cases)
will be allowed to continue with Mathematical Statistics at a second year level.
The practical component of the WST121 module utilises the programming language R for practical
applications and illustration of theoretical concepts.
If you are registered in the Actuarial programme, please note:
In order to qualify for the actuarial exemption, you must complete WST111, WST121 (first year) as well
as WST211, WST221 (second year) under the following conditions:
● You must achieve an average mark of 60% across the four subjects’ examination papers.
● You must score a subminimum of 55% in each examination paper.
Because you can score a maximum of 50% in a supplementary examination, it does not count.
Note, even if you miss the exemption requirements, but still pass all your modules, you will still qualify
to get the degree. If you miss this exemption, please note that you can still obtain it by writing one of
the Actuarial board exams (see https://www.actuarialsociety.org.za/ for more information).
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© 2023 University of Pretoria: Adapted for NAS
3.6
Code of conduct
We are not only facilitating learning in a module, we are also preparing you for the world of work. We
expect you to adhere to the code of conduct as spelled out in the Escalation policy of UP.
3.6.1 Communication via email (wst121stats@gmail.com)
When you send an email to the lecturer, you have to use a respectful tone and include all the following
aspects:
● A clear and explanatory heading (e.g. “Sick note for ST1”) in the subject line;
● Your full name, surname and student number at the end of the mail;
● Short and clear message.
● Harassment/violent/abusive messages sent via email will not be tolerated and will be
escalated to Faculty for disciplinary action.
3.6.2 Compliments and complaints
You are more than welcome to express your appreciation to your lecturer or tutor and supply feedback
about aspects of the course that you enjoy and find valuable.
If you have a query or complaint, you have to submit it in writing with specifics of the issue or the
nature of the complaint. It is imperative that you follow the procedure outlined below in order to
resolve your issues:
1. Consult the lecturer concerned about your complaint/concerns.
If the matter has not been resolved,
2. consult the class representative (The primary function of the Class Representative is to serve
as a two-way communication channel between the class and the lecturer).
If the matter has not been resolved,
3. consult the module co-ordinator (large modules with multiple lecturers)
If the matter has not been resolved,
4. consult the Head of Department (cc the module co-ordinator)
If the matter has still not been resolved,
5. consult with the Dean of the Faculty (cc the Head of the Department)
Other University Policies and Regulations: https://www.up.ac.za/article/2754069/up-policies-andother-important-documents
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4
Module information
4.1
Purpose of the module
The goal of the WST111 module was to present a solid undergraduate foundation in statistical theory
while the WST121 module provides an indication of the relevance and importance of the theory in
solving practical problems in the real world. Topics that are covered include the following: Sampling
distributions. Statistical inference: Point and interval estimation. Hypothesis testing with applications
in one and two-sample cases. Linear models and estimation by least squares. Analysis of variance.
Introductory categorical data analysis. Distribution-free testing methods. Identification, use,
evaluation and interpretation of statistical computer packages and statistical techniques.
4.2
Module outcomes
The student must be able to use the sampling distributions such as the normal distribution, the 𝜒2distribution, the 𝑡-distribution and the 𝐹-distribution to calculate probabilities associated with sample
statistics.
The student must be able to understand the importance of estimation and the difference between
the point estimation and interval estimation of a specific population parameter.
The student must be able to understand the importance of hypothesis testing and to be able to make
scientifically accountable decisions concerning the population mean, 𝜇, the population proportion, 𝑝,
the population variance, 𝜎2, the difference between two population means, 𝜇1– 𝜇2, the difference
between two population proportions, 𝑝1– 𝑝2, and the equality of two population variances, 𝜎2 = 𝜎2.
1
2
The student must be able to use the method of least squares to fit a linear model to an experimental
response.
The student must be able to solve inferential problems associated with the linear statistical model
such as estimation and tests of hypotheses relating to the model parameters.
The student must be able to use the technique of analysis of variance to compare the means of more
than two populations.
The student must be able to identify a multinomial experiment and calculate cell probabilities
associated with the experiment.
The student must be able to use nonparametric techniques to analyse data where response
measurements are difficult to quantify or assumptions underlying standard methodology are not met.
