Statistical Methods for Decision Analysis

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This Course Guide has been taken from the most recent presentation of the course. It would be useful for reference purposes but please note that there may be updates for the following presentation.


Statistical Methods for Decision Analysis

Welcome to MATH S280 Statistical Methods for Decision Analysis! This is a course that aims to bring you a wide-ranging and realistic introduction to modern practical statistics.

Few things in the world are clear-cut: it is rare that, when information is sought to shed light on a question of interest, a definite and definitive answer emerges. Instead, a myriad of complications and provisos and apparently random elements combine to muddy the waters. Making sense of data, and hence obtaining useful and well-founded answers to important questions, is the major goal of statistics in general, and hence of this course. (Deciding what data to collect and how best to do so are very important element of statistics.)

The statistical analysis of data is usually based on some kind of modelling of the situation of interest. However, large amounts of data and realistic models together lead to many and difficult calculations. This is where the power of computers comes in. The computer can take the drudgery out of the computational side of statistics, and thus leave the analyst free to concentrate on understanding and interpreting the results. In this course, you will use two statistical software packages. The first of these is a commercial statistical software package called MINITAB. This is used throughout the course. The second software package, which is called SUStats (Software for Understanding Statistics), has been developed specifically to help you to develop your understanding of key statistical ideas and concepts. This is used first in Unit A3.

It is hoped that you will find the course stimulating and satisfying. Many students initially find some of the statistical ideas and concepts a little difficult. But once you have grasped them, you will be able to explore and enjoy some fascinating topics: a beauty of statistics is that the need for it is ubiquitous, and hence that its applications are very many and varied. Most of the data sets that you will have the opportunity to analyze in this course have arisen from real-world examples addressing real-world problems.

2.1 Preparatory work

The recommended prerequisite for this course is MATH S122 A Foundation in Applied Mathematics. You need to be able to understand simple tables and graphs. Familiarity with some basic differentiation and integration techniques may be of some help in a few places. The course relies principally on computer-aided analysis of data as well as on pencil-and-paper arithmetic, but you will not be expected to follow complicated algebraic arguments or to produce a great deal of algebra in your assignments and in the examination.

You are expected to be familiar with the basics of Windows. If you have not used a computer operating a version of Windows, then you should spend some time familiarizing yourself with Windows before the course starts.


2.2 Calculator

You will need a calculator. If you studied MATH S122, the calculator you used then will be adequate. Otherwise, you should obtain one with basic mathematical functions and a memory. Basic statistical functions (sample summaries and linear regression) are also a recommended attribute of your calculator but are not essential. However, your calculator must be one of the models on the list of approved calculations. The list will be posted on the OLE (see Section 5.2).


2.3 Computer

You will need access to a computer capable of running the two course software packages.

MINITAB and SUStats are provided, on a CD-ROM, with the course. Installation instructions are given in the Introduction to Computer Book A. You should install the software before the course begins. The reason for not leaving the installation until your study of Unit A1 is in case there are difficulties with getting the software properly installed on your computer. In this way, you can get any problems sorted out before you are expected to use the software.

3.1 Study Units

The course is made up of four blocks of study. Each block is divided into several units, bound together, and has an associated computer book containing software-based activities. These four blocks are followed by a review unit, which has been designed to help you with revising for the examination.

Each unit, including its associated computer book chapters, is timetabled for two weeks' or, in some cases, three weeks' study (see the Presentation Schedule).

The title of each unit is given below:

  1. Unit A1 Exploring data
  2. Unit A2 Interpreting data
  3. Unit A3 Modelling variation
  4. Unit A4 Sample and populations
  5. Unit A5 Events occurring at random
  6. Unit B1 The normal distribution
  7. Unit B2 The central limit theorem
  8. Unit B3 Confidence intervals
  9. Unit C1 Testing hypotheses
  10. Unit C2 Nonparametrics
  11. Unit C3 The modelling process
  12. Unit D1 Estimation
  13. Unit D2 Regression
  14. Unit D3 Related variables

(The syllabus of each unit can be found in Appendix A of this Guide.)


3.2 Computing Component: MINITAB and SUStats

The place of these two software packages in the course has already been described. The use of MINITAB is supported by a booklet entitled Guide to MINITAB.


