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Tutorial Letter 101/3/2013
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.
STA2601/101/3/2013
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 .............................................................................5
3.1 Lecturer(s)............................................................................................................................5
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 ...............................................................................................................7
6 MODULE-SPECIFIC STUDY PLAN.......................................................................................8
7 MODULE PRACTICAL WORK AND WORK-INTEGRATED LEARNING .................................8
8 ASSESSMENT.....................................................................................................................8
8.1 Assessment plan ..................................................................................................................8
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.....................................................................................10
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....................................................................................................................20
ADDENDUM B: SECOND SEMESTER ASSIGNMENTS..................................................................24
B.1 Assignment 01....................................................................................................................24
B.2 Assignment 02....................................................................................................................27
B.3 Assignment 03....................................................................................................................30
3
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 afile!
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 difficult 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.
5
3 LECTURER(S) AND CONTACT DETAILS
3.1 Lecturer(s)
The lecturer responsible for this module is as follows:
Ms S. MuchengetwaE-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 contact details of the department will be sent to all students in a later tutorial letter.
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, 4th Edition (2007).
You have to buy this book. Please consult the list of official booksellers and their addresses listed inmy Studies @ Unisa. Prescribed books can be obtained from the University’s official booksellers. Ifyou have difficulty locating your book(s) at these booksellers, please contact the Prescribed Books
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Section at 012 429 4152 or e-mail [email protected]. If you cannot find 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: Statistiese Tabelle / Statistical Tables third Edition (1977).
Foreign students may have difficulty 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 find 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.
Please consult the publication my Studies @ Unisa which you received with your study material formore information on myUnisa.
7
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 17 FebruaryStart writing your assignment 18 February 24 February
Assignment 2 Chapter 3 to Chapter 5 25 February 10 MarchStart writing your assignment 11 March 17 March
Assignment 3 Chapter 6 to Chapter 8 18 March 31 MarchStart writing your assignment 1 April 7 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 4 AugustStart writing your assignment 5 August 11 August
Assignment 2 Chapter 3 to Chapter 5 12 August 25 AugustStart writing your assignment 26 August 1 September
Assignment 3 Chapter 6 to Chapter 8 2 September 15 SeptemberStart writing your assignment 16 September 22 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 final 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 final 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 reflection of a typical examination question. There are fixedsubmission dates for the assignments and each assignment is based on specific 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 first 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 Learning
Assignments 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 find 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:
9
Assignment for Sections from the following Due DateSEMESTER 1 Chapters are covered1 Chapters 1 and 2 of Study Guide 24 February 2014
and Workbook2 Chapters 3, 4 and 5 of Study Guide 17 March 2014
and Workbook3 Chapters 6, 7, and 8 of Study Guide 7 April 2014
and Workbook
Assignment for Sections from the following Due DateSEMESTER 2 Chapters are covered1 Chapters 1 and 2 of Study Guide 11 August 2014
and Workbook2 Chapters 3, 4 and 5 of Study Guide 1 September 2014
and Workbook3 Chapters 6, 7, and 8 of Study Guide 22 September 2014
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 13–23. The assignments for Semester 2 are in Appendix B, pages24–33. 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 on myUnisa.
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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 final 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 yourfinal 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 first assignment before a fixedgiven 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 fifteen weeks. This means that
• if you are registered for the first semester, you will write the examination in May/June 2014and should you fail and qualify for a supplementary examination, that supplementary exami-nation will be written in October/November 2014.
• if you are registered for the second semester, you will write the examination in October/November2014 and should you fail and qualify for a supplementary examination, that supplementaryexamination will be written in May/June 2015.
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 final mark of less than50%, it implies that you failed the module STA2601. However, should your results be within aspecified percentage (usually from 40% to 49%), you will be given a second chance in the formof a supplementary examination on the dates as specified in 10.2. If you fail the examination
11
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 final 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 definitely run into trouble if you skip sections!
You are automatically admitted to the exam on the submission of Assignment 01 by a specific 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 lecturers
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ADDENDUM A: FIRST SEMESTER ASSIGNMENTS
A.1 Assignment 01
ONLY FOR SEMESTER 1 STUDENTSASSIGNMENT 01
Unique Nr.: 864608Fixed closing date: 24 FEBRUARY 2014
QUESTION 1
(a) Probabilities for the random variable X are defined by:
P (x) =x2 − 1
nfor x = 2, 3, 4, 5
(i) Determine n.
