UM_QM2-IB_Ordner

UNISEMINAR

Theory

Practice

Exams

Extras

Sem

inar

Introduction

Quantitative Methods II (IB) Academic Year 2011/2012, Block 3

Uniseminar – Quantitative Methods II Introduction

Welcome to Uniseminar!

IInnttrroodduuccttiioonn

Uniseminar offers EExxaamm PPrreeppaarraattiioonn SSeemmiinnaarrss,, SSuummmmaarryy SSccrriippttss aanndd

LLeeaarrnniinngg CCaarrddss for students of the Maastricht University. It is our goal to op-‐

timally prepare you for your exams and to make your own exam preparation as

efficient as possible. In order to achieve this goal, we have developed a system of

seminars in combination with an extensive summary script, which is proven for

several years by now.

In university it is often the case that there is a lot of material available for a

course and that the importance of this material is hard to evaluate. Since we, as

students, have made this experience as well, you are provided with a Uniseminar

Summary Script of the corresponding course. This folder contains all exam-‐

relevant material and it gives you a good summary of all course topics. The con-‐

tent of the folder is created by experienced Master or PhD students, who have

taught this course already several times. As a consequence, it is possible for you

to concentrate on the actual exam preparation, rather than spending hours

searching and printing the right material.

At the end of week 6 of your block, normally during the weekend, our EExxaamm

PPrreeppaarraattiioonn SSeemmiinnaarrss take place. These seminars are taught by above-‐

average students, who have already mastered their studies at the Maastricht

University and have a great deal of experience in tutoring. Since they have stud-‐

ied and taught at the Maastricht University they know exactly where potential

problems may lie and are therefore able to optimally teach you the whole theory

of the course and practice perfectly tailored examples with you. Furthermore you

can bring in your own questions during the seminar and discuss individual prob-‐

lems during the breaks.

You are able to pick up your SSuummmmaarryy SSccrriipptt aanndd LLeeaarrnniinngg CCaarrddss in ad-‐

vance of the Seminar in order to already start preparing so that you can discover

your own difficulties early enough. Later in the Seminar you will then know what

Introduction Uniseminar – Quantitative Methods II

your weaknesses are and be able to pay special attention to these sections or ask

questions about it. Our Summary Script and Learning Cards are updated every

year according to the current course’s content and we are always trying to opti-‐

mize the folder as much as possible.

AAbboouutt UUss

Uniseminar was founded 5 years ago by two students at the University of St.

Gallen in order to make Exam Preparation more efficient and coherent. Since

2005 we have expanded our vision and are now offering seminars and material

for an efficient exam preparation in Switzerland, the Netherlands, Italy and Ger-‐

many.

Thanks to this longstanding experience, we were able to build up a team of highly

qualified tutors and editors and are therefore able to guarantee high quality of

exam preparation.

The team of Uniseminar is grown strongly over the years and comprehends sev-‐

eral mathematicians, statisticians and economists, who all bring a great didactical

experience. All tutors of Uniseminar have been teaching their field for years and

know exactly what is important in order to optimally prepare and pass the exam.


SSuummmmaarryy SSccrriipptt This aim of this folder is to support you with your exam preparation for ‘Quanti-‐

tative Methods II’ as much as possible. Usually it consists of five different sec-‐

tions. As follows, a short overview of the content of this folder:

11.. TThheeoorryy:: The Theory Script summarizes the whole theory of the course in

a simple and understandable way. Concepts are explained with the help of

demonstrative examples. It is structured according to the seven weeks of

the course and is one of the most important parts of your exam prepara-‐

tion.

2. PPrraaccttiiccee:: The Practice part contains practice exercises to each week and

therefore to each chapter of the theory script. By this, you can deepen your

theoretical knowledge with practical exercises and you can go through the

exercises of these topics again, which you have not understood so well un-‐

til now.

3. EExxaammss:: In this part you will find old exams of the Maastricht University,

as well as one practice exams constructed by Uniseminar. During the sem-‐

inar you will then receive a further practice exam.

4. EExxttrraass:: In the Extras part, you will find a formula sheet as well as an ex-‐

planation of how to read off the statistics tables.

5. SSeemmiinnaarr:: In this part, we have provided you with some notepaper so that

you can take notes during the seminar. Furthermore you will receive a

fourth practice exam during the seminar, which you can file in here. In case

you have not subscribed for the Quantitative Methods II seminar yet, you

can do so on our website -‐ www.uniseminar.nl -‐ at any time.


QQuuaannttiittaattiivvee MMeetthhooddss IIII

The ‘QM2’ course treats two different main fields: Mathematics and Statistics. It

is the continuation of QM1, you have attended and hopefully passed in your first

block of this academic year. The topics of the course build on QM1. Although we

integrated some minor parts of the QM1 course, it is essential that you have un-‐

derstood the major theories and practices of QM1 in order to master QM2. De-‐

pending on your difficulties, you should put your focus on certain fields or topics,

however, do not forget that the exam is equally distributed in terms of questions

per topics. It does not make any sense to concentrate on Mathematics only, be-‐

cause this knowledge alone will not be sufficient to pass the exam.

The exam consists of 40 multiple choice questions and you will have 3 hours time

to calculate it. As mentioned, the questions are equally distributed, i.e. you will

have 20 math questions and 20 statistics questions.

HHiinnttss aanndd TTrriicckkss

Here are some tips that may be helpful for your exam preparation. Many students

make typical errors when preparing for their first exams at university. We there-‐

fore want to help you to avoid these mistakes, so that you can focus on the essen-‐

tial stuff, rather than wasting your time with a preparation into the wrong direc-‐

tion!

HHooww ddoo II ooppttiimmaallllyy pprreeppaarree ffoorr aann eexxaamm??

The exams of the university are created in such a way that every student can pass

them with an average preparation time. Since there is a lot of content and time is

limited, planning is the basis of your success. You don’t need to be a genius to

pass the exam, but you should still take care of a few things and try to develop a

certain discipline in the following weeks. The subsequent hints may be helpful

for you:


TTaakkee yyoouurr ttiimmee!!