3.3 Exercise booklets

There are four booklets of additional pencil-and-paper exercises. You might like to use the exercises to reinforce your understanding of the text or when revising for the examination.


3.4 Supplementary Exercise booklet

This Additional Supplementary Booklet provides students with many practical exercises and experience in analyzing of various decision problems that the study units are not covered. The practical problems/exercises in the booklet will emphasize on how to apply the statistical techniques in the followings areas: (1) Risk Management, (2) Financial Engineering and (3) Insurance.


3.5 Course Handbook

A Handbook is provided to give you a convenient source of basic definitions and formulae for use throughout the year of your study. You will not be allowed to bring the Course Handbook to the exam. Another copy of the handbook will be given to you together with the exam paper.


3.6 DVD

Seven programmes are associated with the course and are supplied on DVD. A DVD Guide gives notes on the structure and purpose of these programmes. You can get an idea of what to expect of each programme from reading the programme notes before viewing.


3.7 Presentation Schedule

The Presentation Schedule sets out an overall schedule for the study weeks for each unit. Assignment cut-off dates and the dates for tutorial sessions are also incorporated. (Detailed arrangements for tutorial sessions are posted on the OLE.) Although the suggested study week for each unit is just for your reference, it is important to keep to schedule. For most assignments, the cut-off date is very soon after the end of the study week for the last of the relevant units. We recommend that you try to finish the assignment questions for each unit as soon as you finish the unit, otherwise before the cut-off dates for the assignments you will have a lot of work to do in only a few days.


3.8 Stop Presses

The Stop Presses act as a sort of course newsletter, containing useful information of various types. You should always read these as soon as they are posted to OLE.

4.1 Continuous assessment

You will be awarded a grade for each written assignment and each multiple choice assignment which you submit (and zero for each assignment which is not submitted). The assignments of the course are arranged as below. (See the Presentation Schedule for their cut-off dates.)

MATH S280 Assignment 01 Block A Unit 1 - 4
MATH S280 Assignment 02 Block A Unit 5 and Block B
MATH S280 Assignment 03 Block C
MATH S280 Assignment 04 Block D
MATH S280 Assignment (multiple choice) 41 Block A and B
MATH S280 Assignment (multiple choice) 42 Block A to D

The course uses the 100% rule as defined in the Assessments and Examinations section of the Student Handbook. All assignment (multiple choice) scores and the best three assignment scores are used to calculate the overall continuous assessment score.

Overall, the continuous assessment contributes 30% of your final score for the course. The weightings of assignments and assignments (multiple choice) are 80:20. See the Assignment Booklet for a detailed breakdown of marks within each assignment.


4.2 The examination

There is a three-hour examination at the end of the course which is based upon the whole course. You will be sent a specimen examination paper which you should work through carefully before the examination. Sample solutions will be provided. You will be allowed to take your calculate into the examination and no other course materials will be allowed.

The examination constitutes the remaining 70% of your overall score for the course.


4.3 How to pass the course

The assessment comprises two components: the continuous assessment and the examination. The contribution of each type of assessment is shown in the following table:


Assessment typeWeightings
4 assignments (count the best 3 of the 4)24%
2 assignments (multiple choice) (all required)6%
Examination (3 hours)70%


The preliminary overall course score is calculated as 24% of your overall assignment score, plus 6% of your overall assignment (multiple choice) score, plus 70% of your final examination mark. If each of your overall continuous assessment score and your final examination score are higher than 40%, plus you have paid your fees for the course, then you will be certain to pass MATHS 280.

5.1 From your tutor

Your tutor is there to help you understand the ideas in the course and the best way for him/her to do this is through the comments put on your assignments. Go through the script and take note of the comments written by your tutor; they will help you avoid similar errors in later assignments and in the examination. Try to attend tutorials because there you will have the opportunity to talk to your tutor directly and, just as important, to talk to other students.

In addition, tutors make themselves available for you to telephone them for immediate help or advice. Personal particulars of your tutor should be sent to you by the Registry before the course starts. Keep your tutor's address since you have to send your assignments to him/her directly.