(ii) Construct the probability distribution of X.
(iii) Calculate the mean value of X.
(iv) Calculate the standard deviation of X.
(v) Would you say that the distribution is symmetrical?
(b) Let X1, X2, X3, X4, X5 and X6 be a random sample of size 6 drawn from a normal populationwith mean µ and variance σ2. Consider the following two estimators for µ.
W1 =X1 + 2X2 + 3X3 + 4X4 −X5 +X6
10
W2 =3X1 + 2X2
5
(i) Show that both W1 and W2 are unbiased estimators of µ.
(ii) Which estimator would you prefer and why?
13
QUESTION 2
Suppose that X1, X2, ...., X12 is a random sample from a n(50; 100) distribution and that
X =1
12
12∑i=1
Xi and Y =12∑i=1
[Xi −X
σ
]2.
Suppose that we also define T1 =6∑i=3
[(Xi − 50) /10]2
T2 =12∑i=7
[(Xi − 50) /10]2
T3 =12∑i=1
[Xi − 50]2
100
(a) Write down an expression for fX7 (x7) .
(b) Find P (X1 > 65) .
(c) Find P (35 < X1 < 70) .
(d) What is E (T3)?
(e) What is V ar (Y ) (where Y =12∑i=1
[Xi −X
σ
]2)?
(f) What is the distribution of U =T1/4
T2/6?
(g) What is the distribution of1
U?
(h) Find a value a such that P (U < a) = 0.95.
(i) Find a value a such that P(
1
U< a
)= 0.95.
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QUESTION 3
Complete the following statements in your answer book (i.e. give the missing words and do notwaste time rewriting everything):
(a) The statistic T is called an unbiased estimator for the parameter θ if .....................
(b) Let X1; X2; ...; Xn be a random sample from a population with unknown expected valueµ and unknown variance σ2. An unbiased estimator for the population variance is σ2 =.....................
(c) The efficiency of two estimators of the same parameter is a function of their .......................
(d) When multiple measurements or observations are made on each of the individuals or units ina sample the assumption of ....................... is violated.
(e) If random variable Y has a n (0; 1) distribution, Y 2 has a ......................... distribution with......................... degree(s) of freedom.
(f) The significance level of a statistical test is defined as α = ..........................
(g) The hypothesis “The blood platelet count of male cancer patients is lower than the bloodplatelet count of healthy men”, requires a ......................-sided test of significance.
QUESTION 4
Let X1, X2, ..., Xn be a random sample of size n from a discrete distribution with probability function
P (X = r) =λre−λ
(1− e−λ) r!for r = 1; 2; ...
[Please note:
In a real life situation a random sample will result in, for example, X1 = 2; X2 = 3; X3 = 3; X4 =10; X5 = 2 etc ... . Do not fall into the trap to argue that X1 = 1; X2 = 2; ...; Xn = n becausethis is only one very specific outcome out of the millions of other possibilities. Denote the sampleoutcome by X1 = r1; X2 = r2; ...; Xn = rn.]
(a) Find the likelihood function for the sample.
15
(b) Show that∂lnL (λ)
∂λ= −n− ne−λ
(1− e−λ)+
n∑i=1
ri
λ
(c) Find the maximum likelihood estimator for λ. [Hint: Your answer will be an implicit solutionfor λ. In other words, λ is solved in terms of λ and however hard you may try it is impossibleto solve explicitly for λ. This means that you cannot “take λ to the left” and equate it to aknown expression.] Do not feel discouraged by “all the calculus” in this problem. One cannotbe a good statistician without a sound knowledge of mathematics!
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A.2 Assignment 02
ONLY FOR SEMESTER 1 STUDENTSASSIGNMENT 02
Unique Nr.: 877902Fixed closing date: 17 March 2014
QUESTION 1The time taken in minutes to complete a race by a random sample of n = 100 athletes is shownbelow:
95 78 108 123 109 90 106 102 105 8786 107 80 91 96 119 111 102 107 10187 104 81 105 118 106 94 110 101 9992 82 92 106 72 67 84 77 104 8426 105 76 86 115 111 89 125 115 8794 93 123 104 112 103 109 84 105 10162 122 114 93 104 133 101 116 90 8596 103 138 98 141 128 95 113 100 8066 100 71 117 106 90 120 100 100 10529 117 82 118 113 109 106 124 111 90
(a) Use Excel but do not include the output to complete this table.
