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Tutorial Letter 101/3/2016

Applied Statistics II

STA2601

Semesters 1 & 2

Department of Statistics

IMPORTANT INFORMATION:

This tutorial letter contains important

information about your module and

includes the assignment questions

for both semesters.

STA2601/101/3/2016

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....................................................................................................................18

A.3 Assignment 03....................................................................................................................22

ADDENDUM B: SECOND SEMESTER ASSIGNMENTS..................................................................32

B.1 Assignment 01....................................................................................................................32

B.2 Assignment 02....................................................................................................................37

B.3 Assignment 03....................................................................................................................41

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 statistical

software package SAS JMP. This means that you must have access to a suitable computer for

a component of practical work.

This module will equip you with a proper basis in statistical knowledge, introduce you to a statistical

package and highlight the value of thorough statistical know-how that the business and outside

world require of students who major in Statistics! Knowledge of statistics will enable you to conduct

quantitative research and statistical literacy will enable you to understand research reports you

might encounter as a scientist in your everyday life or enable you to understand statistical reports

you 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 and

wish 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 them

all 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

file!

Some of this tutorial matter may not be available when you register. Tutorial matter that

is not available when you register will be posted to you as soon as possible, but is also

available 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 brochure

entitled my Studies @ Unisa.

Check the study material that you have received against the inventory letter. You should have

received all the items listed in the inventory, unless there is a statement like “out of stock” or “not

available”. If any item is missing, follow the instructions on the back of the inventory letter without

delay.

Shortly after registration The Department of Despatch should supply you with the following tutorial

matter for this module:

• Tutorial letter 101. Read it and save it as it contain important information as well as your

assignments 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 prescribed

sections, 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 modules

for 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 the

statistical software SAS JMP to do the computations and interpret the results for decisions regard-

ing tests for normality, independence and hypothesis concerning means, proportions, variances

and regression. Students should be able to solve applied statistics problems arising in government

and 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 and R.

• 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. Muchengetwa

GJ GERWEL (C-Block), Floor 6, Office 6-05

Tel: (011) 670-9253

Cell: 074 065 9020

E-mail address: muches@unisa.ac.za

You might also want to write to us. Letters should be sent to:

Ms S. Muchengetwa

Department of Statistics

PO Box 392

UNISA

0003

All queries that are not of a purely administrative nature but are about the content of this module

should be directed to me. Please have your study material with you when you contact me. E-mail

address 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, please

consult the publication My Studies @ Unisa that you received with your study material. This

brochure contains information on how to contact the University (e.g. to whom you can write for

different queries, important telephone and fax numbers, addresses and details of the times certain

facilities 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 to

Statistics and Data Analysis Using JMP, 5th Edition (2012).

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You have to buy this book. Please consult the list of official booksellers and their addresses listed in

my Studies @ Unisa. Prescribed books can be obtained from the University’s official booksellers. If

you have difficulty locating your book(s) at these booksellers, please contact the Prescribed Books

Section at 012 429 4152 or e-mail vospresc@unisa.ac.za. If you cannot find the book you can buy

the latest edition.

You need to purchase one other publication. The publication is a book of tables containing the

normal, t-, chi-squared and F-tables.

STOKER, DJ: Statistical Tables / Statistiese Tabelle 4th Edition (1997).

Foreign students may have difficulty in obtaining this book. If you are unable to obtain this book

you may use any other book of tables, but keep in mind that the tables used in the examination will

be 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 addresses

of 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 resources

and information at the University. The myUnisa learning management system is Unisa’s online

campus that will help you to communicate with your lecturers, with other students and with the

administrative 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 then

click on the “Login to myUnisa” link on the right-hand side of the screen. This should take you to

the myUnisa website. You can also go there directly by typing in http://my.unisa.ac.za.

7

Please consult the publication my Studies @ Unisa which you received with your study material for

more information on myUnisa.

5.1.3 Discussion classes

There are no discussion classes offered in this module. Should the need for discussion classes

arise 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 To

Study Guide and Workbook

Assignment 1 Chapter 1 to Chapter 2 Registration 07 March

Start writing your assignment 08 March 14 March

Assignment 2 Chapter 3 to Chapter 5 15 March 22 March

Start writing your assignment 23 March 29 March

Assignment 3 Chapter 6 to Chapter 8 30 March 11 April

Start writing your assignment 12 April 18 April

Note: For the text book, you will see the instructions

in you workbook on which pages to read.

SEMESTER 2 Study units for preparing your From To

assignments

Study Guide and Workbook

Assignment 1 Chapter 1 to Chapter 2 Registration 12 August

Start writing your assignment 13 August 19 August

Assignment 2 Chapter 3 to Chapter 5 20 August 26 August

Start writing your assignment 27 August 02 September

Assignment 3 Chapter 6 to Chapter 8 03 September 23 September

Start writing your assignment 24 September 30 September

Note: For the text book, you will see the

instructions in you workbook on which

pages 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 semester

mark is composed of 30% of assignment 1, 35% of assignment 2 and 35% of assignment 3 of the

marks you receive. An assignment submitted late or not at all will give you 0%. If you do well in

your assignments you have a good semester mark and that can make all the difference between a

pass 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 fixed

submission dates for the assignments and each assignment is based on specific chapters in the

study guide. You have to adhere to these dates as assignments are only marked if they are received

on or before the due dates.

