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Tutorial Letter 101/3/2015
Applied Statistics II
STA2601
Semesters 1 & 2
Department of Statistics
IMPORTANT INFORMATION:
This tutorial letter contains importantinformation about your module andincludes the assignment questions
for both semesters.
STA2601/101/3/2015
CONTENTS
Page
1 INTRODUCTION ..................................................................................................................4
1.1 Tutorial matter.......................................................................................................................4
2 PURPOSE OF AND OUTCOMES FOR THE MODULE ..........................................................5
2.1 Purpose ...............................................................................................................................5
2.2 Outcomes.............................................................................................................................5
3 LECTURER(S) AND CONTACT DETAILS .............................................................................6
3.1 Lecturer(s)............................................................................................................................6
3.2 Department ..........................................................................................................................6
3.3 University .............................................................................................................................6
4 MODULE RELATED RESOURCES.......................................................................................6
4.1 Prescribed books..................................................................................................................6
4.2 Recommended books ...........................................................................................................7
4.3 Electronic Reserves (e-Reserves)..........................................................................................7
5 STUDENT SUPPORT SERVICES FOR THE MODULE ..........................................................7
5.1 Contact with Fellow Students.................................................................................................7
5.1.1 Study Groups .......................................................................................................................7
5.1.2 myUnisa...............................................................................................................................7
5.1.3 Discussion classes ...............................................................................................................8
6 MODULE-SPECIFIC STUDY PLAN.......................................................................................8
7 MODULE PRACTICAL WORK AND WORK-INTEGRATED LEARNING .................................8
8 ASSESSMENT.....................................................................................................................9
8.1 Assessment plan ..................................................................................................................9
8.2 General assignment numbers ................................................................................................9
8.2.1 Unique assignment numbers .................................................................................................9
8.2.2 Due dates for assignments ....................................................................................................9
8.3 Submission of assignments .................................................................................................10
8.4 Assignments.......................................................................................................................10
9 OTHER ASSESSMENT METHODS.....................................................................................11
10 EXAMINATION...................................................................................................................11
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10.1 Examination Admission .......................................................................................................11
10.2 Examination Period.............................................................................................................11
10.3 Examination Paper..............................................................................................................11
10.4 Previous Examination Papers ..............................................................................................12
10.5 Tutorial Letter with Information on the Examination ...............................................................12
11 FREQUENTLY ASKED QUESTIONS ..................................................................................12
12 SOURCES CONSULTED....................................................................................................12
13 CONCLUSION ...................................................................................................................12
ADDENDUM A: FIRST SEMESTER ASSIGNMENTS ......................................................................13
A.1 Assignment 01....................................................................................................................13
A.2 Assignment 02....................................................................................................................17
A.3 Assignment 03....................................................................................................................21
ADDENDUM B: SECOND SEMESTER ASSIGNMENTS..................................................................27
B.1 Assignment 01....................................................................................................................27
B.2 Assignment 02....................................................................................................................31
B.3 Assignment 03....................................................................................................................35
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1 INTRODUCTION
Dear Student
Welcome to this module. We trust your studies will be rewarding and successful!
The module is called APPLIED STATISTICS II. The module is the follow-up on the module STA1502(Statistical Inference I). The name Applied Statistics was chosen because of its double meaning:Data analysis is in effect applied statistical theory and you will learn how to apply the statisticalsoftware package SAS JMP. This means that you must have access to a suitable computer fora component of practical work.
This module will equip you with a proper basis in statistical knowledge, introduce you to a statisticalpackage and highlight the value of thorough statistical know-how that the business and outsideworld require of students who major in Statistics! Knowledge of statistics will enable you to conductquantitative research and statistical literacy will enable you to understand research reports youmight encounter as a scientist in your everyday life or enable you to understand statistical reportsyou might encounter as a manager in your business.
We trust that you will work seriously and continuously. We hope that you will enjoy this module andwish you all the best!
1.1 Tutorial matter
Take note that every tutorial letter you will be receiving is important and you have to read themall immediately and carefully. Some information contained in these tutorial letters may be urgent,while others may, for example, contain examination information. So, it is wise to keep them all in a�le!
Some of this tutorial matter may not be available when you register. Tutorial matter thatis not available when you register will be posted to you as soon as possible, but is alsoavailable on myUnisa.At the time of registration, you will receive an inventory letter that will tell you what you have re-ceived in your study package and also show items that are still outstanding. Also see the brochureentitled my Studies @ Unisa.
Check the study material that you have received against the inventory letter. You should havereceived all the items listed in the inventory, unless there is a statement like �out of stock� or �notavailable�. If any item is missing, follow the instructions on the back of the inventory letter withoutdelay.
Shortly after registration The Department of Despatch should supply you with the following tutorialmatter for this module:
� Tutorial letter 101. Read it and save it as it contain important information as well as yourassignments for the semester.
� A study guide written by a lecturer to guide you through the relevant sections in the pre-scribed book. Use it together with the textbook as the guide indicates the relevant prescribedsections, explaining dif�cult concepts in more detail, giving additional examples and exer-cises, etc.
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� Other tutorial letters to further assist you with your studies, will be dispatched to you through-out the semester.
If you have access to the Internet, you can view the study guide and tutorial letters for the modulesfor which you are registered on the University's online campus, myUnisa, at http://my.unisa.ac.za.
There are two types of tutorial letters:
� The 100-series (e.g. Tutorial letter 101, 102, 103, etc.) containing general information, as-signment questions, information about your lecturer or the examination, a trial paper, etc.
� The 200-series (e.g. Tutorial letter 201, 202, 203, etc.) containing the solutions to the assign-ments and the trial paper.
2 PURPOSE OF AND OUTCOMES FOR THE MODULE
2.1 Purpose
Students credited with this unit standard, will be able to identify the correct technique, manage thestatistical software SAS JMP to do the computations and interpret the results for decisions regard-ing tests for normality, independence and hypothesis concerning means, proportions, variancesand regression. Students should be able to solve applied statistics problems arising in governmentand industry.
2.2 Outcomes
Qualifying students will be able to:
� describe various probability distributions and illustrate their applications as probabilities as-sociated with critical values from the tables.
