QM1 Sample

Quantitative Methods I

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uniseminar.nl – Quantitative Methods I


Table of contents page

1. Introduction 1 2. Theory 3 A complete summary of the exam relevant theory 2.1 Maths 2.2 Statistics 3. Solutions to in-class exercises 55 Detailed solutions to all exercises you had to prepare during the block. 3.1 Maths 3.2 Statistics 4. Formula Sheet 130 4.1 Maths 4.2 Statistics 5. Trial Exams (incl. extensive solutions) 135


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1. Introduction Uniseminar.nl was founded in September 2008 by two students from the University of Maastricht for students from the University of Maastricht, who influence the seminars with their own impressions and experience of the courses. Uniseminar.nl cooperates with uniseminar.ch, which is the proven counterpart of uniseminar.nl at the University of St. Gallen Switzerland.

As you, we had and wanted to pass the exams and from our own experience we know how difficult it can be to deal with problems like time pressure and organization of the study material.

Quantitative Methods I The QM1 exam you are facing will be a Multiple Choice exam. As the exam the seminar is separated into a Mathematics part and a Statistics part. The exam consists of 40 questions, 20 for each subject The required literature for the exam is: For Maths Sydsæter, Knut & Hammond, Peter J. (2008) - Essential Mathematics for Economic Analysis For Statistics Sharpe, Norean R., De Veaux, Richard D., & Velleman, Paul F. (2010) - Business Statistics For Both: - Lectures How to prepare the exam: To optimally prepare the exam we provided you with a uniseminar.nl Summary Scrit. After the seminar you should first try to understand the whole theory of the seminar (with your notes) and the theory script. The tasks incl. solutions from the tutorials will help you to understand the theory ( in-class exercises). In the last days before the exam you should mainly focus on the trial exams, with the help of the formula sheet. A year’s exam always contains a decent amount of old year’s questions. We hope that we could help you with our seminar wish you GOOD LUCK with your exams!!! Your uniseminar.nl team


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Theory & Examples

In the following part you will find explanations of all the relevant topics of the course. It contains all essential principles concepts, as well as rules you need to know. In addition there are many examples that make it easier to understand it. It is very important that you understand the theory in order to able to apply it later in the exam. Before doing practice exams you should work through this part.

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2.1 Mathematics 1. Previous knowledge

1.1 Algebra

1.1.1 Real Numbers:

Integer Numbers – all whole numbers:

The integer numbers can be divided into two groups, odd and even numbers:

Natural Numbers – all nonnegative and nonzero integer numbers:

Rational Numbers – all real numbers, which can be written as a fraction of integer numbers:

Examples are:

Irrational Numbers – all real numbers, which cannot be written as a fraction of integer numbers:

Examples are: Please note that terms which are mathematically not defined, such as , or , are not part of the irrational numbers!

1.1.2 Rules of algebra:

These rules are the basic set-up of algebra, which allow to make simple transformation of terms.

I. II. III.


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IV. V. VI. VII. VIII. IX. X. XI. XII. XIII.

One of the most often applied and unfortunately not always most straightforward to see operations is referred to as factoring. Factoring basically uses what is stated by formula XII. and XIII. and the following will give two examples:

Additionally there are some important formulas, which also result from XIII. and are known as the binomial formulas:

I. II. III.

1.1.3 Fractions:

Fractions simplify calculations with terms, which are divided by another term. The following rules will make calculations involving fractions easier:

I.

II. III. IV. V. VI. VII. VIII.

IX.


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As in I. it is possible to cancel out common factors. However, note that this is only possible with multiplications, not with sums and therefore:

1.1.4 Powers:

The notation of powers is used to shortcut the multiplication of the same number/constant with itself:

In these cases we call the basis, while or is called the exponent.

Additionally there are certain properties which simplify the calculations involving powers:

I. II. III. IV. V.

Please note that powers can only be simplified if the basis or the exponent are the same! Moreover, powers are about the products, therefore:

1.1.5 Growth with powers:

Powers can be used to easily express the growth of a certain variable, for example the money stored in a bank account. In this case an amount of 1000,-€ will grow as follows over three years if the growth rate (in this case the interest rate) is equal to 5%.