You will only get examination admission if you submit the first assignment by its due date. You

should 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 the

questions 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 tutors

or do research, you are actively engaged in learning. Looking at the assessment criteria given

for each assignment, and the feedback you receive in your marked assignment, will help you to

understand 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 be

written on the cover of your assignment upon submission. The unique numbers are given later on

in 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 Date

SEMESTER 1 Chapters are covered

1 Chapters 1 and 2 of Study Guide 14 March 2016

and Workbook

2 Chapters 3, 4 and 5 of Study Guide 29 March 2016

and Workbook

3 Chapters 6, 7, and 8 of Study Guide 18 April 2016

and Workbook

Assignment for Sections from the following Due Date

SEMESTER 2 Chapters are covered

1 Chapters 1 and 2 of Study Guide 19 August 2016

and Workbook

2 Chapters 3, 4 and 5 of Study Guide 02 September 2016

and Workbook

3 Chapters 6, 7, and 8 of Study Guide 30 September 2016

and Workbook

8.3 Submission of assignments

For detailed information on assignments, please refer to the my Studies @ Unisa brochure, which

you 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 brochure

my 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 are

enrolled for and do the set of assignments for that semester only. The assignments for Semester

1 are in Appendix A, pages 13–31. The assignments for Semester 2 are in Appendix B, pages

32–48. Solutions to the assignments will be posted to ALL students registered for this module a

while 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 your

final mark. This 20% contribution comes from the marks you obtained for the three assignments

and is called your semester mark. If you do well in your assignments you have a good semester

mark and that can make all the difference between a pass or fail or between a distinction or simply

a pass!

Currently admission to the examination is only based on the proof that you are actively involved

in your studies. This proof is based on the submission of your first assignment before a fixed

given date. Admission therefore does not rest with the department and if you do not submit that

particular assignment in time, we can do nothing to give you admission. Although you are most

probably a part time student with many other responsibilities, work circumstances will not be taken

into 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 2016

and should you fail and qualify for a supplementary examination, that supplementary exami-

nation will be written in October/November 2016.

• if you are registered for the second semester, you will write the examination in October/November

2016 and should you fail and qualify for a supplementary examination, that supplementary

examination will be written in May/June 2017.

The examination section will provide you with information regarding the examination in general,

examination venues, examination dates and examination times. Eventually, your results will also

be 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 assignment

questions, but there will also be questions on theory. Should you have a final mark of less than

50%, it implies that you failed the module STA2601. However, should your results be within a

specified percentage (usually from 40% to 49%), you will be given a second chance in the form

of a supplementary examination on the dates as specified in 10.2. If you fail the examination with

less than 40%, the semester mark will not count to help you pass.

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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 typical

examination 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 questions

are 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 lecturer

want to discuss any matter about the examination, it will be included in this tutorial letter. In the

study guide you are given clear indications of the sections in the textbook that you have to know

and can be tested on in the examination. Remember that you have to work continuously and do

not treat statistics as any other subject, where it may be possible to study only selected sections of

the 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 ALL

enquiries about exam admission should be directed by e-mail to exams@unisa.ac.za.

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 be

dedicated, work consistently and practice, practice and practice some more! We hope that you will

enjoy 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 01

Unique Nr.: 871099

Fixed closing date: 14 March 2016

QUESTION 1

Suppose that the probability distribution for X is as shown below.

x fX (x)

0 110

1 310

2 25

3 15

(a) Calculate the mean value of X. (3)

(b) Calculate the variance of X. (4)

(c) Would you say that the distribution is symmetrical? (7)

[15]

QUESTION 2

Suppose that X1, X2, ...., X7 is a random sample from a n(9; 25) distribution and that

X =1

7

7∑i=1

X i and Y =7∑

i=1

[X i − X

σ

]2

.

Suppose that we also define V1 =3∑

i=1

[(X i − 9) /5

]2V2 =

7∑i=4

[(X i − 9) /5

]2W =

7∑i=1

[X i − 9]2

25

.

13

(a) Is fX5(x5) = fX6

(x6)? (2)

(b) Write down an expression for fX6(x6) . (2)

(c) Find P (X1 < 8) or P (X1 > 10) . (5)

(d) Is Z =X5 − 9

5∼ n (0; 1)? (3)

(e) What is E (W )? (2)

(f) Let X =1

7

7∑i=1

X i and T =

√7(X − µ

)√6(X i − X

)26

. Determine P (T ≥ 2.447) . (2)

(g) What is the distribution of U =V1/3

V2/4? (2)

(h) Find a value a such that P (U < a) = 0.95. (2)

[20]

QUESTION 3

Let X1, ...., Xn be a random sample from a n(µ; σ 2) distribution with σ 2 unknown but µ known.

(a) Prove that the two statistics A1 =1

n

n∑i=1

(X i − µ)2 and A2 =1

n − 1

n∑i=1

(X i − X)2 are both

unbiased estimators of σ 2. (10)

(b) Which one of the two estimators is the most efficient?

