ABM Student Guide V1.0.pdf

ABM Student Guide V1.0

ADVANCED BUSINESS MATHEMATICS

Student Guide

Page 2 of 76 ABM Student Guide V1.0

Modification History

Version Date Revision Description

V1.0 May 2011 For Release

© NCC Education Limited, 2011 All Rights Reserved

The copyright in this document is vested in NCC Education Limited. The document must not be reproduced by any means, in whole or in part, or used for manufacturing purposes, except with the prior written permission of NCC Education Limited and then only on condition that this notice is included in any such reproduction.

Published by: NCC Education Limited, The Towers, Towers Business Park, Wilmslow Road, Didsbury, Manchester M20 2EZ, UK.

Tel: +44 (0) 161 438 6200 Fax: +44 (0) 161 438 6240 Email: [email protected] http://www.nccedu.com

http://www.nccedu.com/�

Title Here


CONTENTS 1. Module Overview and Objectives .............................................................................. 7

2. Learning Outcomes and Assessment Criteria .......................................................... 7

3. Syllabus ....................................................................................................................... 7

4. Related National Occupational Standards ................................................................ 9

5. Teaching and Learning ............................................................................................... 9

5.1 Lectures ........................................................................................................................ 9

5.2 Tutorials ........................................................................................................................ 9

5.3 Private Study ................................................................................................................ 9

6. Further Reading List ................................................................................................. 10

7. Assessment .............................................................................................................. 11

Topic 1: Introductory Management Statistics ....................................................................... 13

1.1 Learning Objectives .................................................................................................... 13

1.2 Timings ....................................................................................................................... 13

1.3 Private Study Exercises .............................................................................................. 14

1.4 Tutorial Exercises ....................................................................................................... 16

Topic 2: Probability Distributions .......................................................................................... 19


2.2 Timings ....................................................................................................................... 19



Topic 3: Inferential Statistics 1 ............................................................................................... 25


3.2 Timings ....................................................................................................................... 25



Topic 4: Inferential Statistics 2 ............................................................................................... 29


4.2 Timings ....................................................................................................................... 29



Topic 5: Differentiation 1 ........................................................................................................ 33


5.2 Timings ....................................................................................................................... 33



Topic 6: Differentiation 2 ........................................................................................................ 37

Title Here



6.2 Timings ....................................................................................................................... 37



Topic 7: Regression Analysis 1 .............................................................................................. 41


7.2 Timings ....................................................................................................................... 41



Topic 8: Regression Analysis 2 .............................................................................................. 45


8.2 Timings ....................................................................................................................... 45



Topic 9: Time Series Analysis ................................................................................................ 49


9.2 Timings ....................................................................................................................... 49



Topic 10: Linear Programming ................................................................................................. 55


10.2 Timings ....................................................................................................................... 55



Topic 11: Linear Programming with Solver Routines ............................................................. 59


11.2 Timings ....................................................................................................................... 59



Topic 12: Decision Tree Analysis ............................................................................................. 63


12.2 Timings ....................................................................................................................... 63



Equations and Statistical Tables ................................................................................................. 67

Management Statistics ........................................................................................................ 67

Probability ........................................................................................................................... 68

Title Here


Hypothesis Testing .............................................................................................................. 69

Fitting Data .......................................................................................................................... 70

Differentiation ...................................................................................................................... 71

Time Series ......................................................................................................................... 72

Probabilities under the Normal Distribution Curve ............................................................... 73

Percentage Points of the Student Distribution ..................................................................... 75

Critical Values for the χ2 Distribution.................................................................................... 76

Title Here


Overview


1. Module Overview and Objectives Mathematical models and data analysis are foundational within numerous disciplines of management thought. Whether the focus is on understanding the broad economic environment, carrying out market research, optimising the supply chain, diversifying financial risk or another area of business practice, the requirement to be able to use analytical techniques is vital.

This module advances existing quantitative skills to a level in which students can formulate, use and interpret mathematical models within a business context. An appreciation of the use of computer software to support such models is also developed.

2. Learning Outcomes and Assessment Criteria

Learning Outcomes Assessment Criteria

The learner will be able to The learner can

1 Use summary and inferential statistics to inform business decisions

1.1 Calculate and interpret standard scores 1.2 Use index numbers when reviewing data 1.3 Compare and contrast sampling methods and techniques 1.4 Use confidence intervals to indicate the reliability of

estimates 1.5 Apply significance testing to business hypotheses

2 Analyse management decisions using optimisation techniques

2.1 Use differentiation in marginal analysis 2.2 Formulate a linear programme 2.3 Interpret the optimal and sensitivity results from the

solution of a linear programme.

3 Understand and apply approaches to business forecasting

3.1 Use Pearson and Spearman correlation analysis 3.2 Perform a simple regression analysis 3.3 Evaluate results from a multiple regression analysis 3.4 Use time series analysis to predict future business

performance

4 Evaluate sequential management decisions

4.1 Formulate a decision tree representation of a management decision

4.2 Solve a decision tree

3. Syllabus

Syllabus Topic No Title Proportion Content

1 Introductory management statistics

1/12 • Summary statistics • Data types • Index numbers

Title Here


2 Probability distributions

1/12 • Simple probability review • Standard scores (Z-scores) • Probability distributions

3 Inferential statistics 1

1/12 • Sampling distributions • Point estimates and confidence intervals • Introduction to hypothesis testing

4 Inferential statistics 2

1/12 • Hypothesis testing with a sample • Sampling approaches • Two sample t-test • Significance, error types, power • Frequency data and the χ2 test

5 Differentiation 1 1/12 • Gradient and the derivative • Rules of differentiation

6 Differentiation 2 1/12 • Partial differentiation • The total derivative

7 Regression analysis 1

1/12 • Pearson correlation • Simple linear regression • Spearman correlation

8 Regression analysis 2

1/12 • Multiple regression analysis • Further regression models

9 Time series analysis

1/12 • Preparing time series for analysis • Time series decomposition

o Additive and multiplicative models o Centred moving averages o Seasonal adjustment

• Forecasting with time series • Exponential smoothing

10 Linear Programming

1/12 • Formulating two variable linear programme • Graphical solution of a linear programme • Sensitivity analysis

11 Linear Programming with Solver routines

1/12 • Multiple variable linear programmes • Solving linear programmes with software • Interpreting the output • Extensions of linear programming: the integer

programme

12 Decision tree analysis

1/12 • Formulating decision trees • Solving decision trees with rollback • Posterior probabilities within decision trees • Uncertainty

Title Here


4. Related National Occupational Standards The UK National Occupational Standards describe the skills that professionals are expected to demonstrate in their jobs in order to carry them out effectively. They are developed by employers and this information can be helpful in explaining the practical skills you have covered in this module.

Related National Occupational Standards (NOS) Sector Subject Area: 15.3 Business management Related NOS: BAD321 Organise and report data BAD322 Analyse and report data BAG127 Solve business problems BAG128 Evaluate and solve business problems

5. Teaching and Learning

Suggested Learning Hours Lectures: Tutorial: Private Study: Total:

12 24 114 150

The teacher-led time for this module is comprised of lectures and tutorials. You will need to bring this Student Guide to all classes for this module. The breakdown of the hours for each topic is given in the topic notes below. Generally, the order of study should be to complete private study work before tutorials as your private study work will be reviewed during the tutorial.

5.1 Lectures

Your lecturer will be presenting the basic knowledge and the theoretical concepts required for the unit during this time. He/she will use PowerPoint slides during the lecture time and you will be expected to take notes.

You will also be encouraged to be active during this time and discuss and/or practice the concepts covered. Lectures will include question and answer elements to promote participation and to allow your lecturer to check whether you understand the concepts they are covering.

5.2 Tutorials

These are designed to deal with the questions arising from the lectures and private study sessions. You should think carefully beforehand about any areas in which you might need additional guidance and support and use this time to discuss these with your teacher.

5.3 Private Study

This Student Guide also contains details of the private study exercises. You are expected to complete these exercises to improve your understanding. Your tutor will set deadlines for the completion of this work and go over the suggested answers with you. The deadlines will usually be before the scheduled tutorials for that topic. Some of the private study tasks may require you to work

http://www.cfa.uk.com/images/standards_1_ba/BAD321%20Organise%20and%20report%20data.pdf�

http://www.cfa.uk.com/images/standards_1_ba/BAD322%20Analyse%20and%20report%20data.pdf�

Title Here


in a small group so you will need to plan your time carefully and ensure that you can meet with your group members to complete the work required before the deadline.