� describe desirable properties of estimators for population parameters and derive these esti-mators through the methods of maximum likelihood and least squares.
� use statistical software SAS JMP.
� do statistical estimation and hypothesis testing for a single population.
� test for normality by employing various techniques.
� do statistical estimation and hypothesis testing involving two populations.
� do statistical estimation and hypothesis testing involving more than two populations.
� measure relationships between variables.
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3 LECTURER(S) AND CONTACT DETAILS
3.1 Lecturer(s)
The lecturer responsible for this module is as follows:
Ms S. MuchengetwaGJ GERWEL (C-Block), Floor 6, Of�ce 6-05Tel: (011) 670-9253Cell: 074 065 9020E-mail address: [email protected]
You might also want to write to us. Letters should be sent to:
Ms S. MuchengetwaDepartment of StatisticsPO Box 392UNISA0003All queries that are not of a purely administrative nature but are about the content of this moduleshould be directed to me. Please have your study material with you when you contact me. E-mailaddress is included above.
PLEASE NOTE: Letters to lecturers may not be enclosed with or inserted into assignments.
3.2 Department
The departmental secretary can be contacted at 011 670-9255 for other queries.
3.3 University
If you need to contact the University about matters not related to the content of this module, pleaseconsult the publication My Studies @ Unisa that you received with your study material. Thisbrochure contains information on how to contact the University (e.g. to whom you can write fordifferent queries, important telephone and fax numbers, addresses and details of the times certainfacilities are open).
Always have your student number at hand when you contact the University.
4 MODULE RELATED RESOURCES
4.1 Prescribed books
The prescribed book for this semester is
Sall, J., Creighton, L., and Lehman, A.: JMP Start Statistics A Guide toStatistics and Data Analysis Using JMP, 5th Edition (2012).
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You have to buy this book. Please consult the list of of�cial booksellers and their addresses listed inmy Studies @ Unisa. Prescribed books can be obtained from the University's of�cial booksellers. Ifyou have dif�culty locating your book(s) at these booksellers, please contact the Prescribed BooksSection at 012 429 4152 or e-mail [email protected]. If you cannot �nd the book you can buythe latest edition.
You need to purchase one other publication. The publication is a book of tables containing thenormal, t-, chi-squared and F-tables.
STOKER, DJ: Statistical Tables / Statistiese Tabelle 4th Edition (1997).
Foreign students may have dif�culty in obtaining this book. If you are unable to obtain this bookyou may use any other book of tables, but keep in mind that the tables used in the examination willbe the ones from Stoker.
4.2 Recommended books
There are no recommended books for this module.
4.3 Electronic Reserves (e-Reserves)
There are no e-Reserves for this module.
5 STUDENT SUPPORT SERVICES FOR THE MODULE
For information on the various student support systems and services available at Unisa (e.g. stu-dent counseling, tutorial classes, language support), please consult the publication my Studies @Unisa that you received with your study material.
5.1 Contact with Fellow Students
5.1.1 Study Groups
It is advisable to have contact with fellow students. One way to do this is to form study groups.Please consult the publication my Studies@Unisa to �nd out how to obtain the addressesof students in your region.
5.1.2 myUnisa
If you have access to a computer that is linked to the internet, you can quickly access resourcesand information at the University. The myUnisa learning management system is Unisa's onlinecampus that will help you to communicate with your lecturers, with other students and with theadministrative departments of Unisa - all through the computer and the internet.
To go to the myUnisa website, start at the main Unisa website, http://www.unisa.ac.za, and thenclick on the �Login to myUnisa� link on the right-hand side of the screen. This should take you tothe myUnisa website. You can also go there directly by typing in http://my.unisa.ac.za.
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Please consult the publication my Studies @ Unisa which you received with your study material formore information on myUnisa.
5.1.3 Discussion classes
There are no discussion classes offered in this module. Should the need for discussion classesarise in future, students will be informed in advance about actual dates and venues.
6 MODULE-SPECIFIC STUDY PLAN
SEMESTER 1 Study units for preparing your assignments From ToStudy Guide and Workbook
Assignment 1 Chapter 1 to Chapter 2 Registration 28 FebruaryStart writing your assignment 28 February 06 March
Assignment 2 Chapter 3 to Chapter 5 07 March 13 MarchStart writing your assignment 14 March 20 March
Assignment 3 Chapter 6 to Chapter 8 21 March 03 AprilStart writing your assignment 04 April 10 April
Note: For the text book, you will see the instructionsin you workbook on which pages to read.
SEMESTER 2 Study units for preparing your From ToassignmentsStudy Guide and Workbook
Assignment 1 Chapter 1 to Chapter 2 Registration 07 AugustStart writing your assignment 08 August 14 August
Assignment 2 Chapter 3 to Chapter 5 15 August 21 AugustStart writing your assignment 22 August 28 August
Assignment 3 Chapter 6 to Chapter 8 29 August 18 SeptemberStart writing your assignment 19 September 25 September
Note: For the text book, you will see theinstructions in you workbook on whichpages to read.
7 MODULE PRACTICAL WORK AND WORK-INTEGRATED LEARNING
There are no practicals for this module.
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8 ASSESSMENT
8.1 Assessment plan
The assessment in this module consists of three assignments and an examination.
Your �nal mark for the module is determined from your semester mark and your examination mark.The semester mark forms 20% and the examination mark 80% of the �nal mark. The semestermark is composed of 30% of assignment 1, 35% of assignment 2 and 35% of assignment 3 of themarks you receive. An assignment submitted late or not at all will give you 0%. If you do well inyour assignments you have a good semester mark and that can make all the difference between apass or fail or between a distinction or simply a pass!
The three assignments prescribed for this module must be seen as part of the learning process.The typical assignment question is a re�ection of a typical examination question. There are �xedsubmission dates for the assignments and each assignment is based on speci�c chapters in thestudy guide. You have to adhere to these dates as assignments are only marked if they are receivedon or before the due dates.
You will only get examination admission if you submit the �rst assignment by its due date. Youshould complete all assignments as well as you can, since
� they are the sole contributors towards your semester mark,
� they form an integral part of the learning process and indicate the form and nature of thequestions you can expect in the examination.