[Hint: Use the properties of the χ2-distribution to answer both questions.] (10)

[20]

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QUESTION 4

Let X1, X2, . . . , Xn be independent random variables such that

E(X i ) = ciθ1 + wiθ2 i = 1, 2, . . . , n

V ar(X i ). = σ 2 i = 1, 2, . . . , n

Find the least squares estimators of θ1 and θ2 if you assume that c1, c2, . . . , cn and

w1, w2, . . . , wn are known constants. (20)

QUESTION 5

Consider the probability density function

fX (x; θ) =1

θe− xθ for x ≥ 0 and

1

θ> 0.

Assume that E(X) = θ and var(X) = θ2.

Let X1, ...., Xn be a random sample from this distribution.

(a) Find the maximum likelihood estimator (m.l.e.) of θ. (6)

(b) Show that the m.l.e. is an unbiased estimator. (4)

(c) What is the variance of this m.l.e.? (3)

[13]

QUESTION 6

Study the three snapshots given below to answer questions regarding the software program JMP.

(a) What is the name of the data table in snapshot 1? (2)

(b) For which column have we selected the Formula option under the menu of Column Properties

(refer to snapshot 1)? (2)

(c) What function did we select from the Formula option (refer to snapshot 2)? (2)

(d) What will the effect be if we press OK in snapshot 2? (Elaborate on the properties of the

population from which we will generate a sample.) (2)

(e) What is the name of the data table in snapshot 3? (2)

(f) How many rows will the column with the name "normal(73,8) contain if press OK in snapshot

3? (2)

15

Snapshot 1

Snapshot 2

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Snapshot 3

[12]

[100]

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A.2 Assignment 02

ONLY FOR SEMESTER 1 STUDENTSASSIGNMENT 02

Unique Nr.: 838126

Fixed closing date: 29 March 2016

QUESTION 1

The following data have been observed in an experiment:

632 752 813 856 948 977 1 023 1 121 1 159 1 168 1 185 1 253 1 296 1 311 1 342

1 356 1 465 1 478 1 503 1 536 1 586 1 609 1 683 1 699 1 712 1 821 1 944 1 982 1 992 2 194

We wish to test the null hypothesis that the observations come from a normal distribution by

using a goodness-of-fit test. The 30 observed values must be classified into six classes with

equal probability for each class interval. Assume that α = 0.10.

The figure below shows the JMP output for the sample.

Figure 1:

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You may assume that

n∑i=1

(X i − X)2 = 4 622 377.467 and that

n∑i=1

(X i − X)3 = 146 534 638.8.

(a) State the null and alternative hypothesis. (3)

(b) Under H0 the distribution is not completely specified and we have to estimate the two un-

known parameters by using the maximum likelihood estimators µ and σ 2 for the goodness-

of-fit test. Calculate the values of the two unknown parameters. (5)

(c) Show that the fourth interval is1379.87 < X ≤ 1549.44. (7)

(d) Use the output to make a conclusion on whether the data follows a normal distribution. Com-

ment using all available information. (4)

(e) Test the sample for:

(i) Skewness (two-sided) at the 10% level. (7)

(ii) Kurtosis (two-sided) at the 10% level. (7)

(f) Would you say that the sample comes from a normal distribution ? Substantiate your answer.

(2)

[35]

QUESTION 2

(a) A study to determine the efficiency of a drug for arthritis resulted in the comparison of two

groups, each consisting of six arthritic patients. One group was inoculated with the serum,

the other group received a placebo (an inoculation that appears the same but actually is

nonactive). After a period of time each patient in the study was asked to state whether

his/her arthritic condition had improved. The following results were observed:

Treated Untreated

Improved 6 2

Not improved 0 4

Does the information above support the theory that the treated patients tend to improve sig-

nificantly more than untreated patients. Test at the 5% level of significance. (10)

19

(b) Mineral compositions were determined for rocks taken from two totally different areas. A ge-

ologist wants to investigate the theory that quarts concentrations are more variable in the Ku-

ruman area than in the Thabazimbi area. The data given below are the quarts concentration

(as a percentage) for 12 rocks from the Kuruman area and for 10 rocks from the Thabazimbi

area. All rocks were taken at random.

Kuruman (X) Thabazimbi (Y )

34.0 30.8

37.5 30.9

21.3 33.8

18.7 24.8

27.9 20.7

36.4 28.0

19.8 21.9

21.6 23.0

7.9 25.7

31.5 20.4

7.0

24.4

(i) Test the claim at the 5% level of significance. (10)

(ii) On what assumption(s) is your test statistics based? (3)

(iii) Derive the appropriate 95% one-sided confidence interval forσ 2

Y

σ 2X

in accordance with the

being tested. (7)

[30]

QUESTION 3

Consider the following sample of size n = 20 from a(µ; σ 2

)distribution:

3 5 6 9 10 11 12 14 14 15

16 17 17 18 19 20 21 22 25 26

(a) Show how you derive a two tailed confidence interval of the variance. (5)

(b) Calculate the 90% confidence interval of the variance. (5)

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(c) Test the claim that the σ 2 > 38 at the 10% level of significance. (10)

[20]

QUESTION 4.