You should also use this time to revise the content of lectures to ensure understanding and conduct extra reading (using the supplementary textbooks or other materials available in the library or online). You should bring any questions to the tutorial for additional guidance and support.

6. Further Reading List A selection of sources of further reading around the content of this module must be available in your Accredited Partner Centre’s library. The following lists provide suggestions of some suitable sources.

There are many textbooks in the business mathematics and related areas. The longer list provides titles of those that include most of the aspects within this module. If possible it is useful to have access to at least three different titles.

Additionally attention is drawn to useful websites in the private study sections for each of the topic areas. Buglear (2004), Quantitative Methods for Business - The A-Z of QM, Butterworth-Heinemann ISBN-10: 0750658983 ISBN-13: 978-0750658980 Burton, Carol and Wall (2001), Quantitative Methods for Business and Economics, 2nd Edition, Prentice Hall ISBN-10: 0273655701 ISBN-13: 978-0273655701 Dewhurst (2006), Quantitative Methods for Business and Management, 2nd Edition, McGraw Hill ISBN-10: 0077109023 ISBN-13: 978-0077109028 Lucey (2002), Quantitative Techniques, 6th Edition, Thomson Learning ISBN-10: 1844801063 ISBN-13: 978-1844801060 Oakshott (2009), Essential Quantitative Methods for Business, Management and Finance, 4th Edition, Palgrave McMillan ISBN-10: 0230218180 ISBN-13: 978-0230218185 Waters (1998), A Practical Introduction to Management Science, 2nd Edition, Prentice Hall ISBN-10: 0201178478 ISBN-13: 978-0201178470 Wisniewski (2010), Quantitative Methods for Decision Makers, 5th Edition, Prentice Hall ISBN-10: 0273742663 ISBN-13: 978-0273742663 Neave (2011), Elementary Statistics Tables, 2nd Edition, Routledge ISBN-10: 041556347X ISBN-13: 978-0415563475 Short list:

Title Here


For topics 1, 2, 3, 4, 7 and 8: Hinton (2004), Statistics Explained, 2nd Edition, Routledge ISBN-10: 0415332850 ISBN-13: 978-0415332859 For topics 5 and 6: Dowling (1990), Schaum’s Outline of Calculus for Business, Economics, and The Social Sciences, Schaum’s Outlines ISBN-10: 0070176736 ISBN-13: 978-0070176737 For topics 9, 10, 11 and 12: Keast and Towler (2009), Rational Decision Making for Managers: An Introduction, Wiley and Sons ISBN-10: 0470519657 ISBN-13: 978-0470519653

7. Assessment This module will be assessed by means of an examination worth 100% of the total mark. This assessment will cover the learning outcomes and assessment criteria given above.

Title Here


Topic 1


Topic 1: Introductory Management Statistics

1.1 Learning Objectives

This topic provides an overview of the key concepts of data types, summary statistics and index numbers.

On completion of the topic, you will be able to:

• Recognise nominal, ordinal, interval and ratio data types • Recognise and use mode, median, mean, range, standard deviation and coefficient of variation • Calculate Laspeyres and Paasche index numbers • Use index numbers to calculate percentage changes and deflate series

1.2 Timings

Lectures: 1 hour

Private Study: 9.5 hours

Tutorials: 2 hours

Title Here


1.3 Private Study Exercises

You should spend approximately 9.5 hours on the Private Study for this topic. You should use this time to complete the exercises below as directed by your lecturer and to review the contents of this topic.

Exercise 1:

Review the lecture material for this topic and discuss any confusion you have with other students. List the points for which you remain uncertain and raise these in the tutorial. [You must still attempt the other exercises below and take your answers to the tutorials.]

Exercise 2:

A shop records its daily sales over a one week period. Calculate the median, mean (µ), population standard deviation (σ) and coefficient of variation (CV);

£148, £262, £164, £382, £246, £328, £126

Exercise 3:

Use MS Excel to check your answers to Exercise 2. Clue: look for the functions AVERAGE, MEDIAN and STDEVP (or in Excel 2010 STDEV.P)

Exercise 4:

Calculate the Laspeyres and Paasche price indices for the following data. Take 2005 as the base year.

year

litre of beer price quantity

litre of whiskey price quantity

litre of wine price quantity

2005 £0.95 200 £19.80 10 £10.50 36

2006 £0.99 150 £20.39 12 £11.15 48

2007 £1.05 120 £20.99 11 £12.35 60

Title Here


Exercise 5:

Investigate the following sites;

http://go.worldbank.org/OK9RCLZQN0 (this is part of the ‘Statistical Manual of the World Bank’ Accessed 3rd March 2011)

www.usablestats.com/tutorials/basicStats (this comes from the ‘Usable Statistics’ site. Accessed 3rd March 2011)

http://go.worldbank.org/OK9RCLZQN0�

http://www.usablestats.com/tutorials/basicStats�

Title Here


1.4 Tutorial Exercises

The tutorials for this topic will last for 2 hours. You can expect to spend some of this time discussing your answers to the Private Study exercises with your lecturer and other students. Your lecturer will then direct you on completing the tasks below.

Exercise 1:

The mean queuing (or waiting) time experienced by a customer with a single server is given by;

𝑡 = 𝑝 × �𝑝

𝑎 − 𝑝� × �

𝐶𝑉𝑝 + 𝐶𝑉𝑎2

�

Where p is the mean time for the customer to be processed, a is the mean inter-arrival time between customers, and CV are the respective coefficient of variations.

A newspaper kiosk takes on average 2 minutes to process a customer, with a standard deviation of 1 minute. A new customer arrives on average every 3 minutes with a standard deviation of 2 minutes. What is the typical time a customer waits to be served?

Exercise 2:

A price index series initially used 2001 as its base year; in 2006 the base year was updated. Link the two series to create a single series with 2006 as the base year. Use the single series to calculate annual percentage changes in price.

Year Price Index (2001 base)

Price Index (2006 base)

2001 100.00 2002 100.30 2003 103.97 2004 105.73 2005 106.33 2006 110.21 100.00 2007 105.79 2008 113.31 2009 121.10 2010 130.30

Exercise 3:

Show the equivalence of the Laspeyres price index formula to the weighted price relative formula.

Title Here


Exercise 4:

The lectures introduced the Laspeyres Price Index and the Paasche Price Index. What do you think the formulae would be for the Laspeyres Quantity Index and the Paasche Quantity Index?

Exercise 5:

Calculate the Laspeyres and Paasche quantity indices for the following data. Take 2005 as the base year.

year

litre of beer price quantity

litre of whiskey price quantity

litre of wine price quantity

2005 £0.95 200 £19.80 10 £10.50 36

2006 £0.99 150 £20.39 12 £11.15 48

2007 £1.05 120 £20.99 11 £12.35 60

Exercise 6:

Review of Private Study Exercises.

Title Here


Topic 2


Topic 2: Probability Distributions


This topic provides a brief reminder of the rules of simple probability and an overview of four important probability distributions.


• Recall the rules of simple probability • Use key probability distributions

o Binomial distribution o Poisson distribution o Exponential distribution o Normal distribution

• Calculate z-scores.

2.2 Timings

Lectures: 1 hour


Tutorials: 2 hours

Title Here




Exercise 1:


Exercise 2:

Investigate the following site.

www.usablestats.com/tutorials/basicStats2 (accessed 15th March 2011)

Exercise 3:

The economy is forecast to boom with probability 0.3, grow steadily with probability 0.5 or experience recession with probability 0.2. The returns on each of two shares in such circumstances are given below. Calculate the expected return (mean) and standard deviation for each share.