It is totally normal that you will be slower in solving exercises in the be-‐

ginning. You can be sure that you will improve your efficiency and velocity

after some practice, however you should start preparing for the exam early

enough. By this you can avoid that you get time pressure in the last days

before the exam.

CCoonncceennttrraattee oonn pprraaccttiiccee!!

Although you should make sure that you have understood all basic con-‐

cepts of the theory, try to do as many exercises and old exams as possible.

You will notice that exam questions repeat a lot and that calculations will

become a lot easier, if you do them the second time.

SSttuuddyy aalloonnee aanndd iinn ggrroouuppss!!

In the beginning you should start studying and practicing alone so that you

know for sure that you have understood most of the content. If you experi-‐

ence problems you should ask a friend and you should try to solve the

problem together. Group work can be very helpful, especially for students

with different capabilities. Nevertheless, often it can also be very ineffi-‐

cient and you would do much better if you learned alone.

DDoonn’’tt cchheecckk tthhee ssoolluuttiioonnss iimmmmeeddiiaatteellyy!!

Take your time when doing an exercise or a practice exam and try first to

solve it on your own. It is important for your learning process that you also

make mistakes, because it shows you where you still have problems and

on which topics you should concentrate. It is better to do the mistakes dur-‐

ing your preparation than in the exam.


QQuueessttiioonnss && MMiissttaakkeess

As soon as you discover a mistake or misunderstanding in the text you can sub-‐

mit a question on our webpage www.uniseminar.nl. Just log in under ‘My Ac-‐

count’ with your email-‐address and your password and go to the section ‘Ques-‐

tions & Feedback’. Your questions will be forwarded to the tutor so that he can

optimally tailor the seminar to your needs. If it is an urgent question, we will

send you an answer before the seminar as soon as possible.

Theory

Practice

Exams

Extras

Sem

inar

T

Theory


Theory Uniseminar – Quantitative Methods II

TThheeoorryy

The Theory Script summarizes the whole theory of the course in a simple and

understandable way. Concepts are explained with the help of demonstrative

examples. It is structured according to the seven weeks of the course and is one

of the most important parts of your exam preparation. Although practice is very

important, it is even more crucial to understand the basic concepts of the course

in order to be able to calculate and understand all different kinds of exercises and

exam questions.

TTaabbllee ooff CCoonntteennttss

MMaatthheemmaattiiccss 11

1 Series of payments and discounting 1

2 Matrices, Determinants and Systems of Equations 6

3 Linear Programming 19

SSttaattiissttiiccss 3377

1 Recap from QM1 – Hypothesis Testing 37

2 Statistic inference based on two samples 39

3 Statistic inference based on more than two samples 48

4 Simple Regression 55

5 Multiple Regression 63

6 Regression assumptions 76

Theory – Mathematics Uniseminar – Quantitative Methods II

6

22 MMaattrriicceess,, DDeetteerrmmiinnaannttss aanndd SSyysstteemmss ooff EEqquuaattiioonnss

This section introduces matrix algebra, which is in itself a branch of mathematics.

In general, matrices and their determinants can be used to solve or simplify a

large variety of problems. One of the big applications is the solving of systems of

linear equations. However, there are other fields when you will encounter

matrices that are not part of this course. One example some of you may

encounter in later courses is that Econometricians use matrices to represent

variables and parameters in regression analysis.

22..11 MMaattrriicceess

WWhhaatt iiss aa mmaattrriixx??

In general a mmaattrriixx is simply a rectangular array of numbers. Thereby the order

of the numbers and the format of the matrix are important and cannot simply be

changed. The use of matrices is that they facilitate many otherwise more difficult

mathematical operations. To make this clear just consider the following system

of equations:

𝑎𝑎 𝑥𝑥 + 𝑎𝑎 𝑥𝑥 = 𝑏𝑏

𝑎𝑎 𝑥𝑥 + 𝑎𝑎 𝑥𝑥 = 𝑏𝑏

In matrix notation this can be written as follows:

𝐴𝐴𝐴𝐴 = 𝑏𝑏 𝑤𝑤𝑤𝑤𝑤𝑤ℎ 𝐴𝐴 =𝑎𝑎 𝑎𝑎𝑎𝑎 𝑎𝑎 , 𝑏𝑏 = 𝑏𝑏

𝑏𝑏 𝑎𝑎𝑎𝑎𝑎𝑎 𝑥𝑥 =𝑥𝑥𝑥𝑥

Thereby, A is a matrix of format 2x2, while b and x are matrices of format 2x1.

Thereby, the first number always refers to the row of a matrix, while the second

refers to the columns. As b and x only have one column they can also be called

vectors.

Uniseminar – Quantitative Methods II Theory – Mathematics

19

33 LLiinneeaarr PPrrooggrraammmmiinngg

LLiinneeaarr pprrooggrraammiinngg attempts to optimize a linear function given a linear set of

constraints. That means that we need to maximize or minimize a certain

objective function, e.g. profits or costs, given a certain set of constraints, e.g.

capacity constraints or demand constraints. In general there are two ways to

solve such a problem. First, easy problems, which involve only two decision

variables, e.g. the production level of two different goods, can be solved with the

help of a graph. Second, for more complex problems Excel needs to be employed

to solve them. To do so, Excel does nothing else than trial and error to find the

optimal value of the objective function.