Ten judges were asked to give a rating on the firmness and comfort of a mattress on a scale of 0

to 7.0. Higher ratings imply greater firmness and comfort. The following were obtained:

Firmness X i Comfort Yi

2.5 5.03.0 4.75.0 3.04.0 4.23.5 4.52.0 3.03.3 5.95.2 5.52.8 3.25.8 6.1

(a) Calculate the correlation coefficient, R. manually. Show all calculations. (7)

(b) Test the hypothesis that ρ > 0.65 at the 5% level of significance. (8)

[15]

[100]

21

A.3 Assignment 03

ONLY FOR SEMESTER 1 STUDENTSASSIGNMENT 03

Unique Nr.: 842643

Fixed closing date: 18 April 2016

QUESTION 1

Sixteen newly born babies were weighed at birth and then again when they were one month old.

(The weights are given in kg.) Do the data indicate that the variance in weight is increased by age?

Weight at birth (X) Weight one month later (Y )

3.90 4.70

2.85 3.30

2.90 3.60

2.45 4.25

3.60 4.40

4.32 5.20

3.78 3.40

3.85 4.35

2.95 3.50

3.25 4.20

3.95 5.00

3.80 4.40

3.75 4.45

3.25 4.30

3.40 4.10

4.00 4.85

Let µ denote the mean birth weight of a new born baby, and assume that σ 2 is unknown.

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Figure 2:

You make use of the JMP output given above.

(a) What assumption(s) is/are necessary in order to conduct the statistical test specified in (b)

below? Are they met? Give a brief discussion. (4)

(b) Test H0 : µ = 3.6 against H1 : µ > 3.6 at the α = 0.05 level of significance. (8)

(c) How will the test procedure change, if in fact you know that σ = 0.50 kg? (6)

(d) Find a 90% confidence interval for µ. Can you use this interval to confirm your conclusion in

(a)? Justify your answer. (5)

23

(e) Use JMP to do a paired t-test to determine the difference between the means, µX and

µY . Submit all relevant output. (7)

[30]

QUESTION 2

The tolerance to pain is measured for 18 females and 15 males (of more or less the same age

and the same physical condition) in order to test the hypothesis that females have a lower average

score. (The lower the score the more tolerant you are to pain.) Assume that females = population

1 and that males = population 2.

Study the following JMP output and answer the questions given below :

0

5

10

15

pain

female male

gender

male­femaleAssuming equal variancesDifferenceStd Err DifUpper CL DifLower CL DifConfidence

3.569290.961205.529681.60890

0.95

t RatioDFProb > |t|Prob > tProb < t

3.71334831

0.0008*0.0004*0.9996 ­4 ­3 ­2 ­1 0 1 2 3 4

t Test

Figure 3: Graphical representation and t-test

24

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0.0

1.0

2.0

3.0

Std 

Dev

female male

gender

femalemale

Level1815

Count2.4987203.026044

Std Dev1.7401232.580071

MeanAbsDif to Mean1.7284442.561333

MeanAbsDif to Median

O'Brien[.5]Brown­ForsytheLeveneBartlettF Tes t 2­sided

Tes t0.70631.96982.23730.54861.4666

F Ratio1111

14

DFNum313131

.17

DFDen0.40710.17040.14480.45890.4489

p­Value

Welch Anova testing Means Equal, allowing Std Devs Not Equal

13.3077F Ratio

1DFNum

27.198DFDen

0.0011*Prob > F

3.6480t Test

Tests that the Variances are Equal

Figure 4 : Tests that the variances are equal

(a) What is your conclusions regarding the test of H0 : µ1 = µ2 against H1 : µ1 < µ2 at the 1%

level of significance? (Justify your answer.) (4)

(b) On what assumption(s) do you base your test statistic? (Justify your answer.) (5)

(c) Give a 95% two-sided confidence interval for (µ1 − µ2). (3)

[12]

QUESTION 3

We wish to test H0 : µ = 20 against H1 : µ 6= 20, using a sample of size n from a normal

population with mean µ and variance σ 2. Find the smallest sample size to ensure that the

power of the test, at the 5% level of significance, will be at least 0.90 if µ = 20±√

2 σ . [Hint: You

have to solve this by trial and error using table F of the study guide.) [5]

25

QUESTION 4

Two random samples of sizes 15 and 18, drawn from n(µ1; σ

21

)and n

(µ2; σ

22

)populations

respectively, with σ 21 6= σ

22, yielded the following statistics:

n1 = 15 X1 = 40 S21 = 30

n2 = 18 X2 = 42 S22 = 56

Test the null hypothesis that the two population means are equal. Perform a two-sided test at

the 5% level of significance.

Clearly show how you find the critical value. [11]

QUESTION 5

The JMP data set Car Physical Data.jmp (if you are interested see SAS\Support Files English\Sample

Data\Car Physical Data.jmp) contains a sample of 116 cars classified according to Model; Country;

Type; Weight;..etc.

The table below shows a snippet of this data table (the first 22 lines showing only the third (Type)

and fourth (Weight) columns) to give you an idea of the data we used to perform an Analysis of

Variance.