Share 1 Share 2 State Probability Return Return Boom 0.3 30% 15% Grow 0.5 5% 10%

Recession 0.2 -20% -10%

Exercise 4:

Type each of the following into a cell in Excel. Use the Excel Help facility to identify what the function calculates.

a. =POISSON(3, 6, FALSE)

b. =POISSON(3, 6, TRUE)

c. =BINOMDIST(6, 10, 0.6, FALSE)

d. =BINOMDIST(6, 10, 0.6, FALSE)

e. =NORMSDIST(0)

f. =NORMSDIST(-2.5)

http://www.usablestats.com/tutorials/basicStats2�

Title Here


Exercise 5:

Use Excel to produce a normal distribution table, an example is shown over page. The table should show P(Z ≤ z), with Z a random variable. Read z to the first decimal place from first column with the second decimal place read along top row.

Then investigate the following sites;

http://econ.lse.ac.uk/ie/iecourse/ec220course_statstables0203.pdf (accessed 15th March 2011)

www.reading.ac.uk/ssc/publications/stats_tables.pdf (accessed 15th March 2011)

z 0 1 2 3 4 5 6 7 8 9 -3.00 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010 -2.90 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014 -2.80 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019 -2.70 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026 -2.60 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036 -2.50 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048 -2.40 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064 -2.30 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084 -2.20 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110 -2.10 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143 -2.00 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183 -1.90 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233 -1.80 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294 -1.70 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367 -1.60 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455 -1.50 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559 -1.40 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681 -1.30 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823 -1.20 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985 -1.10 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170 -1.00 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379 -0.90 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611 -0.80 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867 -0.70 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148 -0.60 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451 -0.50 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776 -0.40 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121 -0.30 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483 -0.20 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859 -0.10 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247 -0.00 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641 0.00 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.10 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.20 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.30 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.40 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.50 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.60 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.70 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.80 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.90 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.00 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 1.10 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 1.20 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 1.30 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 1.40 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1.50 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 1.60 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1.70 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1.80 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 1.90 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 2.00 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817

http://econ.lse.ac.uk/ie/iecourse/ec220course_statstables0203.pdf�

http://www.reading.ac.uk/ssc/publications/stats_tables.pdf�

Title Here


2.10 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 2.20 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 2.30 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 2.40 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936 2.50 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 2.60 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 2.70 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 2.80 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 2.90 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 3.00 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990

Title Here




Exercise 1:

A new employee sits a preliminary Health and Safety examination. The examination consists of 10 multiple choice questions, wherein each question has four possible answers from which to choose. The pass mark is 40%. What is the probability that the employee passes by guessing only?

Exercise 2:

Clients arrive at a point of service at a rate of 20 per hour. What is the probability of exactly 20 clients arriving in a given hour? What is the probability of 20 to 25 clients arriving in a given hour? Assuming a client has just arrived, what is the probability that the next client arrives within 2 minutes? Assuming a client has just arrived, what is the probability that the next client arrives after 4 minutes have passed? (Assume client arrivals are described by the Poisson distribution.)

Exercise 3:

A company has supplied 10,000 projection lamps. Previous data indicates that the lifetime of the projection lamps is approximately normally distributed with mean of 6000 hours and standard deviation of 1000 hours. How many projection lamps are expected to fail in less than 5000 hours? How many projection lamps are expected to last more than 8000 hours?

Exercise 4:

Review of Private Study Exercises

Title Here


Topic 3


Topic 3: Inferential Statistics 1


This topic provides an overview of introductory inferential statistics.


• Recognise the terms sample statistic and population parameter • Use confidence intervals to indicate the reliability of estimates • Know when approximate large sample or exact confidence intervals are appropriate

3.2 Timings

Lectures: 1 hour


Tutorials: 2 hours

Title Here




Exercise 1:


Exercise 2:

A cruise ship was interested in the typical duration each client spent in the breakfast buffet. The entry and exit times of 30 cruisers was noted. The time spent at breakfast for the 30 were:

Time (minutes) Spent Dining 43 35 36 25 30 35 42 28 18 21 39 43 34 38 27 34 38 41 19 44 34 39 19 36 29 33 24 40 31 18

Calculate an approximate 99% confidence interval for the mean breakfasting time.

Exercise 3:

A country is due to hold a referendum on a constitutional change. A market research company asked 1,112 people whether they would vote ‘yes’ or ‘no’. 667 of the people polled indicated that they would vote ‘yes’. Calculate a 95% confidence interval for the proportion of the electorate that will vote ‘yes’.

Title Here


Exercise 4:

Use Excel to produce a table of percentage points of the Student t-distribution.

1 5.00% 2.50% 1.00% 0.50% 2 10.00% 5.00% 2.00% 1.00% 90.00% 95.00% 98.00% 99.00%

= n-1 1 6.3138 12.7062 31.8205 63.6567

2 2.9200 4.3027 6.9646 9.9248 3 2.3534 3.1824 4.5407 5.8409 4 2.1318 2.7764 3.7469 4.6041 5 2.0150 2.5706 3.3649 4.0321 6 1.9432 2.4469 3.1427 3.7074 7 1.8946 2.3646 2.9980 3.4995 8 1.8595 2.3060 2.8965 3.3554 9 1.8331 2.2622 2.8214 3.2498

10 1.8125 2.2281 2.7638 3.1693 11 1.7959 2.2010 2.7181 3.1058 12 1.7823 2.1788 2.6810 3.0545 13 1.7709 2.1604 2.6503 3.0123 14 1.7613 2.1448 2.6245 2.9768 15 1.7531 2.1314 2.6025 2.9467 16 1.7459 2.1199 2.5835 2.9208 17 1.7396 2.1098 2.5669 2.8982 18 1.7341 2.1009 2.5524 2.8784 19 1.7291 2.0930 2.5395 2.8609 20 1.7247 2.0860 2.5280 2.8453

Exercise 5:

Investigate the following site (sections 1 to 4 and 10).

http://davidmlane.com/hyperstat/confidence_intervals.html (accessed 21st March 2011)

http://davidmlane.com/hyperstat/confidence_intervals.html�

Title Here




Exercise 1:

The mean time taken for 20 workers to each complete a similar task was found to be 1 hour 12 minutes. The sample standard deviation was calculated as 14 minutes 20 seconds. Assuming the data is taken from a normally distributed population calculate an exact 99% CI of the mean.

Exercise 2:

Using the people in your tutorial group calculate 95% and 99% confidence intervals for the height of a human. State any assumptions you make.

Exercise 3:


Topic 4


Topic 4: Inferential Statistics 2


This topic provides an overview of using inferential statistics in hypothesis testing.


• Perform a single sample t-test of the mean • Perform a two sample t-test • Interpret significance probabilities • Perform a chi-squared goodness of fit test

4.2 Timings

Lectures: 1 hour


Tutorials: 2 hours

Title Here




Exercise 1:


Exercise 2:

A company will consider opening a new branch of its restaurant chain within a town if the mean amount spent per head when dining out is greater than £30. A lower figure would not match the company’s offering. One Saturday evening, market researchers successfully questioned 16 people who had dined out in the town and found a sample mean of £32.50 and sample standard deviation of £4.76. Test the following hypotheses at the 5% significant level.

H0: µ ≤ £30

H1: µ > £30

Exercise 3:

A fair trading investigation is being carried out in the European Union. As a first step the selling price (before tax) of a particular car model is being compared in two member countries. The following data have been collected:

Country 1: 22 cars, sample mean 14,500 euros, sample standard deviation 725 euros

Country 2: 18 cars, sample mean 14,000 euros, sample standard deviation 550 euros

The null hypothesis is that the mean selling price is equivalent in both countries. Test the null hypothesis at the 1% and 5% significance levels.

Exercise 4:

Study and critique the following website;

www.usablestats.com/tutorials/twoSamplet (accessed 11th April 2011)

Study the hypothesis testing material at http://stattek.com;

http://stattrek.com/Lesson5/HypothesisTesting.aspx (accessed 11th April 2011)

http://www.usablestats.com/tutorials/twoSamplet�

http://stattrek.com/Lesson5/HypothesisTesting.aspx�

Title Here




Exercise 1:

A software call centre wishes to improve its efficiency in solving clients’ questions.

Eight workers were selected at random to undertake additional training. Before and after training the performance of the workers was monitored. Calculate the mean number of solved problems per day, before and after training. Test the following hypotheses at the 5% significant level;

H0: Mean increase in performance ≤ 0

H1: Mean increase in performance > 0

solved problems per day before training following training Worker 1 50 55 Worker 2 63 66 Worker 3 57 58 Worker 4 45 49 Worker 5 40 38 Worker 6 62 67 Worker 7 64 69 Worker 8 53 50

Exercise 2:

Market research questioned 1085 managers about their preferred fashion brand and their career level. Are these associated?