5.2 Online support

Besides telephone contact, the Online Learning Environment (OLE) will be used to facilitate communication among all the participants in MATH S280 - students, tutors and Course Coordinator. The OLE is an online platform for you to gain access to and use the different Internet tools, and to engage in conversation with other M280 participants.

In your package of course materials, you will be given a User Guide to help you use the OLE. This guide's title is 'Online Learning Environment - User Guide'.

Using the OLE is a required activity in your study of MATH S280. For instance, you have to use the OLE for the submission of assignments (multiple choice). You will find that the discussion board feature is useful for posting requests for help. However, you should not expect an immediate reply when you post any problems in the discussion board.


5.3 From your fellow students

One of the best ways of learning is by talking about your work with fellow students. In this second-level course you may only see them at the infrequent tutorials during the year. That leaves a lot of weeks when you could be on your own. Make sure then that you have the telephone numbers of some MATH S280 students in your living or working area so that you can keep in touch. You might even like to form your own self-help group to meet regularly: this is often a good way of getting people to discuss common difficulties, especially in the assessment questions.

A word of warning about study groups; whereas students are encouraged to discuss their problems with one another, including those relating to assignments, it is essential that you submit your own attempt at an assignment or assignment (multiple choice) and not the one you copied. You must remember that the assignments are part of the teaching and learning process and so it is at your own interest to ensure that you submit your own work.


5.4 From the Course Coordinator

If there are any queries which your tutor cannot settle for you, he/she will probably advise you to contact the Course Coordinator. The Course Coordinator is a full-time staff member of the School of Science and Technology at HKMU and is responsible for the course presentation. Ways of contacting the Course Coordinator can be found in the Letter to Students enclosed in the package of course materials that you have received.