33..11 TThhee ggrraapphhiiccaall aapppprrooaacchh

DDrraawwiinngg tthhee sseett

Whenever you like to solve a problem via the graphical approach it is best to

start with the constraints, which define the so-‐called set. Only this area is of

interest for the optimization later on, as other areas are ruled out by the

constraints. To show this, consider the following example:

max 3𝑥𝑥 + 4𝑦𝑦

𝑠𝑠. 𝑡𝑡. : 2𝑥𝑥 + 4𝑦𝑦 ≤ 20

3𝑥𝑥 + 𝑦𝑦 ≤ 9

𝑥𝑥 + 𝑦𝑦 ≥ 2 𝑥𝑥,𝑦𝑦 ≥ 0

In the problem we can see three constraints subject to which we need to

maximize the objective function. The easiest way to draw these constraints is to

set one of the variables equal to zero to compute the crossing point with one of

the axis and then set the other variable zero to compute the crossing point with

the other axis. Let’s start with the first constraint:

2𝑥𝑥 + 4 ∗ 0 = 20 → 𝑖𝑖𝑖𝑖 𝑦𝑦 = 0, 𝑡𝑡ℎ𝑒𝑒𝑒𝑒 𝑥𝑥 = 10

2 ∗ 0+ 4𝑦𝑦 = 20 → 𝑖𝑖𝑖𝑖 𝑥𝑥 = 0, 𝑡𝑡ℎ𝑒𝑒𝑒𝑒 𝑦𝑦 = 5

Theory – Mathematics Uniseminar – Quantitative Methods II

20

Therefore we know that the first constraint crosses the 𝑥𝑥-‐axis at 𝑥𝑥 = 10 and

the 𝑦𝑦 -‐axis at 𝑦𝑦 = 5 . Applying exactly the same approach to the other two

constraints we can produce the following graph. The numbers identify the

constraints. There is, however, one last thing that needs to be mentioned. The

first two constraints are smaller equal constraints, and therefore the set is

constrained to the area below them. The last constraint is a greater equal

constraint, which requires the feasible set to lie above this constraint. The

feasible set is therefore the light grey area beneath both the first and the second

constraint and above the third. On this area we will later try to maximize our

objective function.

TThhee oobbjjeeccttiivvee ffuunnccttiioonn

Knowing the set, it is now necessary to add the oobbjjeeccttiivvee ffuunnccttiioonn. Contrary to

the constraints, however, the objective function is not an equality and therefore

we cannot simply plug any variable equal to zero. A first step is to set the

objective function equal to a constant 𝑐𝑐:

3𝑥𝑥 + 4𝑦𝑦 = 𝑐𝑐

To be absolutely clear, maximizing the objective function does now mean finding

the largest possible value of 𝑐𝑐, which does not violate any of the constraints. To

be able to draw the objective function we need to rewrite it in terms of 𝑦𝑦:

Uniseminar – Quantitative Methods II Theory – Statistics

63

55 MMuullttiippllee RReeggrreessssiioonn

Once you understood the idea of a simple regression the step to a mmuullttiippllee

rreeggrreessssiioonn is not very hard. The main change is that more explanatory variables

are added to the regression model. By doing this we need to introduce some more

tests to evaluate the performance of the model. Moreover, there are some

problems that can arise and some new types of variables that can be introduced.

55..11 TThhee rreeggrreessssiioonn eeqquuaattiioonn

RReeggrreessssiioonn eeqquuaattiioonnss ffoorr tthhee ppooppuullaattiioonn

As just mentioned we can produce a multiple regression model by adding more

explanatory variables. Obviously the grade of student does not only depend on

the bonus points, but may also depend on the subject he or she is studying, the

gender or the nationality. Another impact might be attributed to the amount of

study time that is spent in group learning rather than self-‐study. By adding all

these variables the population regression model becomes:

𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 = 𝛽𝛽 + 𝛽𝛽 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 + 𝛽𝛽 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + 𝛽𝛽 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 + 𝛽𝛽 𝑓𝑓𝑓𝑓𝑓𝑓𝑐𝑐𝑐𝑐𝑐𝑐 + 𝛽𝛽 𝑔𝑔𝑔𝑔𝑔𝑔 + 𝛽𝛽 𝑜𝑜𝑜𝑜ℎ𝑒𝑒𝑒𝑒

+ 𝛽𝛽 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 + 𝛽𝛽 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔²+ 𝛽𝛽 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∗ 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 + 𝜖𝜖

You do not have to worry if you are unable to understand the purpose of all these

variables by now. This whole section will attempt to explain them one by one.

RReeggrreessssiioonn eeqquuaattiioonnss ffoorr tthhee ssaammppllee

Just as we did in the simple regression case we need to take a sample to estimate

the regression model. This results from the fact that the population is too big to

be observed. Switching to the sample then also means to switch from Greek to

normal letters:

𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 = 𝑏𝑏 + 𝑏𝑏 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 + 𝑏𝑏 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + 𝑏𝑏 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 + 𝑏𝑏 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 + 𝑏𝑏 𝑔𝑔𝑔𝑔𝑔𝑔 + 𝑏𝑏 𝑜𝑜𝑜𝑜ℎ𝑒𝑒𝑒𝑒

+ 𝑏𝑏 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 + 𝑏𝑏 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔²+ 𝑏𝑏 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∗ 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 + 𝑒𝑒

Theory – Statistics Uniseminar – Quantitative Methods II

64

Again, all those sample coefficients are just estimates for the population

coefficients and another sample would result in different coefficients. Moreover,

the last term is still called the residual and it is on average equal to zero.

55..22 TThhee EExxcceell OOuuttppuutt

RReeggrreessssiioonn ssttaattiissttiiccss

First of all note that the rreeggrreessssiioonn oouuttppuutt in this section is for our complete

model as it is given above. We will work with this model throughout this chapter.

It is not a problem if you do not understand some terms yet, as they will be

explained soon. The purpose of this is simply to give an overview of the output, so

that you know where to find what.

The first row labeled mmuullttiippllee RR does not have a very deep meaning in the case

of a multiple regression. In a simple regression it gives the correlation between

the dependent and the explanatory variable. In a multiple regression it is still the

square root of the R², but does not carry much intuition.