Career Level Trainee Junior Middle

Labe

l Fashion Brand 1 245 93 30 Fashion Brand 2 221 110 77 Other 185 122 2

Exercise 3:


Title Here


Topic 5


Topic 5: Differentiation 1


This topic provides an overview of gradient, the definition of the derivative and the rules of differentiation.


• Find the derivatives of variables raised to a power • Use the rules of differentiation • Relate differentiation to optimization

o Obtain the economic order quantity formula

5.2 Timings

Lectures: 1 hour


Tutorials: 2 hours




Exercise 1:


Exercise 2:

For each case differentiate y with respect to x.

a. 𝑦 = 𝑥2 b. 𝑦 = 5𝑥4 c. 𝑦 = 2𝑥12 d. 𝑦 = 4

𝑥 e. 𝑦 = −1

𝑥+ 𝑥 + 7

f. 𝑦 = Ad𝑥

+ B𝑥 + K where A, d, B and K are constants.

Exercise 3:

Watch the following video from the http://mathcentre.ac.uk site;

http://mathcentre.ac.uk/video/1086/ (accessed 7th March 2011)

Exercise 4:

For each case differentiate y with respect to x.

a. 𝑦 = 𝑥12(3𝑥2 + 2𝑥) b. 𝑦 = 𝑥2𝑒−𝑥 c. 𝑦 = 𝑒�𝑥2+𝑥+1� d. 𝑦 = 𝑙𝑛(𝑥2)

e. 𝑦 = (2𝑥 + 5)10 f. 𝑦 = e𝑥

𝑥

Exercise 5:

The following sections of the http://mathcentre.ac.uk (accessed 7th March 2011) site provide useful material for further practice.

http://mathcentre.ac.uk/resources/Refresher%20Booklets/basic%20diff%20refresh1Emathcentre/final0203-ltsn-basicdiff.pdf

http://mathcentre.ac.uk/video/426/

http://mathcentre.ac.uk/video/1086/�

http://mathcentre.ac.uk/resources/Refresher%20Booklets/basic%20diff%20refresh1Emathcentre/final0203-ltsn-basicdiff.pdf�

http://mathcentre.ac.uk/resources/Refresher%20Booklets/basic%20diff%20refresh1Emathcentre/final0203-ltsn-basicdiff.pdf�



http://mathcentre.ac.uk/resources/uploaded/mc-ty-diffxtothen-2009-1.pdf


http://mathcentre.ac.uk/resources/uploaded/mc-ty-firstppls-2009-1.pdf


http://mathcentre.ac.uk/resources/uploaded/mc-ty-maxmin-2009-1.pdf

There is much other helpful material on the http://mathcentre.ac.uk site.

http://mathcentre.ac.uk/resources/uploaded/mc-ty-diffxtothen-2009-1.pdf�


http://mathcentre.ac.uk/resources/uploaded/mc-ty-firstppls-2009-1.pdf�


http://mathcentre.ac.uk/resources/uploaded/mc-ty-maxmin-2009-1.pdf�

http://mathcentre.ac.uk/�




Exercise 1:

EdTec Ltd sells basic oscilloscopes to educational customers. A combined project by the marketing and operations team has identified the following expressions to represent demand and total cost curves (in GBP).

price = 5000− 4𝑥

total cost = 5000 + 3000𝑥

Write down an expression for the annual profit, and show this on a graph. Differentiate profit to determine the annual quantity, x, that EdTec produce to maximise profit? What is that profit?

Exercise 2:

Assume that average revenue has a downward sloping linear curve;

𝐴𝑅 = c − k𝑥,

Where c and k are constants and the variable x is the quantity of goods sold. Use differentiation to show that the marginal revenue curve, MR, is downward sloping with twice the gradient of the AR curve.

Exercise 3:


Topic 6


Topic 6: Differentiation 2


This topic provides an overview of partial differentiation and, briefly the total differential.


• Carry out partial differentiation • Relate partial differentiation to optimization • Calculate partial point elasticities • Recognise the total differential

6.2 Timings

Lectures: 1 hour


Tutorials: 2 hours




Exercise 1:


Exercise 2:

For each case find the partial derivatives of z with respect to x and y.

a. 𝑧 = a𝑥2 + b𝑥 + c𝑥𝑦 + d𝑦2

b. 𝑧 = e2𝑥+3𝑦2

c. 𝑧 = (𝑥+𝑦)2

(3𝑥+𝑦)

Exercise 3:

For each case find the partial derivatives of z with respect to x and y.

a. 𝑧 = 𝑥2 + 𝑦2 − 10𝑥 − 10𝑦 + 50

b. 𝑧 = −𝑥2 + 𝑦2 + 10𝑥 − 10𝑦 + 50

c. 𝑧 = −𝑥2 − 𝑦2 + 10𝑥 + 10𝑦 + 50

Find the values of x and y such that both partial derivatives are zero. Use Excel to produce 3D surface plots of each case, and interpret the result.

Exercise 4:

The following sites (accessed 11th March 2011) provide useful material for further practice.

www.metalproject.co.uk/METAL/Resources/Films/Differential_equations/#film501

www.analyzemath.com/calculus.html#multivariable

http://www.metalproject.co.uk/METAL/Resources/Films/Differential_equations/#film501�

http://www.analyzemath.com/calculus.html#multivariable�




Exercise 1:

Use differentiation to find a relationship between marginal revenue and own price elasticity.

Exercise 2:

Use partial differentiation to find expressions for own and cross price elasticities for the demand function;

𝑞1 = 250 − 8𝑝1 + 5𝑝2

Assuming brand 2 has a price of p2 = £80, evaluate the own price elasticity for p1 = £20, £40 and £60.

Exercise 3:

Find own and cross price elasticities for the demand function;

𝑞1 = a𝑝1α𝑝2β

where a, α and β are constants.

Exercise 4:

A company sells a product at price P per unit, whether drop shipped or from held inventory. The annual demand, D, for the product is known for sure. Inspired by the classic economic order quantity model, the company has built an initial model to determine what proportion of the demand for product, x, should be met from inventory, and the order quantity, Q. The company has collected data for

Cd – the cost of purchasing a unit of the product inclusive of drop shipping

Ci – the cost of purchasing a unit of the product from a wholesaler plus shipping from inventory to end customer

Co – ordering cost

Ch – annual holding cost for one unit.

The initial model for profit, π, is described by the equation;


π = PD − 𝑥DCi − (1 − 𝑥)DCd −𝑥DCo𝑄

−𝑄Ch

2

By differentiating profit with respect to x and Q find expressions for the optimum fraction of demand met from inventory and the optimum order quantity.

Exercise 5:


Topic 7


Topic 7: Regression Analysis 1


This topic provides an overview of simple linear regression and correlation.


• Understand a straight line fit to bivariate data • Calculate and interpret Pearson’s correlation coefficient • Calculate and interpret Spearman’s correlation coefficient

7.2 Timings

Lectures: 1 hour


Tutorials: 2 hours




Exercise 1:


Exercise 2:

A hand car wash service has been monitoring its daily demand as it has changed price. Calculate the simple linear regression line and the Pearson correlation coefficient.

Price per Wash Number Cars £5.00 105 £5.50 113 £6.00 99 £6.50 94 £7.00 87

Exercise 3:

a. An advertising company decides which adverts to place in particular magazines. To aid this decision the company studies the response of subjects to adverts. One study examines the relationship between subject age and response. For one advert design, the results of a very small pilot study are below. Calculate the rank coefficient of correlation.

Person Age Band Attitude Anastasia 15-29 very good Beryl 25-34 neither good nor poor Cheryl 35-44 good Deborah 45-54 very poor Elizabeth >54 poor

b. A separate study for another advert resulted in the following data. Calculate the rank coefficient of correlation.

Person Age Band Attitude Alan 15-29 very good Bertie 25-34 neither good nor poor Christopher 35-44 good Dennie 25-34 good Eric 45-54 poor Frank >54 poor


Exercise 4:

Calculate the simple regression line and the coefficient of determination for each of the following four data sets (Anscombe, 1973). Describe your findings.