Converting this into a more general scheme yields the following formula:

Where is the initial amount, is the current amount, is the growth rate and is the amount of years.


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1.1.6 Fractional Powers:

There is a combination of fractions and powers in a way that fractions are used to express powers. In these cases the powers then refer to a certain root of the term.

Assuming that and it is possible to solve this:

Additionally the general rules for powers still apply with fractional powers. Therefore it holds true that:

and

1.1.7 Summations:

The summation notation is widely used in order to quickly write up a longer series of sums. It is generally defined that:

In the case of an infinite sum it can be further generalized:

The summation notation can be applied in two ways. First, it is possible to write up a certain sum with a summation sign in front:

Second, a summation sign can be converted back into a normal sum and solved:


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1.1.8 Inequalities:

Inequalities are used to make statements about which number is larger. It is possible to identify to groups of inequalities. The next three cases are called strict inequalities because the case in which the two numbers are unequal is rules out completely.

Contrary, the next two are known as non strict inequalities, because in both cases the two numbers are also allowed to be equal.

In addition certain laws for inequalities can be constructed which should be straightforward:

I. If and → and II. If → III. If and → IV. If and → V. If and → VI. If and →

1.1.9 Intervals:

First, two types of boundaries of intervals need to be identified. An interval can be open and closed, where open means that the boundary is not part of the interval and closed means that the boundary is included. Consider the following three examples and pay attention to the different sort of brackets used:

= all numbers between 2 and 10, not including 2 and 10 as the brackets are round. Therefore the interval is open.

= all numbers between 4 and 6, this time including 4 and 6 because the brackets are square. This interval is therefore closed.

= all the numbers between and , including but excluding . This interval is called half-open.


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1.1.10 Absolute value:

The absolute value of a number ignores the sign the number might carry. It therefore holds that:

and more general:

1.2 Equations

1.2.1 Linear equations:

This is simplest case of an equation, which can therefore also be solved using the easiest tools. The section will continue introducing ever-harder types of equations, which then require more advanced tools to be solved.

An equation can be solved, by pooling all the unknown terms on one side of the equation and the known terms on the other side of the equation. Whenever we change an equation to achieve this we display the respective steps directly behind the row to make the transformations clear.

Example 1:

Example 2:


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1.2.2 Equations with parameters:

The real extension now is that the unknown of the equation cannot simply be expressed in numbers, but also involves parameters. A parameter can be treated like any other number with the only difference that it could take any value.

Example 1:

Example 2:

1.2.3 Quadratic equations:

So far all of the equations were linear with respect to the unknown . This section will relax this assumption and show how to solve an equation if the unknown is given in quadratic terms. With respect to the result this has one direct consequence because most quadratic equations do not have a single solution for , but two solutions, which satisfy the equation.

Additionally some of the equations require the use of the pq-formula (ABC-formula for the Dutch). Make sure you remember this formula by heart as it will otherwise be impossible to solve some questions:

pq-formula:

Example 1:

or


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Example 2:

A product can only be zero if at least one of the terms is zero, it therefore holds that:

or

Example 3:

or

1.2.4 Linear equations with two unknowns:

For each unknown in we need an equation to be able to solve for the unknown. Until now we only treated one equation with one respective unknown . This section will show how to solve a system of two equations with two unknowns and .

Example 1:

(I) (II)

It is easiest to isolate the in (II), which then reads:

(II)


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This can then be plugged in for in (I)

(I)

The result for can then be used to solve for from (II).

Example 2:

(I) (II)

Again we need to isolate one of the unknowns. Factoring out in (I) results in:

(I)

This result can then be plugged into (II):

(II)

The solution for again allows to solve for in (I):


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1.2.5 Non-linear equations:

In the section about quadratic equations a special case of a non-linear equation has already been discussed. However, there are plenty more types of equations which are non-linear. The following examples will discuss three of the more common applications.

Example 1:

Example 2:

Example 3:

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