Assignments and LearningAssignments are seen as part of the learning material for this module. As you do the assignment,study the reading texts, consult other resources, discuss the work with fellow students or tutorsor do research, you are actively engaged in learning. Looking at the assessment criteria givenfor each assignment, and the feedback you receive in your marked assignment, will help you tounderstand what is required of you more clearly.
8.2 General assignment numbers
The three assignments are numbered 01, 02 and 03 for each semester.
8.2.1 Unique assignment numbers
Please note that each assignment has its unique six-digit assignment number which has to bewritten on the cover of your assignment upon submission. The unique numbers are given later onin this tutorial letter; you will �nd them in the heading of each set of assignment questions.
8.2.2 Due dates for assignments
The closing dates for the submission of the assignments are:
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Assignment for Sections from the following Due DateSEMESTER 1 Chapters are covered1 Chapters 1 and 2 of Study Guide 06 March 2015
and Workbook2 Chapters 3, 4 and 5 of Study Guide 20 March 2015
and Workbook3 Chapters 6, 7, and 8 of Study Guide 10 April 2015
and Workbook
Assignment for Sections from the following Due DateSEMESTER 2 Chapters are covered1 Chapters 1 and 2 of Study Guide 14 August 2015
and Workbook2 Chapters 3, 4 and 5 of Study Guide 28 August 2015
and Workbook3 Chapters 6, 7, and 8 of Study Guide 25 September 2015
and Workbook
8.3 Submission of assignments
For detailed information on assignments, please refer to the my Studies @ Unisa brochure, whichyou received with your study package.To submit an assignment via myUnisa:
� Go to myUnisa.
� Log in with your student number and password.
� Select the module.
� Click on assignments in the menu on the left-hand side of the screen.
� Click on the assignment number you wish to submit.
� Follow the instructions.
For general information and requirements as far as assignments are concerned, see the brochuremy Studies @ Unisa which you received with your study material.
8.4 Assignments
This tutorial letter 101 contains the assignments for both semesters, so select the semester you areenrolled for and do the set of assignments for that semester only. The assignments for Semester1 are in Appendix A, pages 13�26. The assignments for Semester 2 are in Appendix B, pages27�43. Solutions to the assignments will be posted to ALL students registered for this module awhile after the closing date of the relevant assignment. Solutions will also be available onmyUnisa.
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9 OTHER ASSESSMENT METHODS
There are no other assessment methods for this module.
10 EXAMINATION
10.1 Examination Admission
You need to have a �nal mark of 50% to pass this module and 75% to obtain a distinction.
In this module a maximum of 20 marks is added to your examination mark (out of 80) to form your�nal mark. This 20% contribution comes from the marks you obtained for the three assignmentsand is called your semester mark. If you do well in your assignments you have a good semestermark and that can make all the difference between a pass or fail or between a distinction or simplya pass!
Currently admission to the examination is only based on the proof that you are actively involvedin your studies. This proof is based on the submission of your �rst assignment before a �xedgiven date. Admission therefore does not rest with the department and if you do not submit thatparticular assignment in time, we can do nothing to give you admission. Although you are mostprobably a part time student with many other responsibilities, work circumstances will not be takeninto consideration for exemption from assignments or the eventual admission to the examination.
No concession will be made to students who do not qualify for the examination.
10.2 Examination Period
This module is offered in a semester period of �fteen weeks. This means that
� if you are registered for the �rst semester, you will write the examination in May/June 2015and should you fail and qualify for a supplementary examination, that supplementary exami-nation will be written in October/November 2015.
� if you are registered for the second semester, you will write the examination in October/November2015 and should you fail and qualify for a supplementary examination, that supplementaryexamination will be written in May/June 2016.
The examination section will provide you with information regarding the examination in general,examination venues, examination dates and examination times. Eventually, your results will alsobe processed by them and sent to you.
10.3 Examination Paper
Your examination will be a 2 hour examination. The questions will be similar to the assignmentquestions, but there will also be questions on theory. Should you have a �nal mark of less than50%, it implies that you failed the module STA2601. However, should your results be within aspeci�ed percentage (usually from 40% to 49%), you will be given a second chance in the formof a supplementary examination on the dates as speci�ed in 10.2. If you fail the examination
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with less than 40%, the semester mark will not count to help you pass. Please note also that thesemester mark does not apply in the case of a supplementary examination. The �nal mark after asupplementary examination is simply the mark you achieved in that examination, expressed as apercentage.
10.4 Previous Examination Papers
Previous examination papers are available to students on myUnisa. In addition, you will re-ceive a trial paper towards the end of the semester that you can use as an indication of typicalexamination questions. Solutions to this trial paper is also sent out in a follow-up tutorial letter. Re-member that the examples, exercises, activities in the guide as well as your assignment questionsare also indicators of typical examination questions.
10.5 Tutorial Letter with Information on the Examination
As mentioned before, you will receive a tutorial letter containing a trial paper. Should the lecturerwant to discuss any matter about the examination, it will be included in this tutorial letter. In thestudy guide you are given clear indications of the sections in the textbook that you have to knowand can be tested on in the examination. Remember that you have to work continuously and donot treat statistics as any other subject, where it may be possible to study only selected sections ofthe work. All the topics are interlinked and you will de�nitely run into trouble if you skip sections!
You are automatically admitted to the exam on the submission of Assignment 01 by a speci�c date� see Section 8.1. Please note that lecturers are not responsible for exam admission, and ALLenquiries about exam admission should be directed by e-mail to [email protected].
11 FREQUENTLY ASKED QUESTIONS
The my Studies @ Unisa brochure contains an A-Z guide of the most relevant study information.Please refer to this brochure for any other questions.
12 SOURCES CONSULTED
Several books were consulted in preparing this tutorial letter.
13 CONCLUSION
Remember that there are no "short cuts" to studying and understanding statistics. You need to bededicated, work consistently and practise, practise and practise some more! We hope that you willenjoy studying this module and we wish you success in your studies.