The student must be able to identify the nature of the data in a specific problem and on the basis of
that decide on an appropriate analysis technique.
4.3
Articulation with other modules in the programme
The WST121 module is typically included in BSc programmes such as BSc in Actuarial & Financial
Mathematics, Mathematical Statistics, Mathematics, Physics, IT or Computer Science and BCom
programmes such as BCom degrees in Statistics or Econometrics.
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© 2023 University of Pretoria: Adapted for NAS
4.4
Module structure
Linear Models
Sampling
distributions
Estimation
Hypothesis testing
Analysis of Variance
Analysis of Categorical Data
Alternative approaches
4.5
Learning presumed to be in place
A WST111 FINAL MARK (combined semester and examination mark) of at least 50% is the prerequisite
for WST121.
4.6
Credit map and notional hours
This module carries a weighting of 16 credits, indicating that a student should spend an average of 160
hours to master the required skills (including time spent preparing for tests and examinations). This
means that you should devote an average of 12 hours of study time per week to this module. The
scheduled contact time is approximately five hours per week, which means that at least another 7
hours per week of own study time should be devoted to the module.
The number of credits allocated to a module give an indication of the volume of learning required for
the completion of that module and is based on the concept of notional hours. Given that this module
carries a weighting of 16 credits, it follows that you should spend an average of 10x16 hours of study
in total on the module (1 credit = 10 notional hours). This includes time for lectures, assignments, tests
and exams. This means that you should spend approximately 160 hours/14 week = 12 hours per week:
Independent Work*
Lectures/
Live Tutorial &
Textbook
Revision
Practical
Tutorials
exercises
Practical Sessions
5 hours
3.5 hours
1 hour
1.5 hours
1 hour
* Values are estimates. Distribution of hours will vary per student.
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© 2023 University of Pretoria: Adapted for NAS
4.7
Units
Unit: Statistics and Sampling Distributions
Statistics and their distributions
(DB p.285—294, Example 6.3 excluded)
You must be able to:
● define a statistic
● define and understand the concept of a sampling distribution
● define a random sample
● derive sampling distributions using probability rules
● use simulation experiments to identify:
o the general shape of the sampling distribution of a sample mean when sampling
from normal and non-normal distributions
o the effect of the sample size and number of repetitions on the sampling distribution
of the sample mean
Key concepts: statistic, sampling distribution, inference, goodness of the estimate, sample mean,
sample variance, sample size.
The distribution of the sample mean
(DB p.296-302)
You must be able to:
● identify the relationship between the sampling distribution of the sample mean parameters
and population parameters
● identify the importance of the normal distribution in statistical inference regarding 𝜇 (with
𝜎2 known) and regarding 𝑇0
● use the distribution and parameter values of the sample mean and total obtained from a
normal distribution
● know and use the central limit theorem
Key concepts: expected value of the sample mean, variance of the sample mean, expected value of the
sample total, variance of the sample total.
The mean, variance and MGF for several variables
(DB p.306-309, p. 311-312) (Proof, Proposition, Example 6.16 on p.310 excluded)
You must be able to:
● define a linear combination
● give the expected value and variance of a linear combination of independent random
variables
● identify the importance of the normal distribution in statistical inference regarding
● 𝜇1 − 𝜇2 (with n1 and n2 very large)
● give the MGF of a linear combination of independent random variables
Key concepts: linear combination, independent variables, difference between two variables, momentgenerating function for linear combinations, case of normal random variables.
Distributions based on a normal random sample
(DB p. 315-325, Proof p.321 excluded, Moments of 𝑡 excluded)
You must be able to:
● give the distribution of the sum of squares of independent, standard normal variables
● identify the importance of the 𝜒2-distribution in statistical inference regarding 𝜎2
● identify the distribution of a function of the sample variance
16
© 2023 University of Pretoria: Adapted for NAS
●
●
●
describe the relation between the sample mean and sample variance
identify the relation between the t-distribution, the standard normal and 𝜒2-distribution
use the normal distribution, the 𝜒2-distribution, the t-distribution and the F-distribution to
calculate probability expressions in terms of statistics with their associated sampling
distributions.