Type WeightSmall 2700Medium 3265Medium 2935Compact 2670Compact 2790Compact 2895Medium 3640Medium 2880Large 3350Large 3325Medium 3465Compact 2640Large 4285Large 3545Medium 3480Large 4025Compact 2655Sporty 3110Sporty 3320Large 3855Compact 2485Sporty 3280

Figure 5:

26

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Study the following JMP output and answer the questions given below :

1500

2000

2500

3000

3500

4000

4500W

eigh

t

Compact Large Medium Small Sporty

Type

All PairsTukey­Kramer0.05

Figure 6: Graphical representation

TypeErrorC. Total

Source4

111115

DF239182489079325

32997573

Sum of Squares5979562

81796

Mean Square73.1036F Ratio

<.0001*Prob > F

Analysis of Variance

CompactLargeMediumSmallSporty

Level2217302225

Number2815.683657.353304.832250.232812.60

Mean60.97569.36552.21660.97557.200

Std Error2694.93519.93201.42129.42699.3

Lower 95%2936.53794.83408.32371.12925.9

Upper 95%

Std Error uses a pooled es timate of error variance

Means for Oneway Anova

Figure 7: Computations for ANOVA

27

0

100

200

300

400

Std 

Dev

Compact Large Medium Small Sporty

Type

CompactLargeMediumSmallSporty

Level2217302225

Count170.8473293.4872318.5811221.4534360.2665

Std Dev146.1983231.0381262.1444149.6901275.2960

MeanAbsDif to Mean143.8636232.0588258.8333144.3182274.6000

MeanAbsDif to Median

O'Brien[.5]Brown­ForsytheLeveneBartlett

Tes t3.01562.92713.06653.4887

F Ratio4444

DFNum111111111

.

DFDen0.0210*0.0241*0.0194*0.0074*

Prob > F

Welch Anova testing Means Equal, allowing Std Devs  Not Equal

84.8252F Ratio

4DFNum

52.251DFDen

<.0001*Prob > F

Tests that the Variances are Equal

Figure 8: Tests that the variances are equal

28

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1.98157t

0.05Alpha

LargeMediumCompactSportySmall

­194.4180.5658.7666.6

1224.1

180.5­146.3330.1338.8895.5

658.7330.1

­170.9­162.6394.6

666.6338.8

­162.6­160.3396.7

1224.1895.5394.6396.7

­170.9

Abs (Dif)­LSDLarge Medium Compact Sporty Small

Pos itive values show pairs  of means that are significantly different.

LargeMediumCompactSportySmall

LevelA

BCC

D

3657.35293304.83332815.68182812.60002250.2273

Mean

Levels not connected by same letter are significantly different.

LargeMediumLargeLargeCompactSportyMediumMediumLargeCompact

LevelSmallSmallSportyCompactSmallSmallSportyCompactMediumSporty

­ Level1407.1261054.606844.753841.671565.455562.373492.233489.152352.520

3.082

Difference1224.12895.53666.60658.66394.58396.70338.76330.08180.48

­162.59

Lower CL1590.1341213.6821022.9101024.679736.329728.042645.704648.227524.563168.751

Upper CL<.0001*<.0001*<.0001*<.0001*<.0001*<.0001*<.0001*<.0001*<.0001*0.9707

p­Value

Comparisons for each pair using Student's t

Figure 9: Multiple Comparisons

(a) Do you think it is reasonable to assume that the five groups may be considered as indepen-

dent groups? (2)

(b) Do you think it is reasonable to assume that the five groups may be considered as coming

from normal populations? (2)

(c) Do you think it is reasonable to assume that the five groups have equal population variances?

(Justify your answer.) (3)

(d) Refer to Figure 7.

(i) State the appropriate null and alternative hypotheses for the ANOVA problem. (2)

29

(ii) State reason(s) why we could not perform this test manually (if we used all 116 lines of

data values). (2)

(iii) Can you conclude that the mean weights for the different types of cars (i.e. the popu-

lation means of the five groups) differ significantly? (Justify your answer.) (2)

(e) What can you conclude from the results of the multiple comparisons on all pairs of means

and how does this tie in with the graphical output? (4)

[17]

QUESTION 6

Executives in the movie industry of America are interested in predicting the amount of money that

action movies will generate worldwide based on the amount that these movies generate in America.

The following sample data show the box-office amount in America (in billions of dollars) (X i ) and

the box-office amount Worldwide (in billions of dollars) (Yi ) for a random sample of n = 30 action

movies. (Please note that this is an adaptation of an existing JMP data set purely for academic

purposes!)

Action MovieBox-office amount in America

(in billions of dollars)

Box-office amount Worldwide

(in billions of dollars)

A 100 215

B 102 152

C 170 340

D 90 122

E 110 236

F 250 539

G 254 575

H 280 574

I 359 796

J 410 857

K 320 697

L 105 193

M 190 356

N 300 600

O 280 564

P 180 420

Q 370 752

R 210 396

S 190 380

T 270 556

Please note that you are welcome to use JMP to solve the problem!