Set A Set B Set C Set D x y x y x y x y

10 8.04 10 9.14 10 7.46 8 6.58 8 6.95 8 8.14 8 6.77 8 5.76

13 7.58 13 8.74 13 12.74 8 7.71 9 8.81 9 8.77 9 7.11 8 8.84

11 8.33 11 9.26 11 7.81 8 8.47 14 9.96 14 8.1 14 8.84 8 7.04 6 7.24 6 6.13 6 6.08 8 5.25 4 4.26 4 3.1 4 5.39 19 12.5

12 10.84 12 9.13 12 8.15 8 5.56 7 4.82 7 7.26 7 6.42 8 7.91 5 5.68 5 4.74 5 5.73 8 6.89

Reference:

Anscombe, FJ. (1973). ‘Graphs in Statistical Analysis’ The American Statistician 27(1),17-21.

Exercise 5:

Investigate the Excel function ‘TREND’.

Investigate the Excel options for adding trend lines to scatter graphs. (First produce a scatter graph. Excel 2010: Single click on scatter graph. Drop down menu from Chart tools / Layout / Trendline. Excel 2003: Single click on scatter graph. Chart / Add Trendline.)

Exercise 6:

Study the following pages at http://stattrek.com

http://stattrek.com/AP-Statistics-1/Correlation.aspx (accessed 9th April 2011)

http://stattrek.com/AP-Statistics-1/Regression.aspx (accessed 9th April 2011)

http://stattrek.com/AP-Statistics-1/Correlation.aspx�

http://stattrek.com/AP-Statistics-1/Regression.aspx�




Exercise 1:

Use differentiation to minimise the SSE and so derive the simple regression formulae for the gradient m and the intercept c.

Suggested Answer:

Exercise 2:

Show that the total variation can be expressed as the sum of unexplained and explained variation.

Exercise 3:


Topic 8


Topic 8: Regression Analysis 2


This topic provides an overview of multiple regression and its extension with dummy variables. The topic is set at a level suitable for a future general manager, rather than a statistician or econometrician.


• Evaluate results from regression analysis • Interpret results from regression analysis • Recognize the possibility to extend regression analysis

8.2 Timings

Lectures: 1 hour


Tutorials: 2 hours




Exercise 1:


Exercise 2:

Data has been collected for advertised house prices in regions over a city.

Property Price Area Land School Region 1 507 156 1690 3 R1 2 488 167 888 1 R1 3 314 104 759 3 R1 4 231 73 792 4 R1 5 280 110 630 2 R2 6 503 210 655 1 R2 7 288 95 1000 2 R2 8 418 164 902 2 R2 9 291 121 1185 3 R3

10 356 155 780 1 R3 11 342 178 600 3 R3 12 490 200 1234 1 R3

Give an expected value for the advertised house price of a property with the following characteristics: ‘Area’ = 150, ‘Land’ = 850, ‘School’ = 1.5 and ‘Region’ = R3. Explain your answer carefully.

Exercise 3:

Plastic Sheds Ltd. wish to develop a demand function for their acrylic garden sheds. Develop the demand function based on the data below. All other things being equal, what would be the effect of a £100 decrease in the price of the competitor product?

Quarterly Quantity

Own Price

(£)

Competitor Price (£)

Median Income (£)

1990 750 860 18025


1960 600 730 18151 2090 840 820 18278 2080 790 770 18406 2030 660 730 18535 2100 830 870 18665 2110 730 700 18796 2170 850 790 18928 2160 760 690 19060 2090 600 630 19193 2100 630 690 19327 2190 740 680 19462

Exercise 4:

Two alternative presentations of multiple regression are at;

www.statsoft.com/textbook/multiple-regression (accessed 14th April 2011)

http://davidmlane.com/hyperstat/prediction.html (accessed 14th April 2011)

A more advanced presentation can be found out;

http://simon.cs.vt.edu/SoSci/converted/MRegression/ (accessed 14th April 2011)

http://www.statsoft.com/textbook/multiple-regression�

http://davidmlane.com/hyperstat/prediction.html�

http://simon.cs.vt.edu/SoSci/converted/MRegression/�




Exercise 1:

A healthcare service provider calculates a performance index in a way mandated by government. Each staff member within the organization is classed as either ‘Supervisory’ or ‘Frontline’. As a consultant you have been asked to develop a model relating the performance index to the numbers of ‘Supervisory’ and ‘Frontline’ staff. You have been provided with data from 10 of the service provider’s centres.

Centre Performance Index

Supervisory Staff

Frontline Staff

A 36 310 7010 B 34 388 4405 C 28 240 3501 D 37 359 3619 E 46 723 3098 F 45 470 5136 G 30 471 1260 H 57 657 4517 I 40 510 4319 J 57 677 5104

Exercise 2:

For the set of data investigated in Exercise 1, create a third variable ‘FxS’ by multiplying the number of ‘Frontline’ and ‘Supervisory’ staff for each case. Construct two more regression models. For the first include the variables ‘Frontline’ and ‘FxS’, for the second include ‘Supervisory’ and ‘FxS’.

Exercise 3:


Topic 9


Topic 9: Time Series Analysis


This topic provides an overview of time series analysis, with particular focus on decomposition and a brief introduction to exponential smoothing.


• Understand calendar, inflation and population adjustments • Deseasonalise data using additive or multiplicative decomposition. • Apply simple exponential smoothing.

9.2 Timings

Lectures: 1 hour


Tutorials: 2 hours




Exercise 1:


Exercise 2:

A fruit stall operates at the exit of a London tube station. Its sales figures are given below.

08:30 to 11:30 11:30 to 14:30 14:30 to 17:30 Day 1 £218.00 £112.00 £59.00 Day 2 £212.00 £116.00 £66.00 Day 3 £200.00 £115.00 £67.00 Day 4 £193.00 £92.00 £55.00

Use a 3 point moving average to smooth the data. Plot the time series, Y, and the trend, T. On a separate graph plot Y-T. Calculate appropriate seasonal adjustments.

Exercise 3:

Over the past four years the hotel zone of a popular Latin American tourist destination has been increasing its number of rooms. During this period of change the local tourism office has tracked occupancy rates. Extract the trend-cycle and seasonality adjustments from the data. (Use CMA4 and multiplicative decomposition). Plot and discuss any graphs that you think are relevant. What other data would be useful to characterise quantitative changes in the hotel zone?

Y (Occupancy Rate %) 2007 Q1 86

Q2 85

Q3 83

Q4 91

2008 Q1 87

Q2 77

Q3 78

Q4 89


2009 Q1 79

Q2 67

Q3 78

Q4 78

2010 Q1 73

Q2 69

Q3 59

Q4 73

Exercise 4:

Study the following sites.

The Australian Bureau of Statistics (www.abs.gov.au) discusses time series decomposition and the importance of seasonality. In particular the section ‘The Basics’ is a useful read at:

www.abs.gov.au/Ausstats/[email protected]/7d12b0f6763c78caca257061001cc588/af6b411c65fbe6b8ca2571090080a510!OpenDocument accessed 2nd April 2011.

A brief overview of exponential smoothing is given in the ‘Engineering Statistics Handbook’ of NIST, an agency of the US Department of Commerce, at:

www.itl.nist.gov/div898/handbook/pmc/section4/pmc431.htm

http://www.abs.gov.au/�

http://www.abs.gov.au/Ausstats/[email protected]/7d12b0f6763c78caca257061001cc588/af6b411c65fbe6b8ca2571090080a510!OpenDocument�

http://www.abs.gov.au/Ausstats/[email protected]/7d12b0f6763c78caca257061001cc588/af6b411c65fbe6b8ca2571090080a510!OpenDocument�

http://www.itl.nist.gov/div898/handbook/pmc/section4/pmc431.htm�




Exercise 1:

The number of rats captured in a grain store are summarised below. Use simple exponential smoothing with α = 0.2 and α = 0.7 to forecast the number of rats that will be caught in week 7.

Week Rats week 1 216 week 2 224 week 3 217 week 4 233 week 5 245 week 6 229

Exercise 2:

Calculate the mean squared error (MSE) and the Mean Absolute Error (MAE) for the sales data and forecast figures below.