Theory – Statistics Uniseminar – Quantitative Methods II

76

66 RReeggrreessssiioonn aassssuummppttiioonnss

In terms of the regression itself the theory is so far covered. There are, however,

certain problems that can arise with a regression model. These then violate the

regression assumptions. This section attempts to name those problems and

explain how to tackle them. One example how to deal with the problem of a non-‐

constant variance is the llooggaarriitthhmmiicc rreeggrreessssiioonn. It uses the fact that a

logarithm can convert absolute in relative changes. Unfortunately, this also

changes the interpretation of the coefficient and therefore needs some additional

attention.

66..11 LLiinneeaarriittyy

HHooww ttoo cchheecckk tthhee aassssuummppttiioonn

A regression measures the linear relationship between variables. If, for any

reason, the relationship is not linear, then we cannot do a linear regression. There

are two ways to check the assumption. The first one is to look at all scatterplots

between the dependent and any explanatory variable. If any of these scatterplots

shows a non-‐linear relationship between the variables, e.g. a quadratic

relationship, you got a violation. A second approach is to check the residual plots

for all explanatory variables. A non-‐constant relationship between the dependent

and the particular explanatory variable would produce a clear pattern in the

residual plot. If the residual plot looks like a nice cloud everything is fine.

PPootteennttiiaall vviioollaattiioonnss

Some of you might have noticed that we already encountered a violation of the

linearity assumption just within our model. Recall that we added two group

terms, namely the simple group term and a group² term. The reason for this was

that the group term had a quadratic relationship with the grade variable. The first

way to check this is to produce a scatterplot between grade and group. Moreover,

Uniseminar – Quantitative Methods II Theory – Statistics

77

we can also check the residual plot. Estimating a model without the group²

variable leads to the residual plot that is displayed on the left. In that graph you

should be able to clearly see the quadratic pattern. Nevertheless, by including the

group² variable we can solve the problem of non-‐linearity as can be seen from the

two residual plots on the right. In those graphs the pattern is gone because the

group² term is able to pick up the quadratic effect. Therefore, our final model

does not violate the linearity assumption anymore.

66..22 CCoonnssttaanntt vvaarriiaannccee

HHooww ttoo cchheecckk tthhee aassssuummppttiioonn

Throughout most of the test we use the distribution of the error term. If this

distribution is different for different values of the explanatory variable we run

into problems as the tests would collapse. It is therefore required that the

variance of the standard error is equal for all levels of the explanatory variable.

The easiest way to check this is to produce a residual plot and check whether the

variance of the residuals stays the same or not.

Practice

Exams

Extras

Sem

inar

P

Practice Exercises


Practice Uniseminar – Quantitative Methods II

PPrraaccttiiccee EExxeerrcciisseess

This part contains practice exercises to each week and therefore to each chapter

of the theory script. By this, you can deepen your theoretical knowledge with

practical exercises and you can go through the exercises of these topics again,

which you have not understood so well until now. Although you may think that

you already have done enough exercises during the weeks, these exercises are

tailored specifically to your needs and try to teach you the most important topics

of the exam in a practical manner.


MMaatthheemmaattiiccss 11

Exercises 1

Solutions 27

SSttaattiissttiiccss 8833

Exercises 83

Solutions 114

Uniseminar – Quantitative Methods II Practice

1

MMaatthheemmaattiiccss -‐-‐ EExxeerrcciisseess

11 WWeeeekk 11

SSeerriieess,, SSeeqquueenncceess,, IInntteerreesstt RRaatteess aanndd AAnnnnuuiittiieess

11.. CCaallccuullaattee tthhee ffoolllloowwiinngg ccoommppoouunnddiinngg pprroobblleemmss

You put 5000€ into your bank account at an interest rate of 5% per year ii ..

paid annually.

((aa)) What amount will you have after 10 years?

((bb)) How long does it take for the amount to double?

Now assume that this interest is paid biannually.

((cc)) What amount will you have after 10 years?

((dd)) How long does it take for the amount to double?

((ee)) What is the effective interest rate per year?

How much money should you have invested 5 years ago in order to have ii ii ..

100,000€ today? The interest rate is 4%.

The annual interest rate is 9%. What is the effective interest rate, if money ii iiii ..

is compounded

((ff)) annually?

((gg)) biannually?

((hh)) quarterly?

((ii)) monthly?

((jj)) continuously?


2

What is the nominal interest rate, if the effective interest rate is 12% iivv..

compounded

((aa)) biannually?

((bb)) quarterly?

((cc)) monthly?

((dd)) continuously?

Your credit card provider charges you 1.5% interest on the outstanding vv..

balance per month. What is the effective annual rate, which you pay to

your credit card provider?

Assume that the value of your car depreciates at 15% annual rate vvii..

compounded continuously.

((ee)) As compared to the original value, what is value of the car after 2 years?

((ff)) How long does it take for the car to be worth 25% of its original value?

What is the present value of 5000€ which you need to pay in 5 years time vviiii ..

given that the interest rate is 5%

22.. CCaallccuullaattee tthhee ffoolllloowwiinngg iinnffiinniittee ssuummss::

ii ..

1 1 1 11 ...3 9 27 81

+ + + + +

ii ii ..

2 3 4

2 3 4

5 5 5 55 ...8 8 8 8×3 ×3 ×3 ×3+ + + +


19

66 WWeeeekk 66

LLiinneeaarr PPrrooggrraammmmiinngg 22

33.. SSeett uupp tthhee ffoolllloowwiinngg ll iinneeaarr pprrooggrraammmmiinngg pprroobblleemmss,, aanndd ssoollvvee

tthheemm..

In order to ensure safe operations at a casino, it needs sufficient security ii ..

staff. Assume that the casino is open 24/7. The demand for security is dif-‐

ferent for each time of day. Thus, the casino manager has divided each

workday into 6 different security requirements. The security staff re-‐

quirements for each shift are summarized in the table below:

RReeqquuiirreedd SSeeccuurriittyy SSttaaffff 0011::0000 –– 0055::0000 45

0055::0000 –– 0099::0000 15 0099::0000 –– 1133::0000 40

1133::0000 –– 1177::0000 55

1177::0000 –– 2211::0000 90 2211::0000 –– 0011::0000 135

The manager wants to minimize the number of security staff working at the casi-‐

no. The labor union allows this, however, only under the following two condi-‐

tions: Each security officer can only work 8 hours per day. These 8 hours can on-‐

ly be worked consecutively. (N.B., the 6th and the 1st shift are consecutive

((aa)) What are the decision variables?