30

STA2601/101

(a) Compute the equation of the regression line of Y on X. (9)

(b) What is the predicted Box-office amount Worldwide (in billions of dollars) for a Box-office

amount of 150 billion dollars in America? (2)

(c) What is the error of the estimate in (b)? (7)

(d) Test H0 : β1 = 0 against H1 : β1 6= 0. Use α = 0.05. (7)

[25]

[100]

31

ADDENDUM B: SECOND SEMESTER ASSIGNMENTS

B.1 Assignment 01

ONLY FOR SEMESTER 2 STUDENTSASSIGNMENT 01

Unique Nr.: 842391

Fixed closing date: 19 August 2016

QUESTION 1

Suppose that the probability distribution for X is as shown below.

x fX (x)

4 16

6 26

8 16

9 16

12 16

(a) Calculate the mean value of X. (3)

(b) Calculate the variance of X. (4)

(c) Would you say that the distribution is symmetrical? (7)

[15]

QUESTION 2

Suppose X has a n (100; 64) distribution. State for each one of the following statements whether

it is true or false.

(a) X is a continuous variable with E (X) = 100 and V ar (X) = 64. (1)

(b) fX(x) =

1

8√

2πe−

12(x−100)2/64 for −∞ < x <∞. (1)

32

STA2601/101

(c)100∫−∞

fX(x) dx = 0.5 = P (X ≤ 100). (1)

(d) The value of a for which P (X ≤ a) = 0.95 is a = 113.16. (1)

(e) P (X ≤ −113.16) = 0.05. (1)

[5]

QUESTION 3

Let X1, X2, ..., X7 be independent n (100; 64) variables and that

X =1

7

7∑i=1

X i and Y =7∑

i=1

[X i − X

σ

]2

.

Suppose that we also define V1 =6∑

i=1

[(X i − 100) /8

]2V2 =

3∑i=2

[(X i − 100) /8

]2.

(a) What is the distribution of7∑

i=1

(X i − 100)2

64? (2)

(b) What is the distribution of7∑

i=1

(X i − X

)264

? (2)

(c) What is E

[7∑

i=1

(X i − X

)264

]? (2)

(d) What is V ar

[7∑

i=1

(X i − 100)2

64

]? (2)

(e) What is the distribution of(X1 − 100)2

64? (2)

33

(f) Write down the joint density function for fx2x6(x

2, x

6). (3)

(g) Let X =1

7

7∑i=1

X i and Y =7∑

i=1

[X i − X

σ

]2

. Show that fX;Y (u; v) = fX (u) fY (v). (4)

(h) Find P (115 < X1 < 118) . (6)

(i) Is Z =X5 − 100

8∼ n (0; 1)? (3)

(j) What is the distribution of U =V1/6

V2/2and hence of

1

U? (4)

[30]

QUESTION 4

Let X1, X2, X3 and X4 be a random sample of size n = 4 from a n(µ; σ 2

)distribution.

Consider the following two estimators of the population mean µ.

T1 =X1 + X2 + X3 + X4

4(The sample mean.)

T2 =X1 + 2X2 + 2X3 + 3X4

8(A weighted mean.)

(a) Show that both T1 and T2 are unbiased estimators of µ. (6)

(b) Which estimator do you prefer and why? (9)

[15]

34

STA2601/101

QUESTION 5

Given the probability density function

f (x; λ) =1√

2πλe−

x2

2λ −∞ < x <∞ and λ > 0.

Find the maximum likelihood estimator for λ. [10]

QUESTION 6

Let X1; X2; . . . ; Xn be independent random variables such that

E (X1) = cθ1 with c a known constant

and E (X i ) = θ1 + θ2 for i = 2, 3, . . . , n.

Find the least squares estimates of θ1 and θ2. [15]

QUESTION 7

Study the snapshot given below to answer questions regarding the software program JMP.

(a) What is the name of the data table in snapshot 1? (2)

(b) For which column have we selected the Formula option under the menu of Column Proper-

ties?

(2)

(c) What function did we select from the Formula option? (2)

(d) Elaborate on the properties of the population from which we will generate a sample. (2)

(e) What will the effect be if we press OK ? (2)

35

Snapshot

[10]

[100]

36

STA2601/101

B.2 Assignment 02

ONLY FOR SEMESTER 2 STUDENTSASSIGNMENT 02

Unique Nr.: 778198

Fixed closing date: 02 September 2016

QUESTION 1

The following data have been observed in an agricultural experiment:

13.1 15.2 7.7 16.0 16.4 8.6 9.4 9.5 4.6 6.4 8.7 11.4 13.27.2 13.3 10.9 6.9 11.2 9.8 10.1 10.1 10.0 12.2 6.1 10.4 14.712.9 6.8 12.6 10.6 13.5 12.5 6.6 10.6 10.5

Assume that we wish to test the null hypothesis that the observations come from a normal distrib-

ution by specifically using a goodness-of-fit test. The 35 observed values must be classified into

five classes with equal probability for each class interval. Work with α = 0.05.

The figure below shows some of the possible JMP output for the sample.

Figure 1:

37

You may assume that

n∑i=1

(X i − X)2 = 295.7817 and that

n∑i=1

(X i − X)3 = 27.6073.