Sales Forecast 1 Forecast 2 January 2910 2910 2910 February 2990 2910 2910 March 2837 2926 2958 April 3015 2908 2885 May 3069 2929 2963 June 2996 2957 3027 July 3268 2965 3008 August 3580 3026 3164 September 3236 3137 3414 October 3308 3157 3307


Exercise 3:

Locate an inflation adjusted House Price Index, such as that available from the UK’s ‘Nationwide’ building society, www.nationwide.co.uk/hpi/historical.htm. Investigate your time series using Excel and, for example a 12 point Centred Moving Average.

Exercise 4:


http://www.nationwide.co.uk/hpi/historical.htm�


Topic 10


Topic 10: Linear Programming


This topic provides an overview of Linear Programming, with a particular focus on graphical solution of a 2 variable problem.


• Recognize the concept of constrained optimisation • Formulate a two variable linear programme • Find a graphical solution to a two variable linear programme • Appreciate the process of sensitivity analysis

10.2 Timings

Lectures: 1 hour


Tutorials: 2 hours




Exercise 1:


Exercise 2:

Solve the following simultaneous equations;

a. Y = 5, 6X + 7Y = 210

b. Y = 5, 7X + 15Y = 525

c. 7X + 15Y = 525, 5X + 28Y = 700

d. 6X + 7Y = 210, 5X + 28Y = 700

Exercise 3:

Minimise: cost = 9X + 3Y

Subject to the following constraints:

Constraint 1: Y ≥ 5

Constraint 2: 6X +7Y ≥ 210

Constraint 3: 7X +15Y ≤ 525


Constraint 5: X, Y ≥ 0

Which constraints are binding and which are non-binding?

Exercise 4:

Use the web to find and study some presentations of the graphical solution of linear programming problems. For example:


www.purplemath.com/modules/linprog.htm accessed 4th April 2011.

www.msubillings.edu/BusinessFaculty/Harris/LP_Problem1.htm accessed 4th April 2011.

http://www.purplemath.com/modules/linprog.htm�

http://www.msubillings.edu/BusinessFaculty/Harris/LP_Problem1.htm%20accessed%204th%20April%202011�




Exercise 1:

Maximise: profit = 39X + 42Y

Subject to the following constraints:

Constraint 1: Y ≥ 5

Constraint 2: 6X +7Y ≥ 210



Constraint 5: X, Y ≥ 0

Which constraints are binding and which are non-binding?

Exercise 2:

For the previous exercise what happens if constraint 1 is relaxed to Y ≥ 4? What is the shadow price of constraint 1? For what increases and decreases of constraint 1 is this shadow price valid?

Exercise 3:


Topic 11


Topic 11: Linear Programming with Solver Routines


This topic provides an overview of Linear Programming with the Excel Solver.


• To formulate an LP for solution with a Solver • Interpret LP output

o Binding and non-binding constraints o Allowed increases and decreases o Shadow price

• Recognise extensions to LP e.g. integer programming.

11.2 Timings

Lectures: 1 hour


Tutorials: 2 hours




Exercise 1:


Exercise 2:

You are required to prepare a 3 slide presentation on the following problem. The presentation should include; (i) the problem expressed as an objective function and constraint inequalities and, (ii) a management discussion of the Answer and Sensitivity Reports. (First solve the problem without integer constraints so that Answer and Sensitivity Reports can be obtained. After this you might choose to solve the problem with integer constraints).

The small and independent company, ‘Bedroom Furniture Factory’ manufactures five items of bedroom furniture. Each month the company decides how many of each item to manufacture based on three sets of constraints: upstream material supply, in factory process and downstream demand. The company’s monthly labour costs are known and fixed. The contribution for each type of unit of furniture is known.

Profit Information:

Fixed Labour Costs: £55,000

Furniture Item

Contribution (Selling price – Material Costs)

Wardrobe £190 Ottoman £370 Drawers £250 Dressing table £310 Chair £100

Upstream (monthly supply side constraints):

Monthly Availability

Required per Unit Manufactured Material wardrobe ottoman drawers d. table chair Lumber 10 m3 0.03 0.01 0.02 0.015 0.005 Fabric 422 m2 0 3 0 0 1


Internal Process Constraints (monthly):

Hours Available

Hours Required per Unit Manufactured Process wardrobe ottoman drawers d. table chair Shaping 300 0.4 0.4 0.5 0.3 0.3 Plane&sand 300 0.5 0.1 0.5 0.2 0.2 Spray 200 0.5 0.8 0.5 0.1 0.1 Wash&trim 102 0 0.3 0 0 0.3 Assemble 1000 1 1.5 0.5 1.5 0.5 Pack 150 0.1 0.1 0.5 0.1 0.15

Downstream (demand side constraints):

Unit Monthly Demand

Wardrobe 200 Ottoman 50 Drawers 250 D. table 100 Chair 290

Exercise 3:

Browse the following websites:

www.solver.com/optimization.htm accessed 5th April 2011.

www.cheresources.com/refinery_planning_optimization.shtml accessed 5th April 2011.

http://www.solver.com/optimization.htm�

http://www.cheresources.com/refinery_planning_optimization.shtml%20accessed%205th%20April%202011�




Exercise 1:

A company is in the fortunate position of having six projects available with positive Net Present Value. Sadly the company has a current year capital ration of £4m. All other things being equal which projects should the company fund? [Tip: assume the projects are indivisible and use the binary constraint in Excel’s Solver.]

Project Required Current Year Investment Net Present Value A 2 3.94 B 3 7.24 C 2 4.08 D 1 3.07 E 2 7.53 F 3 3.35

Exercise 2:

A company has two production centres P1 and P2. It has four distribution centres D1, D2, D3 and D4. Production centre P1 has a weekly production volume of up to 15,000 units. Production centre P2 has a weekly production volume of up to 25,000 units. Each week the company decides what quantity of units to ship from each production centre to each distribution centre, based on the weekly requirement at each distribution centre, Q, and of the costs of shipping one unit, C:

D1 D2 D3 D4 D1 D2 D3 D4 P1 Q11 Q12 Q13 Q14 P1 C11 C12 C13 C14 P2 Q21 Q22 Q23 Q24 P2 C21 C22 C23 C24

Where,

C11 = £0.12, C12 = £0.15, C13 = £0.20, C14 = £0.10, C21 = £0.25, C22 = £0.20, C23 = £0.15, C24 = £0.15,

Next week’s requirements for the four distribution centres are; 10,000 for D1, 5200 for D2, 7500 for D3, 13,800 for D4. How many units should each of the production centres produce and ship to each of the distribution centres?

Exercise 3:


Topic 12


Topic 12: Decision Tree Analysis


This topic provides an overview of decision tree analysis.


• Represent a sequential management decision with a tree diagram • Use rollback to solve a decision tree and to inform a management decision • Understand the difference between a priori and a posteriori probabilities • Recognise 1st, 2nd and 3rd order uncertainty.

12.2 Timings

Lectures: 1 hour


Tutorials: 2 hours




Exercise 1:


Exercise 2:

Rollback the following decision tree assuming that the numbers represent a positive flow variable such as profit, revenues or net present value.

Exercise 3:

Rollback the decision tree from exercise 2, but now assume that the numbers represent a negative flow variable such as losses or costs. Comment on your answer.

Exercise 4:

A clothing retailer is due to open a new store in a city in which it is not yet represented. From previous experience the retailer knows that 70% of new store openings are profitable within the first year, whilst 30% are not. The company contracts a market research organisation to improve these estimates before opening the new store. The market research organisation estimates the reliability of its survey procedure as follows. If the new store is going to be profitable within the first year then the survey will be positive 50% of the time, neutral 30% of the time and negative 20% of the time. If


the store is not going to be profitable in the first year the survey will be positive 30% of the time, neutral 30% of the time and negative 40% of the time. Calculate the posterior probabilities for the stores first year profitability.

Exercise 5:

A company employs a large number of salespeople. You have been offered the job of travelling salesperson and can choose whether to work in Region 1, Region 2 or Region 3. You will be paid based on your performance. Of the existing salespeople with greater than median pay 30% are in Region 1, 40% are in Region 2. Of the existing salespeople earning less than median pay 40% are in Region 2 and 40% are in Region 3. In which Region would you choose to work and why?