Your lecturer
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ADDENDUM A: FIRST SEMESTER ASSIGNMENTS
A.1 Assignment 01
ONLY FOR SEMESTER 1 STUDENTSASSIGNMENT 01Unique Nr.: 618989
Fixed closing date: 6 March 2015
QUESTION 1
(a) A shipment of six television sets contains two defective sets. A hotel makes a random pur-chase of three of the sets. Let X be the number of defective sets purchased by the hotel.
(i) Construct the probability distribution of X: (3)
(ii) Calculate the mean value of X . (3)
(iii) Calculate the standard deviation of X . (4)
(iv) Would you say that the distribution is symmetrical? (7)
(b) (i) Show that if
T1 D .X1 � �/2
T2 D12
h.X1 � �/2 C .X2 � �/2
iand
T3 D12.X1 � X2/2
they are both unbiased estimators for � 2: (7)
(ii) Which estimator is the most ef�cient?�HINT: T3 D 1
2 .X1 � X2/2 D
�X1 � X
�2C�X2 � X
�2 with X1 C X22
:
�(9)
[33]
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QUESTION 2Suppose that X1; X2; ::::; X7 is a random sample from a n.8I 9/ distribution and that
X D17
7PiD1X i and Y D
7PiD1
"X i � X�
#2.
Suppose that we also de�ne V1 D5PiD3
�.X i � 8/ =3
�2V2 D
7PiD6
�.X i � 8/ =3
�2W D
7PiD1
[X i � 3]2
9
(a) Write down the joint density function for fx2x4.x2; x4/: (3)
(b) Find P .X1 > 11/ : (3)
(c) Find P .7 < X1 < 12/ : (4)
(d) What is E .V2/? (2)
(e) What is Var .Y /
0@where Y D 7PiD1
"X i � X�
#21A? (2)
(f) What is the distribution of U DV1=3V2=2
? (2)
(g) What is the distribution of1U? (2)
(h) Find a value a such that P .U > a/ D 0:05: (2)
[20]
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QUESTION 3
(a) Draw separate freehand sketches of
(i) a n.3I � 2/ p.d.f. (2)
(ii) a negatively skew p.d.f. (2)
(iii) a leptokurtic p.d.f. (2)
(iv) a scatter diagram of an .XI Y / data set for which � < 0: (2)
(b) What is the relationship between a Type II error and the power of a test? (2)
(c) Discuss the connection between two random variables, X1 and X2; being uncorrelated andindependent. (3)
(d) Give the de�nition of the P-value or exceedance probability. (1)
(e) Name two distributions which are symmetric about zero. (2)
[16]
QUESTION 4
(a) Let X1; X2 and X3 be three independent random variables such that
E.X1/ D c1�1 C c2�2I E.X2/ D c1�1 and E.X3/ D c2�2
Find the least squares estimators of �1 and �2 if you assume that c1 and c2 are knownconstants.
(15)
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(b) The probability distribution function (p.d.f.) of the two-parameter gamma distribution (withparameters � > 0 and � > 0) is given by
f .xI�I�/ D1
0 .�/ ��x��1e�x=� for x > 0
D 0 for x � 0
Let X1; X2; :::; Xn be a random sample from this distribution and assume that the parameter� is known, but that the parameter � is unknown.
(i) Show that the likelihood function L .�/ in given by
L .�/ D 0 .�/�n ���n nYiD1X i
!��1e�
PX i=�
(4)
(ii) Find Log L .�/. (3)
(iii) Show that the maximum likelihood estimator (m.l.e) of � equalsPX i=n�. (4)
(iv) Assume the fact that E.X/ D �� if X has a two-parameter gamma distribution. Wouldyou say that the m.l.e. (derived in question (c) above) is an unbiased estimator? (Givethe de�nition that you use and justify your answer,) (5)
[31]
[Total Marks: 100]
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A.2 Assignment 02
ONLY FOR SEMESTER 1 STUDENTSASSIGNMENT 02Unique Nr.: 618993
Fixed closing date: 20 March 2015
QUESTION 1A random sample n D 45 was selected from the marks obtained by STA20601 students for assign-ment 01, and yielded the following:
55 58 67 62 69 79 67 57 51 4855 53 61 66 70 53 43 74 64 5279 51 58 81 56 61 66 91 71 8659 65 73 84 56 64 49 76 70 6047 54 85 44 45
(a) Test the sample for
(i) Skewness (two-sided) at the 10% level. (7)
(ii) Kurtosis (two-sided) at the 10% level. (7)
(b) Would you say that the sample comes from a normal distribution? (1)
[15]
QUESTION 2The manufacturing of a battery designed for a speci�c toy claims that the lifetime of the batteryhas an exponential distribution with a mean of 20 hours. The following data were collected toinvestigate his claim:
Lifetime Number of(in hours) batteries
t � 5 1085 � t � 10 10710 � t � 15 9015 � t � 20 5020 � t � 25 4525 � t � 30 4330 � t � 35 32
t � 35 25Total 500
17
Test the assumption that the data come from an exponential distribution with a mean of 20 hours.Use � D 0:05. [17]
QUESTION 3
(a) In a study of attitude towards early retirement (and partial pension) a special index for work-ing environment was also reported for each interviewed person. The index ranges from 1to 10 with 1 representing a very bad working environment and 10 representing an excellentworking environment.
The table shows the sample classi�cation according to attitude towards early retirement andthe index for working environment.
Index for workingenvironment1-3 4-7 8-10
Attitude Good system 100 50 50towards Mediocre system 40 90 70
early retirement Bad system 10 10 80
We would like to test at the 5% level of signi�cance whether attitude towards early retirementand the index for working environment are independent factors.
(i) Using SAS JMP, construct a Mosaic Plot of the data. (5)
(ii) Interpret the Mosaic Plot. (3)
(iii) State the appropriate null and alternative hypothesis for this test. (2)
(iv) What test statistic is used to test these hypotheses and what is the value of the teststatistic? (2)
(v) Show manually how you would obtain the expected values and the value of the teststatistic? (7)
(iv) Looking at the row percentages in your SAS JMP output, can you draw anyconclusions? (3)
(v) What is your �nal conclusion? Use both the critical value approach and the p-valueapproach. (4)
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(b) The proportions of blood types O; A; B and AB in the general population of a particularcountry are known to be in the ratio 49 : 38 : 9 : 4, respectively. A research team, investigatinga small isolated community in the country, obtained the following frequencies of blood type.