N = 100∑Xi = 10 117 X = ..............∑
(Xi −X)2 = 25 350.11∑
(Xi −X)3 = ...........∑
(Xi −X)4 = .........
(b) (i) Test whether this distribution is symmetric. (Use α = 0.10.)
(ii) Test whether this distribution has a kurtosis of a normal distribution. (Use α = 0.10.)
(iii) Does this distribution originate from a normal population?
(c) Suppose you are given this table and observations in the sample come from a normal distri-bution with mean 100 and standard deviation 16.
Table of observed and expected frequenciesClass interval Oi ei = nπi
X < 75 5 5.9475 ≤ X < 90 18 .......
90 ≤ X < 105 33 ......105 ≤ X < 120 32 27.27120 ≤ X < 135 10 ......
X ≥ 135 2 1.43Totals 100 100
17
(i) Show how intervals 1 and 2 are derived.
(ii) Calculate the 3 missing expected frequencies in the above table.
(iii) At the 0.05 level, use a chi-square goodness-of-fit test to test if the 100 observations inthe sample come from a normal distribution with mean 100 and standard deviation 16.
QUESTION 2
A study is conducted on the relationship between religious affiliation and attitude towards artificialbirth control. The three major religious groups in the area are represented. The following data wasobtained.
Religious GroupAttitude A B CFor birth control 123 64 43Against birth control 77 86 57
Use SAS JMP to determine whether the factors are independent at α = 0.05.
(a) Produce the Mosaic Plot and interpret it.
(b) State the appropriate null and alternative hypothesis for this test.
(c) What test statistic is used to test these hypotheses and what is the value of the test statistic?
(d) Looking at the row percentages in your output, can you draw any conclusions?
(e) What is your final conclusion?
QUESTION 3
A random sample of 12 men were classified according to personality type (introvert or extrovert)and the type of car that they drive (sports model or sedan). The frequencies are as follows:
Type of personality Row
Introvert Extrovert total
Type of Sports model 1 4 5
car Sedan 4 3 7
Column total 5 7 12
Does the information above support the theory that introverts avoid sports models? Test at the 5%level of significance.
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QUESTION 4
(a) Let X1;X2; . . . ;Xn be a random sample from a normal distribution with known mean µ. Use
the distribution of U =n∑i=1
(Xi − µ)2
σ2to show that a 100 (1− α) % two-sided confidence inter-
val for σ2 is given by [ ∑(Xi − µ)2
χ2α2;n
;
∑(Xi − µ)2
χ21−α
2;n
]
(b) LetX1;X2; . . . ;Xn and Y1;Y2; . . . ;Ym be two independent random samples from n (µ1;σ21) and
n (µ2;σ22) distributions respectively. Suppose that the following information is given:
n = 41 m = 31
41∑i=1
Xi = 1 23031∑i=1
Yi = 992
41∑i=1
(Xi − X
)2= 27 362
31∑i=1
(Yi − Y
)2= 20 365
Test H0 : σ21 = σ2
2 against H1 : σ21 6= σ2
2 at the 10% level of significance.
(Clearly show how you compute the left critical value.)
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A.3 Assignment 03
ONLY FOR SEMESTER 1 STUDENTSASSIGNMENT 03
Unique Nr.: 771425Fixed closing date: 7 April 2014
QUESTION 1
A psychologist wants to investigate the properties and the distribution pattern of IQ-scores of a welldefined group of people. He calls this population A. He draws a random sample of size n = 100from the population, performs the necessary IQ tests and ends up with the 100 IQ-scores given intable A.
Table A: Ordered IQ-scores (from small to large)
67 71 71 73 74 77 77 78 78 80
81 81 81 83 84 84 85 85 86 86
87 89 89 90 91 91 92 92 92 93
93 93 94 94 94 95 95 96 96 96
97 97 97 97 98 98 98 99 99 100
101 101 102 102 103 103 104 104 104 105
105 105 106 106 107 107 107 108 108 109
110 110 111 111 112 113 114 114 115 115
115 116 117 117 118 118 119 120 121 121
122 123 123 124 126 126 127 129 134 139∑Xi = 10 091
∑X2i = 1 042 040
(a) Produce a SAS JMP output to answer the following questions.