Key concepts: distributions based on a normal random sample, inference, goodness of the estimate,
sample mean, sample variance, sample size, t-distribution, normal distribution, 𝜒2-distribution, Fdistribution.
Unit: Point estimation
General concepts and criteria
(DB p.331-345) The Bootstrap excluded)
You must be able to:
● define a point estimate and point estimator
● define and calculate the mean square error of a point estimator
● define an unbiased estimator
● define and calculate the standard error of a point estimate
Key concepts: parameters, point estimate, interval estimate, estimator, point estimator, unbiased
estimator, bias, mean square error.
Section 7.2, Section 7.3, Section 7.4 excluded
Some common unbiased point estimators
(Additional material)
You must be able to:
● give and use the expected values and standard errors of common estimators
Unit: Statistical intervals based on a single sample
Basic properties of confidence intervals
(DB p.382-390)
You must be able to:
● calculate the sample size required for a certain interval width
● calculate the bound on the error of estimation
● derive, calculate and interpret a confidence interval
Key concepts: interval estimator, confidence interval, upper and lower confidence limit, confidence
coefficient
Large-sample confidence intervals for a population mean and proportion
(DB p.391-399)
You must be able to:
● derive, calculate and interpret large-sample confidence intervals for the population mean
and proportion
● identify the effect of the sample size, as well as the effect of the confidence coefficient on
the size of a confidence interval
● specify the assumptions required to construct such large-sample confidence intervals.
Key concepts: approximately normal sampling distribution, one-sided confidence limits, upper bound,
lower bound, confidence coefficient, target parameter
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© 2023 University of Pretoria: Adapted for NAS
Intervals based on a normal population distribution
(DB p.401-406)
You must be able to:
● derive, calculate and interpret confidence intervals for the population mean with unknown
variance
● specify the assumptions required to construct such an interval
● derive a prediction interval for a single observation
Key concepts: t-distribution, confidence interval, prediction interval
Confidence intervals for the variance and standard deviation of a normal
population
(DB p.409-410) (Section 8.5 excluded)
You must be able to:
● derive, calculate and interpret confidence intervals for the population variance and
standard deviation
● specify the assumptions required to construct such an interval
Key concepts: confidence interval, confidence level, normality, 𝜒2-distribution
Unit: Tests of hypotheses based on a single sample
Hypotheses and test procedures
(DB p.425-434)
You must be able to:
● name the elements of a test procedure
● define and formulate both the null and the alternative hypothesis for a specific problem
● define a type I error
● define a type II error
● understand the relation between the risks 𝛼 and 𝛽.
Key concepts: research hypothesis, alternative hypothesis, null hypothesis, impossible vs improbable,
test statistic, rejection region, risk, type I error, type II error, probability of error, 𝛼, 𝛽.
Tests about a population mean
(DB p.436-447)
You must be able to:
● identify and apply all three cases of the tests about a population mean
● determine the sample size needed based on 𝛼 and 𝛽
Key concepts: upper-tail and lower-tail alternative, two-tailed alternative, one-tail testing, two-tailed
testing, upper-tail RR, lower-tail RR, two-tailed RR, test statistic, rejection region, sample size, type II
error probability,𝛼,𝛽.
Tests concerning the population proportion
(DB p.450-453)
You must be able to:
● identify and apply a large-sample test
● calculate 𝛽
● determine the sample size required for a given 𝛼 and 𝛽
● identify and apply a small-sample test
Key concepts: upper-tail and lower-tail alternative, two-tailed alternative, one-tail testing, two-tailed
testing, upper-tail RR, lower-tail RR, two-tailed RR, test statistic, rejection region.
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© 2023 University of Pretoria: Adapted for NAS
Tests concerning the population variance
(Additional material)
You must be able to:
● apply a test procedure for the variance
Key concepts: upper-tail and lower-tail alternative, two-tailed alternative, one-tail testing, two-tailed
testing, upper-tail RR, lower-tail RR, two-tailed RR, test statistic, rejection region.
p- Values
(DB p.456-465)
You must be able to:
● define a p-value
● make a decision regarding a hypothesis based on the p-value
● calculate the p-value for a z-test
● calculate the p-value for a t-test
Key concepts: p-value, observed significance value
Section 9.5 excluded.