O n1 D 87
A n2 D 59
B n3 D 20
AB n3 D 4
Test at the 5% level of signi�cance the hypothesis that the proportions in this community donot differ signi�cantly from those in the general population. (11)
[37]
QUESTION 4
(a) In a summer tea-part in Gauteng, Pretoria, a lady claimed to be able to discern, by tastealone, whether a cup of tea with milk had the tea poured �rst or the milk poured �rst. Anexperiment was performed by a researcher to see if her claim is valid. Twelve cups of tea areprepared and presented to her in random order. Six had the milk poured �rst, and six had thetea poured �rst. The lady tasted each one and rendered her opinion.
The results are summarized in a 2 � 2 table below:
Lady says RowTea �rst Milk �rst total
Poured Tea 5 1 6
�rst Milk 1 5 6
Column total 6 6 12
Does the information above support the theory that the lady has no discerning ability. Test atthe 5% level of signi�cance. (7)
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(b) Tyre pressure (in kPa) was measured for the right and left front tyres on a sample of 10vehicles. Assume pressures follow a bivariate normal distribution. The following data wasobtained.
Right tyre Left tyrePressure Pressure
184 185206 203193 200227 213193 196218 221213 216194 198178 180207 210
(i) Find a 95% con�dence interval for �, the population correlation between the pressure inthe right tyre and the pressure in the left tyre. (5)
(ii) Test the hypothesis that � > 0:9: (6)
(b) A psychologist used two well known scales to obtain data on anxiety and frustration tolerancefor a sample of 20 women. (Anxiety is measured on a scale from 1 to 20 being extremelyanxious and frustration tolerance is measured on a scale from 1 to 30 being extremely tol-erant.) He obtained correlation of r D �0:939. A colleague of this psychologist repeatedthe experiment on an independent sample of 30 men and found a correlation coef�cient ofr D �0:783 (between anxiety and frustration tolerance ) for the men.
(i) Construct a 95% con�dence interval for the difference between the two correlations.(10)
(ii) Using the con�dence interval constructed in part (i), test the hypothesis whether thecorrelations are the same. (3)
[31]
[Total Marks: 100]
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A.3 Assignment 03
ONLY FOR SEMESTER 1 STUDENTSASSIGNMENT 03Unique Nr.: 619002
Fixed closing date: 10 April 2015
QUESTION 1
(a) Variance is an important aspect of quality control, in that variability of output is a measure ofconsistency. If a machine making ball bearings is highly variable in its output, much of theproduction run will be unacceptable�either too large or too small in diameter. Suppose thatthe �rst 20 ball bearings have a mean diameter of 6.003 millimeters with a standard deviationof 0.017 millimeters. Determine the 99% con�dence interval for the standard deviation of thediameter of the ball bearings in the run. (5)
(b) In a study of human reaction time in response to a certain stimulus, psychologists used twoindependent samples. Sample one was a random sample of 11 males between the agesof 20 and 40 and sample two was a random sample of 13 females in the same age group.The sample variances of the reaction times were 12m sec2 for the males and 4m sec2 for thefemales. Can the psychologists conclude that the reaction times of males are more variablethan the reaction times of females? Let a D 0:05 (9)
[14]
QUESTION 2The following data gives the weights of 30-full term babies, born at a metropolitan hospital andrecorded to the nearest tenth of a kilogram.
3:3 3:5 3:1 2:8 3:7 3:6 3:7 2:5 3:9 3:23:7 3:5 3:4 3:3 3:5 2:6 3:1 3:1 3:9 3:42:8 3:6 4:3 4:1 3:5 3:9 3:1 3:5 3:0 3:5
A researcher claims that the average birth weight of full term babies is 3.6kg. Let � be the averagebirth weight.
(a) What assumption(s) is/are necessary in order to conduct the statistical test speci�ed in (c)below? (2)
(b) Use SAS JMP to determine whether the assumptions are met. Give a brief discussion. (7)
21
(c) Manually test whether the average birth weight of these babies is signi�cantly less than 3:6kg. Use � D 0:05: (8)
(d) Manually show how this test procedure will change, if in fact you know that � D 0:5 kg? (6)
(e) Manually �nd a 90% con�dence interval for �. Can you use this interval to con�rm yourconclusion in (c)? Justify your answer. (5)
(f) Use JMP to redo the t-Test in (c) and the z-Test in (d). (15)
(g) Use SAS JMP to determine whether there is any reason to reject the null hypothesisH0: � D 0:5?Use a 5% level of signi�cance and test two-sided. (4)
(h) Suppose that another sample of 24 babies was taken from another metropolitan area inde-pendent from the �rst one and the following statistics were obtained.
Y D 3:5125 S2Y D 0:1803
Would you say that the mean birth weight of the second sample is higher than that of the �rstsample? Use � D 0:05.
[Hint: assume that both variances are unknown but equal.] (8)
[55]
QUESTION 3
(a) We wish to test H0 : � D 30 against H1 : � 6D 30; using a sample of size n D 10:from a normal population with mean � and variance � 2: What is the power of the test if� D 30C
p2�? (4)
(b) In response to a complaint that a particular tax assessor (A) was biased, an experimentwas conducted to compare the assessor named in the complaint with another tax assessor(B) from the same of�ce. Eight properties were selected, and each was assessed by bothassessors. The assessments (in thousands of rands) are shown in the table below:
Property Assessor 1 Assessor 21 276:3 275:12 288:4 286:83 280:2 277:34 294:7 290:65 268:7 269:16 282:8 281:07 276:1 275:38 279:0 279:1
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The following SAS JMP output was obtained.
Using the 0.05 level of signi�cance, do the data provide suf�cient evidence to indicate thatassessor A tends to give higher assessments than assessor B. Clearly state the hypothesisimplied by the question and how it can be tested. Give the rejection region and theconclusions. (6)
[10]
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QUESTION 4A national home builder wants to compare the prices per 1000 board feet of standard or bettergrade green Douglas �r framing lumber. He randomly selects �ve suppliers in each of the fourprovinces where the builder is planning to begin construction. The prices are given in the tablebelow:
State1 (10R) 2 (10R) 3(10R) 4 (10R)261 236 250 265255 240 245 270258 225 255 258267 233 248 275270 240 260 275
DO NOT USE SAS JMP. DO THIS MANUALLY:(Regard the data as random samples from normal populations.)