(i) What assumption(s) is/are necessary in order to conduct the statistical test specified in(b) below? Are they met? Give a brief discussion.
(ii) Is there any reason to reject the null hypothesis H0: µ = 100? Use a 5% level of signifi-cance and test two-sided.
(iii) Is there any reason to reject the null hypothesis H0: σ = 15? Use a 5% level of signifi-cance and test two-sided.
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(b) Suppose that the psychologist repeats this whole process but this time he draws a randomsample of size n2 = 60 from a population B, independent of population A, and finds thefollowing statistics:
Y = 105.21 S2Y = 240.6537
Would you say that the mean IQ-score of population B is higher than the mean IQ-score ofpopulation A? Use α = 0.05.
[Hint: assume that both variances are unknown but equal. Assume that S2X = 239.972 for the
sample with n1 = 100 from population A.]
QUESTION 2
An experiment was designed to study the performance of 4 different detergents for cleaning fuelinjectors. The following “cleanness” readings were obtained with specially designed equipment for12 tanks of gas distributed over 3 different models of engines:
EngineDetergent 1 2 3 TotalA 45 43 51 139B 47 46 52 145C 48 50 55 153D 42 37 49 128Total 182 176 207 565
DO NOT USE SAS JMP. DO THIS MANUALLY:
(Regard the data as random samples from normal populations.)
Disregarding engine type, answer the following questions:
(a) What are the values of S21 , S2
2 , S23 , and S2
4?
(b) (i) Compute the “ordinary” average of the four variances computed in (a).
(ii) Compute the MSE according to the definition in the study guide. What do you notice?
(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?
(d) Test at the 5% level of significance whether the population means of the four different deter-gents differ.
(i) State the null and alternative hypotheses.
(ii) State the rejection region and conclusion.
(e) Perform multiple comparisons on all pairs of means. Discuss your results.
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QUESTION 3
A group of 15 subjects is given tests before and after experiencing a learning situation. The datais shown below:
Subject 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Before 27 21 34 24 30 27 33 31 22 27 33 17 25 26 23After 29 32 29 27 31 26 35 30 29 28 36 15 28 26 26
(a) Using the 0.05 level of significance, test the hypothesis that learning actually did take place.Assume that learning is shown by an increase in score. Clearly state the hypothesis im-plied by the question and how it can be tested. Give the rejection region and theconclusions.
(b) Produce a SAS JMP output to support your calculations.
(c) What type of data is this?
QUESTION 4
Using the data in question 2 and disregarding type of detergents, use SAS JMP, submityour output obtained AND discuss the analysis regarding each of the following:
(a) Use Levene’s test to determine if the three groups have equal population variances? Useα = 0.05 level of significance. (State your hypothesis and justify your answer.)
(b) Do these results indicate that there are differences in the engines at the 5% level ofsignificance?)
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?
(c) Can one conclude at the 5% level of significance whether µ1 = µ2 6= µ3. (Justify your answer)
(d) Compare the means of the three engines to determine where the are any differences usingthe Tukey-Kramer HSD method of multiple comparisons. Use all statistics available in the out-put you generated. [Hint: See last year’s tutorial 101 on myunisa under announcementfor the type of outputs you should generate.]
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QUESTION 5
The response time to a stimulus was measured for ten people of varying ages. The results are:
T:Time(sec) 0.9 0.8 1.3 1.1 1.2 1.5 1.4 1.7 1.9 2.3
A:Age(years) 20 23 25 31 33 38 40 53 59 72
(a) Draw a scatterplot of this data set.
(b) The correlation coefficient between response time (T) and age (A) was calculated as r =0.964. Is there a significant positive correlation between response time and age? Use α =0.005.
(c) Calculate the least squares regression line of T on A. Make use of the following:
n∑i=1
ti = 14.1n∑i=1
ai = 394n∑i=1
a2i = 18 122n∑i=1
aiti = 623.4
(d) What would the predicted response time be for a 45-year-old person?
(e) Give an interpretation of the numerical value of the regression coefficient β1.
(f) What is the standard error of your estimate in question (d)?