Unit: Inferences based on two samples
z Tests and confidence intervals for a difference between two population
means
(DB p.484-495)
You must be able to:
● name the assumptions required for the z-test
● derive the test statistic used to test the hypotheses for a difference in means
● apply the test procedure for normal populations with known variance
● apply the large-sample test
● identify the effect of the sample size
● calculate a confidence interval for the difference in means
Key concepts: difference in means, alternative hypotheses, sample size, confidence interval
The two-sample t-test and confidence interval
(DB p.499-505)
You must be able to:
● name the assumptions required for the t-test
● derive the test statistic used to test the hypotheses for a difference in means
● apply the t-test procedure for normal populations with unknown variance
● calculate a confidence interval for the difference in means
● name the assumptions required for the pooled t-test
● apply the pooled t-test procedure for normal populations with unknown variance
Key concepts: upper-tail and lower-tail alternative, two-tailed alternative, one-tail testing, two-tailed
testing, upper-tail RR, lower-tail RR, two-tailed RR, test statistic, rejection region.
19
© 2023 University of Pretoria: Adapted for NAS
Analysis of paired data
(DB p.509-516)
You must be able to:
● name the assumptions required for the paired t-test
● apply the paired t-test procedure for paired data
● calculate a confidence interval for the difference in means for paired data
Key concepts: upper-tail and lower-tail alternative, two-tailed alternative, one-tail testing, two-tailed
testing, upper-tail RR, lower-tail RR, two-tailed RR, test statistic, rejection region.
Inferences about two population proportions
(DB p.519-525)
You must be able to:
● derive and apply the test procedure to test the hypotheses for the difference in proportions
for large samples
● calculate the type II error probability and sample size required
● calculate a large-sample confidence interval for the difference in proportions
Key concepts: upper-tail and lower-tail alternative, two-tailed alternative, one-tail testing, two-tailed
testing, upper-tail RR, lower-tail RR, two-tailed RR, test statistic, rejection region, type II error
probability, sample sizes.
Inferences about two population variances
(DB p.527-531)
You must be able to:
● apply the test procedure to test the hypotheses for the ratio of variances
● calculate p-values for the F-test
● derive and calculate a confidence interval for the ratio of variances
Key concepts: upper-tail and lower-tail alternative, two-tailed alternative, one-tail testing, two-tailed
testing, upper-tail RR, lower-tail RR, two-tailed RR, test statistic, rejection region, p-value.
Section 10.6 excluded.
Unit: Analysis of Variance
Single factor ANOVA
(DB p. 552-563)
You must be able to:
● know the notation and assumptions required for a single factor ANOVA
● define the treatment and error sum of squares and their relationship as well as their
respective distributions
● define the mean square for treatments and error
● derive a test statistic based on sum of squares for treatment and error to compare more
than two population means
● perform a statistical test based on sum of squares for treatment and error to compare the
means of two populations
● construct an ANOVA table
● know the effect of the violation of the assumptions
Key concepts: factor, level, qualitative variable, ANOVA, total sum of squares, sum of squares for
treatments, sum of squares for error, F-distribution.
Section 11.2 excluded.
20
© 2023 University of Pretoria: Adapted for NAS
More on single factor ANOVA
(DB p. 572-573)
You must be able to:
● give the alternative formulation of the single factor model
Section 11.3 from middle p.573-p.580 excluded.
Two-factor ANOVA with Kij=1
(DB p. 583-587)
You must be able to:
● identify a two-factor ANOVA experimental situation
● know the notation and assumptions required for a two-factor ANOVA
● formulate the two-factor ANOVA model
● calculate the relevant sums of squares for the two-factor ANOVA procedure
● calculate the test statistics for the two relevant alternative hypotheses
● construct the corresponding ANOVA table
Key concepts: two factors of simultaneous interest, factor A, factor B, factor levels, SST, SSA, SSB, SSE,
MSA, MSB, MSE, F-ratios.
Section 11.4 from p.590, Randomized Block Experiments excluded.