(a) What are the values of S21 , S22 , S
23 , and S
24? (4)
(b) (i) Compute the �ordinary� average of the four variances computed in (a). (2)
(ii) Compute the MSE according to the de�nition in the study guide. What do you notice?(2)
(c) Do you think it is reasonable to assume that the other two remaining basic assumptions (apartfrom normality that was given as an assumption) of independence and equal population vari-ances are met? (6)
(d) Test at the 5%level of signi�cance whether the population means of the four different groupsdiffer.
(i) State the null and alternative hypotheses.
(ii) State the rejection region and conclusion. (8)
(e) Perform multiple comparisons on all pairs of means. Discuss your results. (8)
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QUESTION 5
Using the data in question 4, use SAS JMP, submit your output obtained AND discuss theanalysis regarding each of the following:
(a) Do you think it is reasonable to assume that the four groups may be considered as indepen-dent groups? (2)
(b) Use Levene's test to determine if the four groups have equal population variances? Use� D 0:05 level of signi�cance. (State your hypothesis and justify your answer.) (8)
(b) Do the data provide suf�cient evidence to indicate that the average price per 1000board feet of Douglas �r differs among the four provinces at the 5% level of signi�cance?)
Justify your answer by giving attention to the following detail:
(i) State the appropriate null and alternative hypothesis for this test.
(ii) What test statistic is used to test these hypotheses?
(iii) What is the value of the test statistic? (9)
(c) Can one conclude at the 5% level of signi�cance whether �2 6D �1 D �4: (Justify youranswer) (9)
(d) Compare the means of the four provinces to determine where the are any differences usingthe Tukey-Kramer HSD method of multiple comparisons. Use all statistics available in theoutput you generated. (5)
[Hint: See last year's tutorial 101 on myunisa under announcement for the type ofoutputs you should generate.]
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QUESTION 6A marketing research experiment was conducted to study the relationship between the length oftime necessary for a buyer to reach a decision and the number of alternative package designs of aproduct presented. Brand names were eliminated from the packages and the buyers made theirselections using the manufacturer's product descriptions on the packages as the only buying guide.The length of time necessary to reach a decision was recorded for 15 participants in the marketingresearch study.
Length of decisionTime, y (sec) 5, 8, 8, 7, 9, 7, 9, 8, 9, 10, 10, 11, 10, 12, 9Number ofalternatives, x 2 3 4
(a) Plot the data to verify that linear regression is a suitable model. (3)
(b) Verify that the linear regression of Y on X is bY D 4:3C 1:5X: (6)
(c) Compile (complete and compute) the following table:
X i Yi bYi Yi � bYi2 5 7:3 �2:32 8 7:3 0:72 8 7:3 0:7:::
::::::
:::4 12 10:3 1:74 9 10:3 �1:3
(5)
(d) Estimate the average length of time necessary to reach a decision when three alternativesare presented, using a 95% con�dence interval. (6)
(e) TestTest H0 : �1 D 0 against
H1 : �1 > 0 at the 1% level of signi�cance.
(6)
(f) Produce a SAS JMP output (Ensure that the correlation of the two variables is included). (7)
[33]
[Total Marks: 175]
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ADDENDUM B: SECOND SEMESTER ASSIGNMENTS
B.1 Assignment 01
ONLY FOR SEMESTER 2 STUDENTSASSIGNMENT 01Unique Nr.: 619012
Fixed closing date: 14 August 2015
QUESTION 1
(a) From a box containing four black balls and two green balls, three balls are drawn in succes-sion, each ball being replaced in the box before the next draw is made. Let X be the numberof green balls.
(i) Construct the probability distribution of X: (3)
(ii) Calculate the mean value of X . (3)
(iii) Calculate the standard deviation of X . (4)
(iv) Would you say that the distribution is symmetrical? (7)
(b) Let X1 and X2 be independent normal variables with mean � and variance � 2:
Suppose that
T1 D 23X1 C
13X2I
T2 D 14X1 C
34X2 and
T3 D 12X1 C
12X2
(i) Show that they are all unbiased estimators of �: (7)
(ii) Which estimator is the most ef�cient and why? (9)
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QUESTION 2Suppose that X1; X2; ::::; X10 is a random sample from a n.100I 100/ distribution and that
X D110
10PiD1X i and Y D
10PiD1
"X i � X�
#2.
Suppose that we also de�ne V1 D5PiD1
�.X i � �/ =�
�2V2 D
10PiD7
�.X i � �/ =�
�2W D
10PiD1
�.X i � �/ =�
�2
:
(a) Is fX1 .x1/ D fX2 .x2/? (2)
(b) Write down the joint density function for fx4x10.x4; x10/: (3)
(c) Find P .X1 > 120/ : (3)
(d) Is Z DX5 � 10010
� n .0I 1/? (2)
(e) What is E .W /? (2)
(f) What is Var .Y /
0@where Y D 10PiD1
"X i � X�
#21A? (2)
(g) What is the distribution of U DV1=5V2=4
? (2)
(h) What is the distribution of1U? (2)
(i) Find a value a such that P�1U< a
�D 0:95: (2)
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QUESTION 3Complete the following statements in your answer book (i.e. give the missing words and do notwaste time to rewrite everything).
(a) The .............................estimators of �1; �2; :::; � k are found by .............................
Q .�1; �2; :::; � k/ DnPiD1.X i � E .X i //2
(i.e. setdQd� j
equal to .............................for j D 1; 2; � � � ; k/: (3)
(b) We commit a...............................error if we do not reject H0 when H0 is false.