(a) State the null and alternative hypothesis. (3)

(b) Under H0 the distribution is not completely specified and we have to estimate the two un-

known parameters by using the maximum likelihood estimators µ and σ 2 for the goodness-

of-fit test. Calculate the values of the two unknown parameters. (5)

(c) Calculate the third interval, that is, find a and b where a < X ≤ b. (7)

(d) Use the output to make a conclusion on whether the data follows a normal distribution. Com-

ment using all available information. (4)

(e) Test the sample for:

(i) Skewness (two-sided) at the 10% level. (7)

(ii) Kurtosis (two-sided) at the 10% level. (7)

(f) Would you say that the sample comes from a normal distribution ? Substantiate your answer.

(2)

[35]

QUESTION 2

(a) We want to test whether there is an association between colour-blindness (achromatopsia)

and a fear of heights (acrophobia). A random sample of 12 people yielded the following data:

Colour-blind Not colour-blind Row total

Fear of heights 2 2 4

No fear of heights 1 7 8

Column total 3 9 12

Do the data supply evidence that colour-blind people tend to fear heights ? Test at the

10% level of significance, i.e. use α = 0.10. (10)

38

STA2601/101

(b) It is believed that the daily number of electrical power failures in Pretoria is a Poisson random

variable with mean 4.2. Test this hypothesis if over 150 days the number of days having i

power failures is as follows:

Failures Number of days

0 0

1 5

2 22

3 23

4 32

5 22

6 19

7 13

8 6

9 4

10 4

11 0

Test the claim at the 5% level of significance. (15)

[25]

QUESTION 3

A circuit board in a personal computer has six layers that must match up reasonably well, or else

the repairing is almost impossible or extremely expensive. (The process that matches layers can

be absolutely perfect on the average yet be so variable that no single board is worth keeping. So

this is one of the rare occasions where testing the variance is necessary.)

Suppose that we are now concerned with one specific layer that will only be acceptable if σ ≤0.01mm. A random sample of 30 of these layers were measured and yielded the following results:

30∑i=1

X i = 30.008mm30∑

i=1

X2i = 30.021686mm2

(a) Test at the 1% level whether the specifications are met. Assume that the sample comes from

a normal distribution.

[Hint: The specifications require that σ ≤ 0.01 which is identical to σ 2 ≤ (0.01)2 and the null

hypothesis usually specifies only one value for a parameter (the “=” part).] (8)

(b) Show how you derive a 95% one sided confidence interval of the variance in accordance with

the hypothesis test. (5)

39

(c) Calculate the 95% one sided one sided confidence interval of the variance in accordance

with the hypothesis test. (3)

(d) Suppose another sample of 20 machines gave a variance of 0.0001745. Test whether the

variances are equal. (9)

[25]

QUESTION 4

Many people believe that the length of a person’s life is highly correlated with the length of the

”lifeline” on the person’s hand. Suppose that the 36 pairs of observations below can be considered

as a random sample of males that died at a specific hospital. The variables are:

X = Age of male at death (to nearest year)

Y = Length of lifeline on left hand in centimeters (to nearest 0,05 cm)

Row X Y Row X Y

1 15 8.25 19 59 10.80

2 22 9.45 20 60 8.30

3 26 7.95 21 62 7.00

4 29 8.25 22 63 8.85

5 33 10.20 23 63 9.00

6 37 8.55 24 65 10.05

7 40 10.10 25 66 8.00

8 42 8.10 26 67 12.40

9 42 7.55 27 69 9.65

10 44 9.75 28 74 10.85

11 46 10.50 29 75 9.50

12 47 10.50 30 75 11.60

13 50 9.15 31 77 8.90

14 51 7.40 32 79 10.50

15 55 9.45 33 80 9.80

16 56 8.85 34 83 8.65

17 58 10.80 35 84 12.05

18 58 13.20 36 92 11.30

Use SAS JMP and submit all relevant output:

(a) Calculate the correlation coefficient, R and interpret it. (9)

(b) Test for the significance of the correlation coefficient at the 5% level of significance. (6)

[15]

[ 100]

40

STA2601/101

B.3 Assignment 03

ONLY FOR SEMESTER 2 STUDENTSASSIGNMENT 03

Unique Nr.: 867426

Fixed closing date: 30 September 2016

QUESTION 1

A random sample of n = 25 packets of peas, all marked 0.500 kg, were weighed on a special

quality control scale and the weights were found to be:

0.503 0.499 0.495 0.471 0.512 0.516 0.440 0.440 0.477 0.474

0.483 0.487 0.488 0.469 0.492 0.514 0.500 0.492 0.499 0.506

0.482 0.499 0.458 0.497 0.493

Let µ denote the mean mass (in kg) of all the packets.

41

Figure 2:

You make use of the JMP output given above.

(a) What assumption(s) is/are necessary in order to conduct the statistical test specified in (b)

below? Are they met? Give a brief discussion. (4)

(b) Is it cheating to mark the packets 0.500kg? (i.e. Is the mean significantly less than 0.500kg?)

Use α = 0.05. (8)

(c) How will the test procedure change, if in fact you know that σ = 0.02 kg? (6)

(d) Find a 90% confidence interval for µ. Can you use this interval to confirm your conclusion in

(b)? Justify your answer. (5)

42

STA2601/101

(e) Use JMP to redo the t-Test in (c) and the z-Test in (b).