Section 11.5 excluded.
Unit: Regression and correlation
The simple and multiple linear regression model
(DB p.613-620,682) (Logistic regression model excluded)
You must be able to:
● define a simple linear regression model
● define a multiple linear regression model
● calculate and use the expected value and variance of 𝑌 for a given 𝑥
Key concepts: deterministic, probabilistic, dependent, independent, regression line
Estimating model parameters
(DB p.624-636)
You must be able to:
● derive and calculate least square estimators and estimates for the parameters
● estimate 
● calculate and interpret the coefficient of determination
Key concepts: fitted (predicted values), error sum of squares, total sum of squares, explained variation,
expected value and variance of least squares estimator(s), unbiased, covariance of least squares
estimators, distribution of least squares estimators.
2
Inferences about the regression coefficient β1
(DB p.640-649) (Fitting of the Logistic Regression Model excluded)
You must be able to:
● derive a statistical test for 𝛽1
● perform and interpret the results of a hypothesis test
● derive a confidence interval for β1
21
© 2023 University of Pretoria: Adapted for NAS
● construct an ANOVA table
Key concepts: upper-tail RR& alternative, lower-tail RR & alternative, two-tailed RR & alternative, tdistribution, p-value, confidence interval.
Inferences concerning the prediction of future y values
(DB p.654-659)
You must be able to:
● calculate a confidence interval for the mean value of the dependent variable when the
explanatory variable has a particular value
● calculate a prediction interval for the dependent variable when the explanatory variable
has a particular value.
Key concepts: confidence interval, prediction interval.
Correlation
(DB p.662-670)
You must be able to:
● calculate an estimate for the population correlation coefficient ρ
● know the properties of r
● test appropriate hypotheses regarding ρ using a t-test
● explain the relation between ρ and β1
● test appropriate hypotheses regarding ρ using a Z-test
● derive the relation between the sample correlation coefficient and the coefficient of
determination
● interpret the values of the correlation coefficient and the coefficient of determination.
Key concepts: bivariate normal distribution, sample correlation coefficient , RR, alternative, coefficient
of determination, explained variation.
Assessing model adequacy
(DB p.674-679)
You must be able to:
● interpret residual plots
● recognize plots that indicate abnormality in the data
suggest remedies for specific abnormalities in the data
Key concepts: residual plot, standardized residuals, nonlinear relationship, non-constant variance,
discrepant observation, observation with large influence, dependence in errors, variable omitted.
Section 12.7 from Estimating Parameters excluded
Section 12.8 excluded
Unit: Goodness-of-fit tests and categorical data analysis
Goodness-of-fit tests when the category probabilities are completely specified
(DB p. 723-730)
You must be able to:
● identify experiments where measurements are qualitative rather than quantitative
● apply a 𝜒2 test procedure to test an hypothesis
● know the relationship between the 𝜒2-test and the z-test
Key concepts: qualitative or categorical measurements, observed counts, expected counts, number of
degrees of freedom.
22
© 2023 University of Pretoria: Adapted for NAS
Goodness-of-fit tests for composite hypotheses
(DB Section 13.2 replaced by additional material)
You must be able to:
● test a hypothesis concerning specified cell probabilities
● test whether a specific model for a population distribution fits the sample data
Key concepts: specified cell probabilities, goodness-of-fit test.
Two-way Contingency tables
(DB p. 744-749) (Excluded from Ordinal Factor & Logistic Regression)
You must be able to:
● estimate row and column probabilities in order to estimate expected cell frequencies under
independence
● perform a statistical test to determine whether two classification variables are
homogeneous
● perform a statistical test to determine whether two classification variables are dependent.
Key concepts: dependency/contingency between two classification criteria, contingency table, row
probabilities, column probabilities, degrees of freedom, independence, expected cell frequencies,
collapsed marginal tables.