� D P .not rejecting H0j H1 is true/
(1)
(c) The probability 1 � � D P .not rejecting H0j H1 is true/ is called the .............................of thetest. (1)
(d) �2 D�4� 4
is the fourth standardized moment and it measures the ............................ of a
distribution. A distribution with �2 > 3 is called..................................... . (2)
(e) The third standardised moment �1 D�3� 3is a measure of the ............................ of a distribu-
tion. A distribution with �1 D 0 is called..................................... . (2)
(f) Repeated measurements on the same individual, for example "paired observations" .X i ; Yi /for i D 1; 2; 3; : : : ; n cannot be considered as ...........................................observations. (1)
(g) If we have k independent random samples .X11; X22; :::; X1n/
.X21; X22; :::; X2n/ � � � .Xk1; Xk2; :::; Xkn/ ; such that the i-th sample comes from a n��i I �
2�distribution where
X i D1n
nPjD1X i j
then SSE DkPiD1
nPjD1............................ is called the error sum of squares and MSE D............................
is called the mean square error.
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To test H0 : �1 D �2 D ::: D �k we use the test statistic F D............................ (4)
[14]
QUESTION 4
(a) Let W1; :::;Wk be independent random variables such that:
E.Wi / D ci�1 C c2i �2 i D 1:::::k
Var.Wi / D � 2 i D 1:::::k
where �1and �2 are unknown parameters and c1:::::ck known constants. Find the leastsquares estimators for �1and �2: (20)
(b) Let Y1; Y2; : : : ; Ym be independent random variables with Yi � n�a C bxi I � 2
�for i D 1I 2I � � � Im.
Assume that aI b and xi are known constants.
(i) Write down the p.d.f. fY .yi /. (4)
(ii) Find the maximum likelihood of � 2: (9)
[33]
[Total Marks: 100]
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B.2 Assignment 02
ONLY FOR SEMESTER 2 STUDENTSASSIGNMENT 02Unique Nr.: 619027
Fixed closing date: 28 August 2015
QUESTION 1The following data have been observed in an experiment:
20 31 8 20 13 17 9 31 34 8 20 24 20 24 2217 28 23 21 23 8 0 28 31 24 10 15 43 13 15
(a) (i) Test the null hypothesis that the sample comes from a normal distribution against thealternative that the distribution is leptokurtic. (Use � D 0:05/:
(ii) Does the population have the skewness of a normal distribution? (Use � D 0:10/:
(15)
(b) Classify the 30 observed values into six classes with equal probability for each classinterval and test the null hypothesis that the observations come from a n .20I 100/ distribution.Let � D 0:10: [Clearly show the derivation of the six equiprobable intervals.] (20)
. [35]
QUESTION 2
(a) Suppose that the temperament of people (how good- or ill-tempered they are) can be mea-sured by a psychological scale and be classi�ed into three distinct groups. In a study todetermine if there exists a relationship between the (true!) colour of hair and the temper ofa person, a random sample of 200 people were subjected to this test and the colour of theirhair noted. The following data were obtained:
Colour of hairBlond Red Brown Black
Good 8 4 30 18Temper Even 24 10 32 34
Bad 8 6 18 8
We would like to test at the 5% level of signi�cance whether there exists a relationshipbetween the colour of hair and the temper of a person.
31
(i) Using SAS JMP, construct a Mosaic Plot of the data. (5)
(ii) Interpret the Mosaic Plot. (3)
(iii) State the appropriate null and alternative hypothesis for this test. (2)
(iv) What test statistic is used to test these hypotheses and what is the value of the teststatistic? (2)
(v) Show manually how you would obtain the expected values and the value of the teststatistic? (7)
(iv) Looking at the row percentages in your SAS JMP output, can you draw anyconclusions? (3)
(v) What is your �nal conclusion? Use both the critical value approach and the p-valueapproach. (4)
(b) During 1977 a random sample of 365 births showed the following distribution over the monthsof a year.
Time Interval Observed number of births
Warm months: January FebruaryNovember December
126
Moderate months: March AprilSeptember October
122
Cold months: May JuneJuly August
117
Test the hypothesis that the number of births is evenly distributed over the year. (Test at the 5%level of signi�cance). (12)
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QUESTION 3
(a) A �rm wants to test two totally different methods of promoting a new product. Two comparableareas are chosen and the two methods are used for a period of two months, one in each area.After this period a random sample of consumers is taken in each area and each person isasked whether he/she knows about the new product. The results of this experiment are asfollows:
Knows about the productYes No
Promotion method Method 1 6 1Method 2 1 4
Do the data supply evidence that method 1 had better results than method 2? Use � D0:05: (7)
(b) At an abattoir the carcass mass (X ) and the front foot mass (Y ) in kg of a random sample ofsize n D 15 of two-year-old Hereford oxen were recorded as:
X i Yi180 1:9220 2:3200 2:1250 2:5280 2:9225 2:2260 2:7190 2:0290 3:0210 2:1220 2:1250 2:6280 2:6260 2:7240 2:3
(i) Find a 95% con�dence interval for �, the population correlation between carcass massand front foot mass. (6)
(ii) Test the hypothesis that � D 0: (7)
33
(b) Suppose that exactly the same experiment (as in (a)) was conducted on a random sample of20 Charolais oxen (also two-year-olds) and it yielded a sample correlation coef�cient of 0.89(call this sample 2).
Test H0 : �1 D �2against H1 : �1 > �2: at the 1% level of signi�cance. (7)
[27]
[Total Marks: 100]
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B.3 Assignment 03
ONLY FOR SEMESTER 2 STUDENTSASSIGNMENT 03Unique Nr.: 619053
Fixed closing date: 25 September 2015
QUESTION 1
(a) When a new production line is being started, management must get an estimate of the meanand variability of the time required to perform tasks in order to time the movement of the line.A sample of 25 workers performs the same task and requires a mean of 4.11 minutes and astandard deviation of 1.85 minutes to do the task. Obtain a 95% con�dence intervals for themean and standard deviation of the time required of all workers to do the task. (11)
(b) The following statistics were computed from data on the hardness of wood stored indoorsand outdoors. Use � D 0:10 and test whether the variability of hardness is affected byweathering.