((bb)) What is the objective Function?

((cc)) What are the constraints?

((dd)) What is the optimal solution?


20

AIXonMobyl produces kerosene for airplanes. AIXonMobyl has 3 different ii ii ..

production facilities (P1, P2, P3), which it uses to supply 4 different air-‐

ports (A1, A2, A3, A4). The costs of 1million gallons of Kerosene from the

production plants to the airports is summarized in the table below

AA11 AA22 AA33 AA44

PP11 10000 14000 13000 20000 PP22 8000 12000 14000 9000

PP33 22000 18000 23000 19000

The supply at each of the production facilities looks as follows

PPllaanntt SSuuppppllyy ((iinn mmiillll iioonn

ggaalllloonnss)) PP11 6000

PP22 4500 PP33 8000

The demand at each of the airports looks as follows

AAiirrppoorrtt SSuuppppllyy ((iinn mmiillll iioonn

ggaalllloonnss)) AA11 1500 AA22 8000

AA33 2000

AA44 3000


27

MMaatthheemmaattiiccss -‐-‐ SSoolluuttiioonnss

11 WWeeeekk 11

SSeerriieess,, SSeeqquueenncceess,, IInntteerreesstt RRaatteess aanndd AAnnnnuuiittiieess

11.. CCaallccuullaattee tthhee ffoolllloowwiinngg ccoommppoouunnddiinngg pprroobblleemmss

You put 5000€ into your bank account at an interest rate of 5% per year ii ..

paid annually.

((aa)) What amount will you have after 10 years?

105000 1.05 8144.47A = × ≈

((bb)) How long does it take for the amount to double?

10000 5000 1.05 2 1.05 ln 2 ln1.05ln 2ln 2 ln1.05 14.21ln1.05

t t t

t t t

= × ⇔ = ⇔ =

⇔ = × ⇔ = ⇔ ≈

Now assume that this interest is paid biannually.

((cc)) What amount will you have after 10 years?

200

.05(1 ) 5000 (1 ) 8193.082

ntt

rA An

= × + = × + ≈

((dd)) How long does it take for the amount to double?

2A0 = A0 × (1+rn)nt ⇔ 2 = (1+ .05

2)2t ⇔ ln 2

ln1.025= 2t

⇔ ln 22× ln1.025

= t ⇔ t ≈14.04

((ee)) What is the effective interest rate per year?

2.05(1 ) (1 )2

neff

rRn

= + −1= + −1= 0.050625


28

How much money should you have invested 5 years ago in order to have ii ii ..

100000€ today. The interest rate is 4%

55

100000100000 1.04 82192.711.04

A A= × ⇔ = ≈

The annual interest rate is 9%. What is the effective interest rate, if money ii iiii ..

is compounded

((aa)) annually?

1.09(1 ) 1 (1 ) 1 0.091

neff

rRn

= + − = + − =

((bb)) biannually?

2.09(1 ) 1 (1 ) 1 0.0920252

neff

rRn

= + − = + − =

((cc)) quarterly?

4.09(1 ) 1 (1 ) 1 0.09314

neff

rRn

= + − = + − ≈

((dd)) monthly?

12.09(1 ) 1 (1 ) 1 0.093812

neff

rRn

= + − = + − ≈

((ee)) continuously?

.091 1 0.09417reffR e e= − = − ≈


67

66 WWeeeekk 66

LLiinneeaarr PPrrooggrraammmmiinngg 22

11.. SSeett uupp tthhee ffoolllloowwiinngg ll iinneeaarr pprrooggrraammmmiinngg pprroobblleemmss,, aanndd ssoollvvee

tthheemm..

In order to ensure safe operations at a casino, it needs sufficient security ii ..

staff. Assume that the casino is open 24/7. The demand for security is dif-‐

ferent for each time of day. Thus, the casino manager has divided each

workday into 6 different security requirements. The security staff re-‐

quirements for each shift are summarized in the table below:

RReeqquuiirreedd SSeeccuurriittyy SSttaaffff 0011::0000 –– 0055::0000 45

0055::0000 –– 0099::0000 15 0099::0000 –– 1133::0000 40

1133::0000 –– 1177::0000 55

1177::0000 –– 2211::0000 90 2211::0000 –– 0011::0000 135

The manager wants to minimize the number of security staff working at the casi-‐

no. The labor union allows this, however, only under the following two condi-‐

tions: Each security officer can only work 8 hours per day. These 8 hours can on-‐

ly be worked consecutively. (N.B., the 6th and the 1st shift are consecutive

((aa)) What are the decision variables?

The decision variables refer to the number of security officers that work in each

shift. Furthermore, since every security officer needs to work his shift consecu-‐

tively, in an 8 hour time period, the decision variables best look as follows:

S01_09 (read: Shift starting at 1 ending at 9)

S05_13 (read: Shift starting at 5 ending at 13)


68

S09_17

S13_21

S17_01

S21_05

((bb)) What is the objective Function?

The objective is to minimize the total number of security guards used in a given

day. Thus, min S01_09 + S05_13 + S09_17 + S13_21 + S17_01 + S21_05

((cc)) What are the constraints?

((dd)) What is the optimal solution?