Block A

Unit A1 Exploring data

1. Data and questions

2. Pie charts and bar charts

2.1 Pie charts

2.2 Bar charts

2.3 Problems with graphics

2.4 Introducing MINITAB

3. Histograms and scatterplots

3.1 Histograms

3.2 Scatterplots

3.3 Histograms and scatterplots using MINITAB

4. Numerical summaries

4.1 Measures of location

4.2 Measures of dispersion

4.3 Symmetry and skewness

4.4 Numerical summaries using MINITAB

Unit A2 Interpreting data

1. Boxplots

1.1 Simple boxplots

1.2 Comparing data sets using boxplots

1.3 Transforming to reduce skewness

1.4 Boxplots using MINITAB

2. Producing useful tables

2.1 Basic table layout

2.2 Including the results of useful calculations

3. Interpreting data in tables

Unit A3 Modelling variation

1. What is probability?

2. Modelling random variables

2.1 Discrete and continuous random variables

2.2 Probability distributions and probability functions

3. Describing probability distributions

3.1 Probability functions

3.2 The cumulative distribution function

4. Bernoulli trials

4.1 The Bernoulli probability model

4.2 The binomial probability model

4.3 Calculations using MINITAB

5. The geometric probability model

5.1 The geometric probability model

5.2 Family patterns

6. The normal distribution

Unit A4 Samples and populations

1. Choosing a probability model

2. The population mean

2.1 The population mean

2.2 Families of distributions: the mean

2.3 The mean of a continuous random variable

3. The population variance

3.1 The variance of a discrete random variable

3.2 Families of distributions: the variance

3.3 The variance of a continuous random variable

4. Two models for uniformity

4.1 The discrete uniform distribution

4.2 The continuous uniform distribution

5. Is the model a good fit?

Unit A5 Events occurring at random

1. The Poisson distribution

1.1 Rare events

1.2 Poisson's approximation for rare events

1.3 Applications of the Poisson approximation

2. Waiting times between events

2.1 Bernoulli processes

2.2 The exponential distribution

3. The Poisson process

3.1 A continuous-time analogue of the Bernoulli process

3.2 Is a Poisson process a good model?

4. Population quantiles

4.1 Quantiles for continuous distributions

4.2 Quantiles for discrete distributions

4.3 Finding quantiles using MINITAB

5. Probability plotting for exponential distributions

Block B

Unit B1 The normal distribution

1. Some early history

2. The normal distribution

2.1 The family of normal distributions

2.2 Calculating probabilities

3. More on means and variances

3.1 Functions of a random variable

3.2 Sums of random variables

4. Calculations using tables

4.1 The standard normal distribution

4.2 Using printed tables

4.3 Standardizing random variables

5. Probability plotting for normal distributions

Unit B2 The central limit theorem

1. The sample mean

2. The central limit theorem

2.1 The distribution of the sample mean

2.2 The sample total and the sample mean

2.3 The central limit theorem

2.4 A corollary to the theorem

3. Normal approximations for discrete distributions

3.1 When is a normal approximation good?

3.2 Normal approximations for binomial distributions

3.3 Normal approximations for Poisson distributions

Unit B3 Confidence intervals

1. Introducing confidence intervals

1.1 Some examples

1.2 A large-sample confidence interval

2. Interpreting confidence intervals

2.1 Repeated experiments

2.2 Plausible ranges

2.3 Investigating the two interpretations

3. Large-sample confidence intervals

3.1 New confidence intervals from old

3.2 Confidence intervals for proportions

3.3 Large-sample confidence intervals using MINITAB

4. The family of t-distributions

5. Confidence intervals for normal means

5.1 Confidence intervals for a normal mean

5.2 Confidence intervals for differences between two normal means

5.3 Confidence intervals for normal means using MINITAB

6. Exact confidence intervals for a proportion

Block C

Unit C1 Testing hypotheses

1. An approach using confidence intervals

2. Introducing significance testing

3. More on significance testing

3.1 Testing a normal mean

3.2 Testing a proportion

3.3 Performing significance tests using MINITAB

4. Two-sample tests

4.1 Testing the difference between two Bernoulli probabilities

4.2 The two-sample t-test

4.3 Performing significance tests using MINITAB

5. Fixed-level testing

5.1 Performing a fixed-level test

5.2 A few comments

5.3 Fisher, Pearson and Neyman

5.4 Exploring the principles of hypothesis testing

6. Power, and choosing sample sizes

6.1 Calculating the power of a test

6.2 Planning sample sizes

6.3 Power and sample size using a computer

Unit C2 Nonparametrics

1. Nonparametric tests

1.1 Early ideas: the sign test

1.2 The Wilcoxon signed rank test

1.3 The Mann–Whitney test

1.4 Nonparametric tests using MINITAB

2. A test for goodness of fit

2.1 Goodness of fit of discrete distributions

2.2 The chi-squared distribution

2.3 The chi-squared goodness-of-fit test

Unit C3 The modelling process

1. Choosing a model: getting started

1.1 Continuous or discrete?

1.2 Which discrete distribution?

1.3 Which continuous distribution?

2. Exploring the data

2.1 Getting a feel for the data

2.2 Interpreting probability plots

2.3 Transforming the data

2.4 Dealing with outliers

3. Statistical modelling with MINITAB

4. Writing a statistical report

4.1 The structure of a statistical report

4.2 Writing the report

Block D

Unit D1 Estimation

1. Principles of point estimation

1.1 Point estimators

1.2 What makes a good estimator?

1.3 Exploring and comparing estimators by computer

2. The method of least squares

3. The method of maximum likelihood

3.1 Discrete probability models

3.2 Continuous probability models

3.3 Properties of maximum likelihood estimators

3.4 Finding maximum likelihood estimates using MINITAB

4. Estimating a normal variance

4.1 Two point estimators of the variance

4.2 The distribution of S2

4.3 Testing the variance of a normal distribution

Unit D2 Regression

1. Regression models

2. Fitting a linear regression model

2.1 The principle of least squares

2.2 The least squares line through the origin

2.3 The least squares line

2.4 Maximum likelihood estimates

3. Checking the assumptions

4. Sampling properties and statistical inference

4.1 The sampling distributions of the estimators

4.2 Confidence intervals

4.3 Predictors and prediction intervals

4.4 Using MINITAB

5. Transforming data

6. The early history of regression

Unit D3 Related variables

1. Correlation

1.1 Are the variables related?

1.2 Correlation and causation

2. Measures of correlation

2.1 The Pearson correlation coefficient

2.2 The Spearman rank correlation coefficient

2.3 Testing for association

2.4 Correlation using MINITAB

3. Contingency tables

3.1 Conditional probabilities

3.2 Association in contingency tables

3.3 The chi-squared test using MINITAB