Unit: Alternative approaches to inference
The Wilcoxon signed-rank test
(DB p. 758-763)
You must be able to:
● perform the Wilcoxon signed rank test to test a hypothesis concerning the mean
● perform the Wilcoxon signed rank test to test a hypothesis concerning the mean for paired
data
● give the assumptions required to perform the Wilcoxon signed rank test
● determine the values for which the null hypothesis should be rejected for a given
significance level
● calculate the p-value for a given value of the test statistic
● adjust the procedure in the presence of ties
Key concepts: absolute value of differences, ranks, ties, Wilcoxon signed rank test, two-tailed test, onetailed test, test statistic, rejection region, critical value, attained significance level, approximately
normally distributed.
The Wilcoxon rank-sum test
(DB p. 766-769)
You must be able to:
● understand how the rank sums of two samples can be used to determine whether two
population distributions differ in location
● calculate the test statistic
● perform the Wilcoxon rank-sum test to test whether two independent population
distributions differ in location
● formulate the appropriate null and alternative hypothesis for two-tailed or one-tailed
testing
● determine the rejection region for any given alternative
23
© 2023 University of Pretoria: Adapted for NAS
Key concepts: compare two populations, independent random samples, rank-sum test, rank combined
set of observations, ties, average ranks, alternative hypothesis, distribution shifted to the right/left,
one-tailed, two-tailed, rejection region.
Section 14.3, 14.4 excluded
5 Support services
Please download a QR code reader on your cellphone. To download a QR code reader, open your
mobile app store (App Store, Google Play or Windows Marketplace) and search for QR code readers.
E-learning support
●
●
●
Report a problem you experience to the Student Help Desk on your campus.
Call 012 420 3837.
Email studenthelp@up.ac.za
Other support services:
FLY@UP:
The Finish
Line is Yours
Disability
Unit
Student
Counselling
Unit
● Think carefully before
dropping modules (after the
closing date for amendments
or cancellation of modules).
● Make responsible choices
with your time and work
consistently.
● Aim for a good semester
mark. Don’t rely on the
examination to pass.
Academic support for students
with learning disabilities:
● Assistive technological
services
● Facilitation of test and
examination
accommodations
● Test and exam concession
applications
● Accessible study venues and
a computer lab
● Referrals for recommended
textbooks in electronic
format
Provides counselling and
therapeutic support to students
www.up.ac.za/fly@up
email: fly@up.ac.za
https://www.up.ac.za/disabilityunit
012 420 2064
email: du@up.ac.za
012 420 2333
24
© 2023 University of Pretoria: Adapted for NAS
Student
Health
Services
Promotes and assists students
with health and wellness
012 420 5233
012 420 3423
The Careers
Office
Provides support for UP students
and graduates as they prepare
for their careers
careerservices@up.ac.za
012 420 2315
24-hour Operational
Management Centre
012 420-2310
012 420-2760
24-hour Operational Manager
Crisis Line
083 654 0476
0800 006 428
Department
of Student
Affairs
Enquiries concerning studies,
accommodation, food, funds,
social activities and personal
problems
012 420 2371/4001
Roosmaryn Building, Hatfield
campus
Centre for
Sexualities,
AIDS and
Gender
Identifies and provides training
of student peer counsellors
012 420 4391
Fees and
funding
http://www.up.ac.za/enquiry
www.up.ac.za/fees-and-funding
012 420 3111
IT Helpdesk
For student IT related queries
Department
of Security
Services
012 420 3051
studenthelp@up.ac.za
Please have a look at the Department of Statistics’s guide on the department’s website.
25
© 2023 University of Pretoria: Adapted for NAS
Annexure 1
DEPARTMENT OF STATISTICS
APPLICATION TO BE EXCUSED FROM PRACTICAL&SEMESTER TESTS 2023
MODULE: WST121 – Mathematical Statistics 121
INITIALS & SURNAME:
STUDENT NO.:
I hereby request that I be excused from the following academic commitment (indicate with X):
Semester Test 1
Semester Test 2
Semester Test 3
Practical Test
Attached is the original copy of my medical certificate or other supporting documentation in
support of my application.
I confirm that I have read and understood the matters relating to the submission of
excuses/apologies as contained in the WST121 Study Guide under Section 3.3.1
I declare that this is a bona fide application and that the medical certificate and/or supporting
documents attached are true.
................................
SIGNATURE
....................
DATE
26
© 2023 University of Pretoria: Adapted for NAS