S01_09 S05_13 S09_17 S13_21 S17_01 S21_05Coefficients 1 1 1 1 1 1 0

Decision Variables:Requirement 01:00 -‐ 05:00 1 0 >= 45Requirement 05:00 -‐ 13:00 1 1 0 >= 15Requirement 09:00 -‐ 17:00 1 1 0 >= 40Requirement 13:00 -‐ 21:00 1 1 0 >= 55Requirement 17:00 -‐ 01:00 1 1 0 >= 90Requirement 21:00 -‐ 05:00 1 1 0 >= 105

Optimal Solution

S01_09 S05_13 S09_17 S13_21 S17_01 S21_05Coefficients 1 1 1 1 1 1 205

Decision Variables: 45 0 40 15 75 30Requirement 01:00 -‐ 05:00 1 45 >= 45Requirement 05:00 -‐ 13:00 1 1 45 >= 15Requirement 09:00 -‐ 17:00 1 1 40 >= 40Requirement 13:00 -‐ 21:00 1 1 55 >= 55Requirement 17:00 -‐ 01:00 1 1 90 >= 90Requirement 21:00 -‐ 05:00 1 1 105 >= 105

Optimal Solution


89

22 WWeeeekk 22

SSttaattiissttiiccaall iinnffeerreennccee bbaasseedd oonn mmoorree tthhaann ttwwoo ssaammpplleess

11.. CCoommppaarriinngg mmoorree tthhaann ttwwoo ppooppuullaattiioonnss ooff qquuaannttiittaattiivvee ddaattaa

You want to compare the average speed of cars at three different points in ii ..

your city. In all of these places the permitted speed is 70km/h, however

you have the feeling that drivers adhere to the speed limit differently at all

three places. The table below shows the results of the samples that you

take at the three different spots.

((aa)) What are the hypotheses for testing the scenario described above?

((bb)) Do the ANOVA assumptions hold?

((cc)) What are SSE, SST, SSTO, MSE and MST?

((dd)) Use an F-‐Test to test the hypotheses.

P1 P2 P3 OverallMean 72.84 74.96 76.00 74.77Median 73.8 78.1 79.2 73.8Variance 231.32 87.80 95.56 127.03Standard Deviation 15.21 9.37 9.78 11.27Min 49.2 56.8 57.6 49.2Max 106.6 92.3 93.6 106.6Count 34 52 45 131


90

You are interested in learning about the grades of your fellow students. In ii ii ..

particular, you wonder on whether grades are the same across all different

nationalities. After sampling, you obtain the following numbers:

((aa)) What are the hypotheses for testing the scenario described above?

((bb)) Do the ANOVA assumptions hold?

((cc)) What are SSE, SST, SSTO, MSE and MST?

((dd)) Use an F-‐Test to test the hypotheses

((ee)) How much of the differences in grades is explained by the factor Nationali-‐

ty?

((ff)) How do the results differ, if you only compare the grades of non-‐German

students?

You receive the following ANOVA output ii iiii ..

German Dutch Belgium Other OverallMean 7.00 5.52 6.76 6.29 6.43Median 7 5.5 7 6.5 6.5Variance 3.20 5.42 2.79 5.01 4.38Standard Deviation 1.79 2.33 1.67 2.24 2.09Min 4 2 4.5 2.5 2Max 10 9.5 9 10 10Count 140 104 60 43 347

Anova: One Factor

SUMMARYGroups Count Sum Average Variance

Var1 41 371 9.05 3.55Var2 58 583 … 2.58Var3 44 … 8.00 1.95

ANOVASource of Variation SS df MS F P-‐Value F Crit

Between Groups 105.7283 … 52.8641 19.8552 0.0000 3.0608Within Groups … … …

Total 478.4755 142


91

((aa)) What are the missing values in the table above?

22.. CCoommppaarriinngg mmoorree tthhaann ttwwoo ppooppuullaattiioonnss ooff ccaatteeggoorriiccaall ddaattaa

You are interested in learning about the market shares of the top brands ii ..

on the German market. You believe that VW with its brands controls 35%,

Mercedes and related brands 20%, Toyota and related brands 15%, and

the remainder by other brands. You take a sample at a busy intersection in

your home town, and count the following frequencies for the respective

brands

((aa)) What are the hypotheses?

((bb)) Use a Chi-‐Square test for goodness of fit to test the null hypothesis

You predict that 50% of students in your year are German, 40% are Dutch ii ii ..

and 10% are of other nationality. You sample randomly and obtain the fol-‐

lowing results



A store sells three different types of tablet PCs, type 1, type 2 and type 3. ii iiii ..

The store manager believes that each of the three types is sold equally fre-‐

quent. The store manager takes a look at the sales records for the preced-‐

ing month, and sees the following numbers

VW Mercedes Toyota Other252 198 162 288

German Dutch Other34 22 12

Type1 Type2 Type3102 109 119


92



You wonder whether the color of a car is dependent on the gender of the iivv..

driver. You observe the following numbers:


((bb)) Use a Chi-‐Square test for independence to test the null hypothesis

An online retailer is interested in learning more about its customers’ shop-‐vv..

ping behaviors. In a first analysis, they classify their customers into first

time customers and repeat customers. The retailer sells products in three

different categories. Shoes, attire and accessories. Of the 60 new customers

buying at the retailer last month, 13 bought shoes, 17 bought attire, and 30

bought accessories. Of the 420 purchases of repeat customers, 100 bought

shoes, 220 bought attire and 200 bought accessories.

((aa)) Set up a contingency table.

((bb)) Use a Chi-‐Square test for independence to test whether the category of

product purchases is independent of customer status.

Black Green RedMale 103 80 25

Female 20 14 15


133

33 WWeeeekk 33

SSiimmppllee RReeggrreessssiioonn

11.. TThhee rreeggrreessssiioonn eeqquuaattiioonn

Before a new movie comes out, a trailer is usually released way in advance. While

the platform TFMP does not have the function of rating movie trailers,

Youtube.com does provide this feature. There, you can rate any video with 0 – 5

stars. You wonder, whether the Youtube trailer rating of a movie is predicting the

final rating of a movie on the TFMP platform. This, you want to examine with the

following regression model:

0 1 _ rat t ratβ β ε= + +

Use the five observations to estimate 0β and 1β ii ..