Sample stored indoors: n1 D 25 X1 D 117 6�X i1 � X1
�2D 8 625
Sample stored outdoors: n2 D 49 X2 D 132 6�X i2 � X2
�2D 27 244
(11)
[22]
QUESTION 2The production yield (in kg) of 20 randomly chosen Macadamia nut trees (growing on a speci�cfarm in the sub-tropical region of Mpumalanga) for the 2006 season was:
4:4 4:0 4:7 4:6 4:1 3:9 4:0 5:2 4:6 3:64:2 3:9 2:8 6:7 5:4 6:2 5:8 3:8 3:9 4:2
Let � denote the mean production (in kg) per tree, and assume that � 2 is unknown.
(a) What assumption(s) is/are necessary in order to conduct the statistical test speci�ed in (c)below? (2)
(b) Use SAS JMP to determine whether the assumptions are met. Give a brief discussion. (7)
35
(c) Manually test H0 : � D 4 against H1 : � > 4 at the � D 0:05 level of signi�cance. (8)
(d) Manually show how this test procedure will change, if in fact you know that � D 0:90 kg? (6)
(e) Manually �nd a 90% con�dence interval for �. Can you use this interval to con�rm yourconclusion in (c)? Justify your answer. (5)
(f) Use JMP to redo the t-Test in (c) and the z-Test in (d). (15)
(g) Use SAS JMP to determine whether there is any reason to reject the null hypothesisH0: � D 0:90kg? Use a 5% level of signi�cance and test two-sided. (4)
(h) Suppose that another sample of 30 randomly chosen Macadamia nut trees for the 2006season was taken and the following statistics obtained:
Y D 4:4567 S2Y D 0:7115
Would you say that the mean production per tree from the �rst sample is signi�cantly differ-ently from that of the second sample? Use � D 0:05.
[Hint: assume that both variances are unknown but equal.] (8)
[55]
QUESTION 3Two lots of insects of the same species, treated with different growth hormones, were weighed after20 days. All other variables like food, temperature, etc. were kept the same. (In other words, theexperiment entailed day-old insects, 22 in total, that were assigned at random to each "treatment".)
Weight of insect (mgs)Growth Hormone A Growth Hormone B
57 89120 30101 82137 50119 39117 22104 5773 3253 9668 31118 88
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When we perform an analysis of variance with only two classes, the F-test as de�ned in theorem7.7 (p. 199) is equivalent to the t-test with two independent groups of equal size. (This is de�nedin section 7.3).
Verify this statement by performing both the mentioned analyses for the data above. In other words:
Use the following SAS JMP output below.
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39
(a) Use SAS JMP output to perform a t-test to test the hypothesis
H0 : �1 D �2 againstH1 : �1 6D �2 Use � D 0:05
Clearly show all your steps and state the necessary assumptions. (5)
(b) (i) Use the SAS JMP output to perform an ANOVA test. Use � D 0:05. (5)
(ii) What assumptions must be satis�ed before this procedure is valid? (2)
(iii) Are they met? (8)
[20]
QUESTION 4Twenty third graders were randomly separated into four equal groups, and each group was taughta mathematical concept using a different teaching method. At the end of the teaching period,progress was measured by a unit test. The scores are shown below.
Group1 2 3 4112 111 140 10192 129 121 116124 102 130 10589 136 106 12697 99 123 119
DO NOT USE SAS JMP. DO THIS MANUALLY:(Regard the data as random samples from normal populations.)
(a) What are the values of S21 , S22 , S
23 , and S
24? (4)
(b) (i) Compute the �ordinary� average of the four variances computed in (a). (2)
(ii) Compute the MSE according to the de�nition in the study guide. What do you notice?(2)
(c) Do you think it is reasonable to assume that the other two remaining basic assumptions (apartfrom normality that was given as an assumption) of independence and equal population vari-ances are met? (6)
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(d) Test at the 5%level of signi�cance whether the population means of the four different groupsdiffer.
(i) State the null and alternative hypotheses.
(ii) State the rejection region and conclusion. (8)
(e) Perform multiple comparisons on all pairs of means. Discuss your results. (8)
[30]
QUESTION 5
Using the data in question 4, use SAS JMP, submit your output obtained AND discuss theanalysis regarding each of the following:
(a) Do you think it is reasonable to assume that the four groups may be considered as indepen-dent groups?
(2)
(b) Use Levene's test to determine if the four groups have equal population variances? Use� D 0:05 level of signi�cance. (State your hypothesis and justify your answer.) (8)
(b) Do the data provide suf�cient evidence to indicate a difference in the average scoresfor the four teaching methods at the 5% level of signi�cance?)
Justify your answer by giving attention to the following detail:
(i) State the appropriate null and alternative hypothesis for this test.
(ii) What test statistic is used to test these hypotheses?
(iii) What is the value of the test statistic? (9)
(c) Can one conclude at the 5% level of signi�cance whether �1 6D �3 D �4: (Justify youranswer) (9)
(d) Compare the means of the four teaching methods to determine where the are any differencesusing the Tukey-Kramer HSD method of multiple comparisons. Use all statistics available inthe output you generated.
(5)
[Hint: See last year's tutorial 101 on myunisa under announcement for the type ofoutputs you should generate.]
[33]
41
QUESTION 6Assume that the simple linear regression model Y D �0 C �1x C � is applicable to the data setgiven below.
X Y1 151 161 183 323 343 355 505 525 547 687 697 73
(a) Plot the data to verify that linear regression is a suitable model. (3)
(b) Compute the least squares regression equation for Y on X , and draw it on the graph in (a).(Marks will be given for formulae and computations.) (6)
(c) Compile (and complete) a table showing the following columns:
X i Yi bYi .Yi � bYi )21 151 16:::
:::7 73
Total
(4)
(d) Verify that
12PiD1.Yi � bYi /2 D 12P
iD1.Yi � Y /2 � 2b�1 12P
iD1.Yi � Y /.X i � X/Cb�21 12P
iD1.X i � X/2
(4)
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(e) Compute a 95% con�dence interval for the expected y-value if x D 6. (5)
(f) Compute a 95% con�dence interval for the expected y-value which we may obtain if, in a newexperiment, x D 6: (6)
(g) Test H0 : �1 D 8 against the alternative H1 : �1 > 8 at the 5% level of signi�cance. (5)
(h) Produce a SAS JMP output (Ensure that the correlation of the two variables is included). (7)
[40]
[Total Marks: 200]
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