(g) Derive a 95% confidence interval for the slope β1.
(h) Submit a SAS JMP output to support your calculations you have done manually.
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ADDENDUM B: SECOND SEMESTER ASSIGNMENTS
B.1 Assignment 01
ONLY FOR SEMESTER 2 STUDENTSASSIGNMENT 01
Unique Nr.: 838399Fixed closing date: 11 August 2014
QUESTION 1
(a) A random variable X is defined for X = 0, 1, 2, 3 and 4. Moreover, P (0) = 0.1, P (2) = 0.4,P (3) = 0.2 and P (1) is twice P (4).
(i) Give the probability distribution function of X.
(ii) Calculate the mean value of X.
(iii) Calculate the standard deviation of X.
(iv) Would you say that the distribution is symmetrical?
(b) Let X1, X2, X3 and X4 be a random sample of size n = 4 from a n (µ; σ2) distribution.
Consider the following two estimators of the population mean µ.
T1 = X1+X2+X3+X4
4(The sample mean.)
T2 = X1+2X2+3X3−2X4
4(A weighted mean.)
(i) Show that both T1 and T2 are unbiased estimators of µ.
(ii) Which estimator do you prefer and why?
QUESTION 2
Comment on the following statements:
(a) The variance of any random variable can never be a negative value.
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(b) The best method of estimating an unknown parameter is the method of least squares.
(c) A type I error is made if H0 is accepted when H1 is true.
(d) One would expect a negative correlation between the length (in cm) and the mass (in kg) ofa child.
(e) r = 0 indicates a lack of relationship between X and Y.
(f) During the 1950s there was a high correlation between the consumption of soft drinks andthe number of polio cases. This indicates that the consumption of soft drinks caused polio.
(g) The only important assumption underlying one-way analysis of variance is that the samplesmust be independent.
QUESTION 3
Suppose that X1, X2, ...., X10 is a random sample from a n(60; 25) distribution and that
X =1
10
10∑i=1
Xi and Y =10∑i=1
[Xi −X
σ
]2.
Suppose that we also define V1 =7∑i=5
[(Xi − 60) /5]2
V2 =10∑i=4
[(Xi − 60) /5]2
V3 =8∑i=1
[Xi − 60]2
25
(a) Is fX2 (x2) = fX5 (x5)?
(b) Write down an expression for fX4 (x4) .
(c) Find P (X3 < 65) .
(d) Find P (55 < X3 < 72) .
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(e) What is E (V3)?
(f) Let Y =10∑i=1
[Xi −X
σ
]2. Determine V ar (Y ) .
(g) Let X =1
10
10∑i=1
Xi and T =
√10(X − µ
)√Σ(Xi −X
)29
. Determine P (T ≥ 3.250) .
(h) What is the distribution of U =V1/3
V2/7?
(i) What is the distribution of1
U?
(j) Find a value a such that P (U < a) = 0.95.
(k) Find a value a such that P(
1
U> a
)= 0.05.
QUESTION 4
(a) Let X1, X2, ..., Xn be independent random variables such that
E (Xi) = θ1 + Ciθ2 for i = 1, 2, ..., n;
V ar (Xi) = σ2 ∀ i
with σ2, θ1 and θ2 unknown parameters and C1, C2, ..., Cn known constants. Find the leastsquares estimators of θ1 and θ2.
(b) Let X1;X2; . . . ;Xn be a random sample from a distribution with p.d.f.
fX (x) = cxc−1 for x > 1
Find the maximum likelihood estimator for the parameter c.
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B.2 Assignment 02
ONLY FOR SEMESTER 2 STUDENTSASSIGNMENT 02
Unique Nr.: 893540Fixed closing date: 1 September 2014
QUESTION 1
A manufacturer is experimenting with a new design for nylon stockings. The main objectives isto maximize the expected service life of a stocking by improving the breaking strength. Thirtystockings of the new design were randomly selected and tested. The breaking strengths of thenylon thread of the thirty stockings are as follows:
(Please note that the sample values are ordered in order of magnitude to easeclassification into intervals.)
7.32 8.13 8.52 9.48 9.56 10.23 10.77 11.21 11.59 11.68
11.85 12.53 12.96 13.11 13.42 13.56 14.69 14.78 15.03 15.36
15.86 16.09 16.83 16.99 17.12 18.21 19.44 20.82 20.92 21.94
The following information is also given:
30∑i=1
Xi = 420.00 and30∑i=1
(Xi −X
)2= 445.33.