((aa)) Plot the values in a plain

((bb)) Draw the best fitting line through the plain

rat t_rat8 3.57 4

9.5 44.5 15.5 2.5


134

((cc)) Find the equation of the best fitting line (remember, b0 is your best esti-‐

mate of 0β and b1 is your best estimate of 1β )

Remember, we are trying to find the coefficients of the “best fitting line”. As you

will remember from your QM1 lecture, you need to minimize the sum of squared

residuals. The residual is defined as: . This reads: the difference be-‐

tween the observed value, , and the predicted value, . While it would be bur-‐

densome to write out the entire sum of the squared residuals, you might remem-‐

ber that there is a shortcut to estimate the coefficients:

and

Thus, to use both equations, we first need to determine and :

Simple math will give you .

Next, you can determine b1 and b0.

and

Determine the R² ii ii ..

R² is the ratio between the explained variation and the total variation. The ex-‐

plained variation is the difference between the sum of square differences be-‐

tween the predicted value, , and the observed value, . The total variation is

î i iy yε = −

iy îy

11

2

1

( )( )

( )

n

i iiXY

nXX

ii

x x y ySSbSS x x

=

=

− −= =

−

∑

∑0 1b y b x= −

x y

3.06.9

xy==

11

2

1

2 2 2 2 2

( )( )

( )

(3.5 3)(8 6.9) (4 3)(7 6.9) (4 3)(9.5 6.9) (1 3)(4.5 6.9) (2.5 3)(5.5 6.9)(3.5 3) (4 3) (4 3) (1 3) (2.5 3)

1.3462

n

i iiXY

nXX

ii

x x y ySSbSS x x

=

=

− −= =

−

− − + − − + − − + − − + − −=− + − + − + − + −

≈

∑

∑

0 6.9 1.3462 3 2.8615b = − × =

îy iy


135

the sum of square differences between the average value of the dependent varia-‐

ble, , and each individual observation, .

Thus, the explained variation is

and the total variation:

Thus,

According to the 5 observations, the explanatory variable t_rat explains 75% of

the change in the dependent variable rat.

Based on your model, which rating would you expect for a movie whose ii iiii ..

trailer has a rating of 3.5?

Plugging in 3.5 for t_rat would yield:

y iy

2

12 2 2

2 2

ˆ( )

(2.8615 1.3462*3.5 6.9) (2.8615 1.3462*4 6.9) (2.8615 1.3462*4 6.9)(2.8615 1.3462*1 6.9) (2.8615 1.3462*2.5 6.9) 11.7788

n

ii

SST y y=

= − =

+ − + + − + + − ++ − + + − =

∑

2 2 2 2 2 2

1( ) (8 6.9) (7 6.9) (9.5 6.9) (4.5 6.9) (5.5 6.9) 15.70

n

ii

SSTO y y=

= − = − + − + − + − + − =∑

2 11.7788 0.75015.70

SSTRSSTO

= = =

∂ 2.862 1.346 _rat t rat= +

∂ 2.862 1.346 3.5 7.573rat = + × =

Extras

Sem

inar

E Exams

Extras

Sem

inar

Exams


Exams Uniseminar – Quantitative Methods II

EExxaammss

You should start early with the calculation of exams, because you need to get a

general feeling of how the exams are built up. You will soon discover how the

exams are constructed and that there are general tendencies, which repeat from

exam to exam. In this part you will find old exams of the Maastricht University, as

well as one practice exams constructed by Uniseminar. During the seminar you

will then receive a further practice exam.


PPrraaccttiiccee EExxaamm 11 ((iinnccll .. ssoolluuttiioonnss)) 11

OOlldd EExxaammss 3333

10/11 Resit

10/11 First Sit

09/10 Resit

09/10 First Sit

08/09 Resit

08/09 First Sit

Uniseminar – Quantitative Methods II Practice Exam 1

1

MMaatthheemmaattiiccss

11..)) What is the value of the following infinite series?

2+32−34 +

38 −

316 +

332 +⋯

a.) 2.50

b.) 2.75

c.) 3.00

d.) 3.25

22..)) Consider the following series of payments. Today you receive 200€, in one

year you receive 400€, in two years again 200€, in three years again 400€,

in four years again 200€ and so on. The project ends after thirteen years

and the yearly interest rate is 5%. What is the present value of this series

of payments?

a.) between 3000 and 3100

b.) between 3100 and 3200

c.) between 3200 and 3300

d.) between 3300 and 3400

33..)) Consider the following investment opportunity: You are offered a payment

of 200€ right now and a second payment of 200€ next year. However, to

get those payments you have to pay 410€ in two years from now. What is

the internal rate of return of this project?

a.) 2.8 %

b.) 1.6 %.

c.) 9.4 %

d.) 11.2 %

Practice Exam 1 Uniseminar – Quantitative Methods II

2

44..)) Consider the following three matrixes: A is of format m x n, B is format k x l

and C is of format p x q. When is the following defined: 𝐴𝐴 ∗ 𝐵𝐵 + 𝐶𝐶?

a.) 𝑘𝑘 = 𝑙𝑙 = 𝑚𝑚 = 𝑞𝑞 and 𝑛𝑛 = 𝑝𝑝

b.) 𝑘𝑘 = 𝑙𝑙 and 𝑚𝑚 = 𝑛𝑛 = 𝑝𝑝 = 𝑞𝑞

c.) 𝑘𝑘 = 𝑙𝑙 = 𝑚𝑚 = 𝑝𝑝 and 𝑛𝑛 = 𝑞𝑞

d.) 𝑘𝑘 = 𝑙𝑙 = 𝑝𝑝 = 𝑞𝑞 and 𝑚𝑚 = 𝑛𝑛

55..)) Consider the following invertible matrix:

𝐴𝐴 =1 2 11 1 02 2 1

The sseeccoonndd column of the inverse 𝐴𝐴 is given as 2𝑥𝑥𝑥𝑥

What are the values of 𝑥𝑥 and 𝑥𝑥 ?

a.) 𝑥𝑥 = 1, 𝑥𝑥 = 0

b.) 𝑥𝑥 = −1, 𝑥𝑥 = 0

c.) 𝑥𝑥 = 0, 𝑥𝑥 = 1

d.) 𝑥𝑥 = 0, 𝑥𝑥 = −1

66..)) Consider the following matrix 𝐴𝐴 = 1 1𝑏𝑏 𝑎𝑎² . It only has an inverse if…

a.) 𝑎𝑎 ≠ 𝑏𝑏

b.) 𝑎𝑎 = −𝑏𝑏

c.) 𝑎𝑎 = 𝑏𝑏 and 𝑎𝑎 = −𝑏𝑏

d.) 𝑎𝑎 ≠ 𝑏𝑏 and 𝑎𝑎 ≠ −𝑏𝑏


3

77..)) When can we be absolutely sure that a certain matrix 𝐴𝐴 is invertible?