Please report only the two different p-values (Submit all relevant output). (12)

[35]

QUESTION 2

It is desired to test H0 : µ = 50 against H1 : µ 6= 50 using a sample of size n = 13 from a n(µ; σ 2

)distribution. Suppose that we know that the true mean is µ = 50+ 0.75σ .

(a) What will the power of the test be if we test at the 5% level of significance? (4)

(b) What will the probability of a Type II error be if the probability of a Type I error is 0.05? (1)

[5]

QUESTION 3

A simple random sample of 400 households from a large community was selected to estimate

the mean residential electricity usage per household during June of 1997. Another simple random

sample of 450 households was selected, independently of the first, to estimate the mean residential

electricity usage during June of 1998. The sample results (expressed in kilowatt hours) were:

1997 1998

n1 = 400 n2 = 450

X1 = 1 252 X2 = 1 330

S1 = 257 S2 = 251

(a) Construct a 99% confidence interval for the difference in mean usage per household between

the two years. (10)

(b) Can it be concluded that mean usage per household has changed significantly between the

two years? (2)

(c) What assumptions did you make in order to construct the confidence interval in (a)? Are they

satisfied? (10)

(d) If 300 of the households selected in 1997 were also included in the sample for 1998, why

would the estimation procedure used in (a) not be appropriate? (3)

[25]

43

QUESTION 4

To test the effect of certain additives in petrol, 24 identical engines were randomly divided into four

groups of six each. Each engine was then filled with 1.0 litre of petrol and 0.1 litre of additive. As

a control, group 1 received 1.1 litre of petrol only. All the engines were switched on and the total

running times were recorded in hours.

Group 1 (Control) 1.1 1.2 1.0 1.3 1.2 1.1 1.0 1.4 1.2 1.3

Group 2 (Additive A) 0.9 0.8 0.85 0.95 0.9 1.05 1.1 0.8 0.95 1.0

Group 3 (Additive B) 0.85 0.95 0.8 0.85 0.75 0.95 0.9 1 0.95 1.0

Group 4 (Additive C) 1.2 1.15 1.3 1.25 1.35 1.2 1.15 1.1 1.3 1.4

Consider the data as random samples from n(µi ; σ

2)

- populations for i = 1, ..., 4.

You can choose between OPTION A and OPTION B to answer this question.

(Please indicate your choice in your assignment answer.)

OPTION A: (Manual ANOVA)

(a) Test at the 5% level whether the samples in fact originate from populations with equal vari-

ances. (5)

(b) Assuming the population variances to be equal, test at the 5% level whether the means of

the populations differ significantly. (10)

(c) Perform multiple comparisons on all pairs of means. Discuss your results. (10)

[25]

44

STA2601/101

OPTION B:

Study the following JMP output and answer the questions given below :

Figure 3: Graphical representation

Figure 4: Computations for ANOVA

45

Figure 5: Tests that the variances are equal

46

STA2601/101

Figure 6: Multiple Comparisons

(a) Do you think it is reasonable to assume that the four groups may be considered as indepen-

dent groups? (2)

(b) Do you think it is reasonable to assume that the four groups may be considered as coming

from normal populations? (4)

(c) Do you think it is reasonable to assume that the four groups have equal population variances?

(Justify your answer.) (4)

(d) Refer to Figure 4:

(i) State the appropriate null and alternative hypotheses for the ANOVA problem. (2)

(ii) Can you conclude that the mean running times for the different types of additives (i.e.

the population means of the four groups) differ significantly? (Justify your answer.) (5)

(e) What can you conclude from the results of the multiple comparisons on all pairs of means

and how does this tie in with the graphical output? (8)

[25]

47

QUESTION 5

The following table gives the rate (Y ) of flow of blood through the kidney for individuals of X years

of age.

Individual Age (X ) Rate of flow (Y )

1 40 467

2 40 573

3 45 430

4 45 476

5 50 466

6 50 375

7 55 352

8 55 426

9 60 340

10 60 405∑xi = 500

∑yi = 4 310

(a) Plot the data to verify that linear regression is a suitable model. (4)

(b) Compute10∑

i=1

(yi − y)2 and10∑

i=1

(xi − x)2. Hint: Make a table. (10)

(c) Verify that the least squares regression equation for Y on X is: Y = 790− 7.18X. (6)

(d) Use the given regression equation to compute yi and show that10∑

i=1

(yi − yi

)2= 17 293.8. (10)

(e) What is the value of10∑

i=1

(yi − y)2 − b21

10∑i=1

(xi − x)2? (3)

(e) Find the predicted rate of flow of blood in the kidney for a 70-year-old person and the approx-

imate standard deviation (standard error of estimate). (6)

(f) Calculate a 99% confidence interval for the rate of flow of blood for x = 60. (6)

(g) Test H0 : β1 = 0 against H1 : β1 6= 0. Use α = 0, 05. (7)

(h) Produce a SAS JMP out ensuring that the correlation is also included in the output. (8)

[60]

[150]

48