(a) Show that the maximum likelihood estimators for the mean and the variance of the breakingstrengths of the new design stockings are
µ = 14.0
andσ2 = 14.8.
(b) Use a chi-squared test with five equiprobable classes to test whether or not the breakingstrengths of the new design stockings are normally distributed. Use α = 0.05. (Use thegiven maximum likelihood estimates.)
(c) Use Excel but do not include the output to complete this table.∑∣∣Xi −X∣∣ = ..............
∑(Xi −X)3 = ...............
∑(Xi −X)4 = ................
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(d) (i) Test whether this distribution is symmetric. (Use α = 0.10.)
(ii) Test whether this distribution has a kurtosis of a normal distribution. (Use α = 0.10.)
(iii) Does this distribution originate from a normal population?
QUESTION 2
In a large firm, salesperson are classified as aggressive, non-aggressive and shy. Their sales forthe month are classified as high, average or low. The following data was obtained.
SalesAggressiveness High Average LowAggressive 64 28 38Non-aggressive 45 22 29Shy 28 29 27
Use SAS JMP to test the hypothesis at α = 0.05, that the relative aggressiveness and salesare independent.
(a) Produce the Mosaic Plot and interpret it.
(b) State the appropriate null and alternative hypothesis for this test.
(c) What test statistic is used to test these hypotheses and what is the value of the test statistic?
(d) Looking at the row percentages in your output, can you draw any conclusions?
(e) What is your final conclusion?
QUESTION 3
A sample of teenagers were divided into males and females on the one hand, and those that areand are not currently dieting on the other. We hypothesize, for example, that the proportion ofdieting individuals is higher among the women than among the men, and we want to test whetherany difference of proportions that we observe is significant. The frequencies are as follows:
Gender RowMale Female total
Dieting 1 4 5Non-dieting 6 1 7Column Total 7 5 12
Does the information above support the theory that the proportion of dieting individuals is higheramong the women than among the men? Test at the 5% level of significance.
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QUESTION 4
(a) In a random sample of 45 observations the sample correlation coefficient is r = −0.75.
Test H0 : ρ = −0.65 againstH1 : ρ 6= −0.65 at the 5% level.
(b) Construct a 95% (two-sided ) confidence interval for ρ.
(c) Let X1;X2; . . . ;Xn be a random sample from a normal distribution with unknown mean µ.
Use the distribution of U =n∑i=1
(Xi −X
)2σ2
to show that a 100 (1− α) % two-sided confidence
interval for σ2 is given by [ ∑(Xi −X
)2χ2
α2;n−1
;
∑(Xi −X
)2χ21−α
2;n−1
]
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B.3 Assignment 03
ONLY FOR SEMESTER 2 STUDENTSASSIGNMENT 03
Unique Nr.: 838374Fixed closing date: 22 September 2014
QUESTION 1
The weights in kgs of a random sample of n = 50 people who took part in a nutrition survey areshown below:
58 65 96 54 63 68 78 79 48 4648 78 75 71 70 62 36 25 55 3839 47 49 58 69 67 89 88 28 2932 61 50 80 70 72 75 76 77 5681 64 66 65 71 73 74 75 68 69
Use SAS JMP to answer the following questions and insert your output.
(a) It is desired to test if the true average weight of the n = 50 people in the study is equal to 65kg at the 0.05 level.
(i) Write down the null and alternative hypotheses for the test.
(ii) What statistical test could be used if we assume that the population variance σ2 is un-known?
(iii) Write down the decision rule for the test.
(b) Based on the SAS JMP output given to you, would it be correct to assume that the trueaverage weight of the so participants in the study significantly differs from 100 kg at the 0.05level?
(c) Interpret the 95% confidence interval for the true average weight µ
(d) What do the figures in the JMP output you produced show about the distribution?
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QUESTION 2
(a) A comparison is made between two innovative teaching methods by using the two methodsfor training two independent groups of pupils, and assessing exam scores obtained by thepupils in the two groups. Assume that σ2
1 = σ22 = σ2 (unknown). Summary of results obtained
from the two samples are shown below:
n1 = n2 = 31; X1 = 40.55; X2 = 46.65; SP = 14.62
(i) Construct a 95% confidence interval for µ1 − µ2.