Exercise 6:

Read the following articles;

Coopersmith E, Dean G, McVean J and Storaune E (2000) ‘Making Decisions in the Oil and Gas Industry’ Oilfield Review 12(4), p2-9. Available at www.slb.com/resources/publications/oilfield_review/en/2000/or2000_win.aspx accessed 1st April 2011.

Bryant I, Malinverno A, Prange M, Gonfalin M, Moffat J, Swager D, Theys P and Verga F. (2002) ‘Understanding Uncertainty’ Oilfield Review 14(3) p2-15 Available at www.slb.com/resources/publications/oilfield_review/en/2002/or2002_aut.aspx accessed 1st April 2011.

http://www.slb.com/resources/publications/oilfield_review/en/2000/or2000_win.aspx�

http://www.slb.com/resources/publications/oilfield_review/en/2002/or2002_aut.aspx�




Exercise 1:

A manager is deciding whether to undertake a detailed seismic survey prior to drilling a well. The alternative to drilling the well is to sell the rights to the tract. Based on existing information, the probability that the well is dry is 0.3 hence the probability that the well is rich is 0.7.

Seismic surveys improve knowledge of the tract but do not provide perfect knowledge.

The probability of a positive survey given that the well is actually rich is 0.5. The probability of a neutral survey given that the well is actually rich is 0.3. The probability of a negative survey given that the well is actually dry is 0.2.

The probability of a positive survey given that the well is actually dry is 0.3. The probability of a neutral survey given that the well is actually dry is 0.3. The probability of a negative survey given that the well is actually dry is 0.4.

Present values have been estimated for each of the rich and dry outcomes. Present costs of a seismic survey are known, as are present sale values for similar drilling rights.

Survey cost 4 Rich well 100 Dry well -120 Tract sale with no survey 30 Tract sale with neutral survey 30 Tract sale with positive survey 60 Tract sale with negative survey 0

Should the manager undertake the seismic survey?

Exercise 2:


Appendix


Equations and Statistical Tables

Management Statistics

Population mean and standard deviation

𝜇 =∑𝑓𝑖𝑥𝑖𝑁

𝜇 = �𝑝𝑖𝑥𝑖

𝜎 = �∑𝑓𝑖(𝑥𝑖−𝜇)2

𝑁 𝜎 = �∑𝑝𝑖(𝑥𝑖 − 𝜇)2

Population Coefficient of Variation

𝐶𝑉 =𝜎𝜇

Sample mean and standard deviation

�̅� =∑𝑓𝑖𝑥𝑖𝑛

𝑠 = �∑𝑓𝑖(𝑥𝑖 − 𝜇)2

𝑛 − 1

Sample skewness

𝑛(𝑛 − 1)(𝑛 − 2)��

𝑥𝑖 − �̅�𝑠

�3𝑛

𝑖=1

Sample Coefficient of Variation

𝐶𝑉 =𝑠�̅�

Simple Index Number

𝑅 =𝑝𝑛𝑝0

× 100

Laspeyres and Paasche Price Index Numbers

𝐿𝑃𝐼 = 100 ×∑𝑞𝑜𝑝𝑛∑𝑞𝑜𝑝𝑜

𝑃𝑃𝐼 = 100 ×∑𝑞𝑛𝑝𝑛∑𝑞𝑛𝑝𝑜

Laspeyres and Paasche Quantity Index Numbers

𝐿𝑄𝐼 = 100 ×∑𝑞𝑛𝑝0∑ 𝑞𝑜𝑝𝑜


𝑃𝑄𝐼 = 100 ×∑𝑞𝑛𝑝𝑛∑𝑞0𝑝𝑛

Probability 𝑃(𝐸1 or 𝐸2) = 𝑃(𝐸1) + 𝑃( 𝐸2) − 𝑃(𝐸1 ∩ 𝐸2)

Theorem of Bayes

𝑃(𝐸|𝐴) =𝑃(𝐴|𝐸)𝑃(𝐸)

𝑃(𝐴)

Theorem of Total Probability

𝑃(𝐴) = �𝑃(𝐴|𝐸𝑖)𝑃(𝐸𝑖)𝑖

Binomial Distribution B(n, p)

𝑃(𝑋 = 𝑥) = 𝑛!𝑥!(𝑛−𝑥)!

𝑝𝑥(1 − 𝑝)(𝑛−𝑥)

𝜇 = 𝑛𝑝 𝜎 = �𝑛𝑝(1 − 𝑝)

Poisson Distribution Po(λ)

𝑃(𝑋 = 𝑥) = e−𝜆𝜆𝑥

𝑥!

𝜇 = 𝜆 𝜎 = √𝜆

Exponential Distribution

𝑓(𝑡) = 𝜆e−𝜆𝑡, 𝑡 ≥ 0 𝑃(𝑇 < 𝑡) = 1 − 𝑒−𝜆𝑡 𝜇 = 1

𝜆 𝜎 = 1

𝜆

Standard Normal Distribution

𝑧 =𝑥 − 𝜇𝜎

𝑓(𝑧) = e−𝑧

22�

√2𝜋


Hypothesis Testing

Distribution of sample means

𝜇�̅� = 𝜇 𝜎�̅� = 𝜎 √𝑛

Large sample confidence interval of the mean (n ≥ 30)

(𝜇−,𝜇+) = ��̅� − 𝑧𝛾𝑠√𝑛

, �̅� + 𝑧𝛾𝑠√𝑛

�

Large sample confidence interval of proportion (np and n(1 - p) ≥ 5)

(𝑝−,𝑝+) = ��̅� − 𝑧𝛾��̅�(1 − �̅�)

𝑛 , �̅� + 𝑧𝛾�

�̅�(1 − �̅�)𝑛

�

Exact confidence interval (underlying population has normal distribution)

(𝜇−,𝜇+) = ��̅� − 𝑡𝛾𝑠√𝑛

, �̅� + 𝑡𝛾𝑠√𝑛

�

Approximate large sample test of the mean

𝑍 =�̅� − 𝜇𝑠√𝑛�

Under the null hypothesis Z ~ N(0, 1), approximately.

Student’s one sample t-test of the mean.

𝑇 =�̅� − 𝜇𝑠√𝑛�

Under the null hypothesis T ~ t (n - 1)

Independent two sample t-test

𝑇 = 𝑥1��−𝑥2��

𝑆𝑝�1𝑛1+ 1𝑛2

𝑆𝑝2 = (𝑛1−1)𝑠12+(𝑛2−1)𝑠22

𝑛1+𝑛2−2

Under the null hypothesis T ~ t (n1 + n2 - 2)


Fitting Data

χ2 Goodness of fit test (Ei ≥ 5 for all i)

𝜒2 = ∑ (𝑂𝑖−𝐸𝑖)2

𝐸𝑖 𝑘

𝑖=1 Under the null hypothesis 𝜒2~𝜒2(𝑘 −𝑚 − 1) k is number of categories, m is number of model parameters estimated from data

χ2 Test of Association (Ei ≥ 5 for all i)

𝜒2 = ∑ (𝑂𝑖−𝐸𝑖)2

𝐸𝑖 𝑘

𝑖=1 Under the null hypothesis 𝜒2~𝜒2�(𝑟 − 1)(𝑐 − 1)� r is number of rows, c is number of columns

Simple Linear Regression

𝑦� = m𝑥i + c is the least SSE straight line where; m = ∑�𝑥i − �̅��𝑦i − 𝑦��

∑�𝑥𝑖 − �̅��2 m = n∑𝑥i𝑦i − ∑𝑥i ∑𝑦i

n∑𝑥i2 − (∑𝑥i)2

c = 𝑦� − m�̅�

The Coefficient of Determination

𝑅2 = 𝑟2 =∑(𝑦 � − 𝑦�)2

∑(𝑦 − 𝑦�)2

The Pearson Correlation Function

R = r = n∑𝑥i𝑦i − ∑𝑥i ∑𝑦i

�(n∑𝑥i2 − (∑𝑥i)2)(n∑𝑦i2 − (∑𝑦i)2)

Spearman’s Rank Correlation (with no ties)

𝑟𝑠 = 1 − 6∑𝑑2

𝑛(𝑛2 − 1)