I.) There exists a matrix 𝐵𝐵 such that 𝐴𝐴𝐴𝐴 = 0

II.) The system of equations 𝐴𝐴𝐴𝐴 = 𝑏𝑏 has a unique solution

a.) In I.) we can be sure, in II.) not

b.) In II.) we can be sure, in I.) not

c.) In both cases we can be sure

d.) Neither in I.) nor in II.) can we be sure

88..)) The determinant of the 3x3 matrix A is equal to 4. First, we multiply the

first row of A with 4. Second, we multiply the second column of A with 3.

Finally, we multiply all elements of A with 2. What will the determinant of

this new matrix be equal to?

a.) 384

b.) 96

c.) 192

d.) 48

99..)) The system 𝐴𝐴𝐴𝐴 = 𝑏𝑏 has a unique solution. What can you say about the

solutions of the system 𝐴𝐴𝐴𝐴 = 𝑐𝑐?

a.) The system will have no solution

b.) The system will have infinitely many solutions

c.) The system will have a unique solution

d.) Nothing can be said on the basis of the given information

Practice Exam 1 Uniseminar – Quantitative Methods II

10

SSttaattiissttiiccss

This trial exam is based on a dataset that contains 315 students. Of these

students multiple characteristics like the QM1 grade, the gender or the

nationality, etc. are collected. The variables are named as follows:

ggrraaddee QM1 grade of a student on a scale from 0 to 10, where 10 is best

bboonnuuss The bonus points a student got for the QM1 exam on a scale from 0

to 8

mmaallee A gender dummy, 1 for male and 0 for female students

IIBB dummy: 1 for IB students, 0 otherwise

EEccoonn dummy: 1 for Economics students, 0 otherwise

FFiissccaall dummy: 1 for Fiscal Economics students, 0 otherwise

Some people claim that the performance in a QM1 exam does solely depend on

the level of mathematical and statistical talent. If this claim was true this would

mean that the QM1 score of a student must be equal to the QM2 score of a

student.

2211..)) Which test can be used to evaluate this claim?

a.) A Chi-‐Square test for Independence to check for the relationship between

talent and QM scores

b.) A two sample t-‐test for QM1 and QM2 scores

c.) A paired sample t-‐test for the difference in QM1 and QM2 grades

d.) ANOVA with the students as the levels and the QM scores as the response

Forget about the claim and focus on something different. Some people claim that

male and female students are not equally successful in QM1. Unfortunately, it is


11

not clear yet who of the two actually performs better. Therefore we could test the

following set of hypothesis about the mean difference in the QM1 bonus scores of

female and male students.

𝐻𝐻 : 𝜇𝜇 − 𝜇𝜇 = 0 𝐻𝐻 : 𝜇𝜇 − 𝜇𝜇 ≠ 0

2222..)) To test this claim, consider the descriptive statistics table on the right.

Against the state alternative, the p-‐value is...

a.) between 5% and 10%.

b.) between 2.5% and 5%.

c.) between 1% and 2.5%.

d.) smaller than 1%.

Obviously gender is not the only characteristic that impacts the grade. Even more

academic discussion focuses on the difference in the performance of IB,

Economics and Fiscal Economics students. With the help of a one way ANOVA we

can test the following hypothesis:

𝐻𝐻 : 𝜇𝜇 = 𝜇𝜇 = 𝜇𝜇

The results of this ANOVA test are given in the Excel report.

2233..)) Which of the following is nnoott true for this ANOVA test?

Extras

Sem

inar

E

Extras


Extras Uniseminar – Quantitative Methods II

EExxttrraass

In this part you find several extras that will be very helpful for your exam

preparation. In this course you will find an extra explanation of how to read-‐off

the critical values of the statistics tables and the formula sheets for math as well

as statistics. We decided to give you the original formula sheets, which you will

also have during your exam, as it is essential to get used to the exam’s

circumstances. However during the seminar the tutor will discuss the formula

sheets and will highlight or extend several sections.


FFoorrmmuullaa SShheeeettss 11

HHooww ttoo rreeaadd ssttaattiissttiiccss ttaabblleess 99

Uniseminar – Quantitative Methods II Extras

9

HHooww ttoo rreeaadd tthhee SSttaattiissttiiccss ttaabblleess

zz-‐-‐ttaabbllee

The 𝑧𝑧-‐scores are given on the outside of the table. The numbers in the inside correspond to the area under the curve, which represents the probability.

zz 00..0000 00..0011 00..0022 00..0033 00..0044 00..0055 00..0066 00..0077 00..0088 00..0099 00..00 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 00..11 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 00..22 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 00..33 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 00..44 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 00..55 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 00..66 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 00..77 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 00..88 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 00..99 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 11..00 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 11..11 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 11..22 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 11..33 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 11..44 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 11..55 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 11..66 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 11..77 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 11..88 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 11..99 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 22..00 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817 22..11 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 22..22 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 22..33 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 22..44 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936 22..55 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 22..66 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 22..77 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 22..88 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 22..99 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 33..00 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990

Date post:	10-Mar-2016
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UM_QM2-IB_Ordner

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