(ii) Comment on the confidence interval for µ1 − µ2
(b) We wish to test H0 : µ = 20 against H1 : µ 6= 20, using a sample of size n from a normalpopulation with mean µ and variance σ2. Find the smallest sample size to ensure that thepower of the test, at the 5% level of significance, will be at least 0.90 if µ = 20±
√2σ. [Hint:
You have to solve this by trial and error using table F of the study guide.)
QUESTION 3
An industrial engineer tests 4 different shop-floor layouts by having each 6 work crews construct asubassembly and measuring the construction times (minutes) as follows:
LayoutCrew 1 2 3 4A 48.2 53.1 51.2 58.6B 49.5 52.9 50.0 60.1C 50.7 56.8 49.9 62.4D 48.6 50.6 47.5 57.5E 47.1 51.8 49.1 55.3F 52.4 57.2 53.5 61.7
DO NOT USE SAS JMP. DO THIS MANUALLY:
(Regard the data as random samples from normal populations.)
Disregarding crew, answer the following questions:
(a) What are the values of S21 , S2
2 , S23 , and S2
4?
(b) (i) Compute the “ordinary” average of the four variances computed in (a).
(ii) Compute the MSE according to the definition in the study guide. What do you notice?
(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?
31
(d) Test at the 5% level of significance whether the 4 floor layouts produce different assemblytimes.
(i) State the null and alternative hypotheses.
(ii) State the rejection region and conclusion.
(e) Perform multiple comparisons on all pairs of means. Discuss your results.
QUESTION 4
Two different toothpastes are tested on two sets of children carefully matched for age, sex, diet,fluoride treatments, and so on. During the test period, the number of new cavities for the subjectsis:
Subject pair 1 2 3 4 5 6 7 8 9 10Toothpaste A 3 1 4 2 5 3 3 1 3 2Toothpaste B 2 1 1 0 4 1 2 1 2 1
(a) Using the 0.01 level of significance, test the hypothesis that there are no differences betweentoothpastes in mean number of cavities. Clearly state the hypothesis implied by the ques-tion and how it can be tested. Give the rejection region and the conclusions.
(b) Produce a SAS JMP output to support your calculations.
QUESTION 5
Using the data in question 3 and disregarding layout, use SAS JMP, submit your outputobtained AND discuss the analysis regarding each of the following:
(a) Use Levene’s test to determine if the six groups have equal population variances? Useα = 0.05 level of significance. (State your hypothesis and justify your answer.)
(b) Do these results indicate that there are differences in assembly times by some of thework crews at the 5% level of significance?)
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?
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(iii) What is the value of the test statistic?
(c) Can one conclude at the 5% level of significance whether µ1 = µ3 6= µ4. (Justify your answer)
(d) Compare the means of the six work crews to determine where the are any differences inmean assembly times using the Tukey-Kramer HSD method of multiple comparisons. Useall statistics available in the output you generated. [Hint: See last year’s tutorial 101 onmyunisa under announcement for the type of outputs you should generate.]
QUESTION 6
An examination consists of two papers. The results for ten candidates are shown below.
Paper X 87 53 59 80 70 73 66 61 52 49
Paper Y 91 59 63 84 67 77 68 65 53 59
(a) Give the formulae used to calculate the estimates β0 and β1 for an estimated regression line.
(b) Use SAS JMP to show that the regression line calculated for the Paper Y mark on the PaperX mark is given by
Y = 9.001 + 0.917X
(c) An eleventh candidate scored 63 in Paper X, but missed the second paper because of illness.What would you estimate her mark in Paper Y to be?
(d) H0 : β1 = 0 against H1 : β1 6= 0 can be tested by the test statistic
T =β1
S/dwhere
d2 =∑
(xi − x)2 and
S2 =1
n− 2
∑(Yi − β0 − β1xi
)2
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Give the critical region where H0 will be rejected if α = 0.05.
(e) The calculated value for the test statistic is T = 10.267. What can you conclude?
(f) Test H0 : β1 = 1 against H1 : β1 6= 1 at the 5% level of significance. Using d2 = 1 420 andS = 3.3656. Give your final conclusion.
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