Differentiation

Definition

𝑓′(𝑥) =𝑑𝑦𝑑𝑥

= lim∆𝑥→0

𝑓(𝑥 + ∆𝑥)− 𝑓(𝑥)∆𝑥

Standard Derivatives

𝒚

𝒅𝒚𝒅𝒙

𝑦 = 𝑎𝑥𝑛 𝑑𝑦𝑑𝑥

= 𝑛𝑎𝑥n−1

𝑦 = ea𝑥 𝑑𝑦𝑑𝑥

= 𝑎ea𝑥

𝑦 = 𝑙𝑛(𝑎𝑥) = 𝑙𝑜𝑔e(𝑥) 𝑑𝑦𝑑𝑥

=1𝑥

Rules of Differentiation

𝑑𝑑𝑥

�𝑎𝑓(𝑥) + 𝑏𝑔(𝑥)� = 𝑎 𝑑𝑓𝑑𝑥

+ 𝑏 𝑑𝑔𝑑𝑥

𝑑𝑑𝑥

�𝑓(𝑥)𝑔(𝑥)� = 𝑓(𝑥) 𝑑𝑔𝑑𝑥

+ 𝑔(𝑥) 𝑑𝑓𝑑𝑥

𝑑𝑑𝑥

�𝑓�𝑔(𝑥)�� = 𝑑𝑓𝑑𝑔

𝑑𝑔𝑑𝑥

𝑑𝑑𝑥

�𝑓(𝑥)𝑔(𝑥)� =

𝑔(𝑥)𝑑𝑓𝑑𝑥−𝑓(𝑥)𝑑𝑔

𝑑𝑥

�𝑔(𝑥)�2

Elasticities of Demand

Own price Cross price Income

𝐸𝑝 =𝜕𝑄1𝜕𝑝1

𝑝1𝑄1

𝐸12 =𝜕𝑄1𝜕𝑝2

𝑝2𝑄1

𝐸𝐼 =𝜕𝑄1𝜕𝐼

𝐼𝑄1

The Total Differential

𝑦 = 𝑦(𝑥1 + 𝑥2 + 𝑥3 +...) 𝑑𝑦 = 𝜕𝑦

𝜕𝑥1𝑑𝑥1 + 𝜕𝑦

𝜕𝑥2𝑑𝑥2 + 𝜕𝑦

𝜕𝑥3𝑑𝑥3 +...

∆𝑦 ≈ 𝜕𝑦

𝜕𝑥1∆𝑥1 + 𝜕𝑦

𝜕𝑥2∆𝑥2 + 𝜕𝑦

𝜕𝑥3∆𝑥3 +...


Time Series

The additive decomposition model

𝑌𝑛 = 𝑇𝑛+𝑆𝑛 + 𝐼𝑛

The multiplicative decomposition model

𝑌𝑛 = 𝑇𝑛 × 𝑆𝑛 × 𝐼𝑛

Three Point Moving Average

Tn = ⅓(Yn-1 +Yn +Yn+1)

Four Point Centred Moving Average

𝑇𝑛 = 18

(𝑌𝑛−2+2𝑌𝑛−1+2𝑌𝑛+2𝑌𝑛+1+𝑌𝑛)

Simple Exponential Smoothing

Ft+1 = αYt + (1 – α)Ft

Errors

𝑀𝑆𝐸 = 1𝑁∑ �𝑌𝑗 − 𝐹𝑗�

2𝑁𝑗=1

𝑀𝐴𝐸 = 1𝑁��𝑌𝑗 − 𝐹𝑗�𝑁

𝑗=1


Probabilities under the Normal Distribution Curve

z 0 1 2 3 4 5 6 7 8 9 -3.50 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 -3.40 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002 -3.30 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003 -3.20 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005 -3.10 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007 -3.00 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010 -2.90 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014 -2.80 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019 -2.70 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026 -2.60 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036 -2.50 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048 -2.40 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064 -2.30 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084 -2.20 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110 -2.10 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143 -2.00 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183 -1.90 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233 -1.80 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294 -1.70 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367 -1.60 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455 -1.50 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559 -1.40 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681 -1.30 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823 -1.20 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985 -1.10 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170 -1.00 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379 -0.90 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611 -0.80 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867 -0.70 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148 -0.60 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451 -0.50 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776 -0.40 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121 -0.30 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483 -0.20 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859 -0.10 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247 -0.00 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641

P(Z < z)

z


z 0 1 2 3 4 5 6 7 8 9 0.00 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.10 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.20 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.30 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.40 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.50 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.60 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.70 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.80 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.90 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.00 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 1.10 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 1.20 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 1.30 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 1.40 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1.50 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 1.60 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1.70 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1.80 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 1.90 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 2.00 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817 2.10 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 2.20 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 2.30 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 2.40 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936 2.50 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 2.60 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 2.70 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 2.80 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 2.90 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 3.00 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990 3.10 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993 3.20 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995 3.30 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997 3.40 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998 3.50 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998


Percentage Points of the Student Distribution

α1 5.00% 2.50% 1.00% 0.50%

α2 10.00% 5.00% 2.00% 1.00% γ 90.00% 95.00% 98.00% 99.00%

df 1 6.3138 12.7062 31.8205 63.6567

2 2.9200 4.3027 6.9646 9.9248 3 2.3534 3.1824 4.5407 5.8409 4 2.1318 2.7764 3.7469 4.6041 5 2.0150 2.5706 3.3649 4.0321 6 1.9432 2.4469 3.1427 3.7074 7 1.8946 2.3646 2.9980 3.4995 8 1.8595 2.3060 2.8965 3.3554 9 1.8331 2.2622 2.8214 3.2498

10 1.8125 2.2281 2.7638 3.1693 11 1.7959 2.2010 2.7181 3.1058 12 1.7823 2.1788 2.6810 3.0545 13 1.7709 2.1604 2.6503 3.0123 14 1.7613 2.1448 2.6245 2.9768 15 1.7531 2.1314 2.6025 2.9467 16 1.7459 2.1199 2.5835 2.9208 17 1.7396 2.1098 2.5669 2.8982 18 1.7341 2.1009 2.5524 2.8784 19 1.7291 2.0930 2.5395 2.8609 20 1.7247 2.0860 2.5280 2.8453 21 1.7207 2.0796 2.5176 2.8314 22 1.7171 2.0739 2.5083 2.8188 23 1.7139 2.0687 2.4999 2.8073 24 1.7109 2.0639 2.4922 2.7969 25 1.7081 2.0595 2.4851 2.7874 26 1.7056 2.0555 2.4786 2.7787 27 1.7033 2.0518 2.4727 2.7707 28 1.7011 2.0484 2.4671 2.7633 29 1.6991 2.0452 2.4620 2.7564 30 1.6973 2.0423 2.4573 2.7500 31 1.6955 2.0395 2.4528 2.7440 32 1.6939 2.0369 2.4487 2.7385 33 1.6924 2.0345 2.4448 2.7333 34 1.6909 2.0322 2.4411 2.7284 35 1.6896 2.0301 2.4377 2.7238 36 1.6883 2.0281 2.4345 2.7195 37 1.6871 2.0262 2.4314 2.7154 38 1.6860 2.0244 2.4286 2.7116 39 1.6849 2.0227 2.4258 2.7079 40 1.6839 2.0211 2.4233 2.7045

t

α1 α1

-t

γ


∞ 1.6449 1.9600 2.3263 2.5758

Critical Values for the χ2 Distribution

0.05 0.01

αR 5.00% 1.00% df

1 3.841 6.635

2 5.991 9.210

3 7.815 11.345

4 9.488 13.277

5 11.070 15.086

6 12.592 16.812

7 14.067 18.475

8 15.507 20.090

9 16.919 21.666

10 18.307 23.209

11 19.675 24.725

12 21.026 26.217

13 22.362 27.688

14 23.685 29.141

15 24.996 30.578

16 26.296 32.000

17 27.587 33.409

18 28.869 34.805

19 30.144 36.191

20 31.410 37.566

21 32.671 38.932

22 33.924 40.289

23 35.172 41.638

24 36.415 42.980

25 37.652 44.314

26 38.885 45.642

27 40.113 46.963

28 41.337 48.278

29 42.557 49.588

30 43.773 50.892

αR

Documents

ABM Student Guide V1.0.pdf