Syllabus Information

1.1. Introduction

The subject Statistics II is given by the Department of Economic Structure and History and Public Economy, and belongs to the knowledge area of Quantitative Methods for Economy and Business. This module is compulsory in the second year. It has a strong quantitative profile, and provides the basic tools for the treatment of information and the quantification of economic phenomena. It forms the basis of the statistical method in decision making in the economy and business field, describing in detail and justifying the methods and techniques used in the analysis and interpretation of economic data. The module focuses mainly on inferential methods, providing the student with the ability of estimating and testing models that describe and explain economic phenomena.

1.2. Recommendations to take this course

There are no previous requirements for studying this subject. However, it is recommended to have completed the courses Mathematics I, Mathematics II and Statistics I to facilitate the learning process and reduce the level of effort. A basic knowledge of office software, mainly a text editor and a spreadsheet, is also recommended in order to carry out the practices correctly and facilitate the acquisition of some relevant competences.

1.3. Context and importance of this course in the degree

The module Statistics II is included in the unit of Quantitative Methods for Business of the curriculum, together with the modules Statistics I, Operational Research and ICTs in Business. This module combines the analytical tools with the module of Foundations of Mathematical Analysis and Tools (Mathematics I, Mathematics II, Econometrics and Econometric Applications in Business).

On the one hand, it uses the knowledge and the skills acquired in the modules Mathematics I and II and Statistics I, as a summary description of the economic phenomenon under study while, at the same time, it provides the student with the techniques and methods needed to understand that phenomenon and explain it by means of a model. In the module, inferential procedures are addressed for the first time; that is, given some observed results, a model that fits that economic phenomenon and explains it is created. The modules Econometrics and Econometric Applications in Business will tackle the same procedures with applications to more specific and sophisticated models, showing the applications of the scientific method to economic problems.

1.4. Activities and key dates

Presentation of the module: In the first session, the objectives and contents of the subject, the teaching methodology and the assessment criteria will be explained in detail.

Intermediate tests: The students will take two intermediate tests using the Microsoft Excel spreadsheet. The first computer-based test (P1) will take place at the end of Unit 4 (November) and the second one (P2) at the end of Unit 6 (Dec-Jan).

Global test: In accordance with the official calendar drawn up by the Center, the students will take a global test during the examination period that will consist of a written exam (WT), in which the competences and skills achieved will be evaluated, having a weight of 60% of the final mark, and a practical test (PT) involving the resolution of a near-real problem with the spreadsheet, weighted 40%.

In the first sitting, the practical test will not be compulsory for the students who have obtained a minimum of three points in each intermediate test. In the second sitting all the students will do both the written and the practical test.

The teaching materials developed during the course, as well as the examination calls and their results will be published in the learning platform: http;//moodle2.unizar.es.

2.1. Learning goals

Passing this course will enable the student to…

1. Understand and use probability as a measure of uncertainty in economic phenomena.
2. Differentiate and apply the most important models of probability, both discrete and continuous.
3. Measure the uncertainty of future results and facts.
4. Use and schedule sampling methods to extract information from economic phenomena.
5. Calculate the sample size needed for making decisions with minimum guarantees.
6. Synthesize the sample information in the more usual statistics. Calculate and assess these statistics, examining the conclusions.
7. Infer properties of theoretical models from sample observations and justify the goodness of these properties.
8. Design tests of statistical hypotheses to corroborate or refute a theory from the sample information.
9. Utilize the more usual statistical tests for proportions, means and variances of random models.
10. Compare and analyze the properties of two random models, detecting the differences between proportions, means and variances.

2.2. Importance of learning goals

The module has a double objective concerning the training of the future professional. The first challenge comes from the instrumental and quantitative character of the module, with the aim of providing the student with the basic tools needed to extract information and to use and interpret it in order to understand an economic reality. The methods and techniques acquired in the module enable the student to develop contents and skills in other modules of the degree. A second goal is to promote a critical spirit in the student when tackling projects in the economic and business field. Statistical methods, which allow to quantify and measure the uncertainty of the information collected, help to guarantee precise and reliable conclusions for taking scientific decisions, providing the students with the criteria needed to understand and examine both their own results and the results provided by external sources.

3.1. Aims of the course

The module and its expected outcomes respond to the following approaches and goals:

The subject Statistics II aims to provide the student with the basic tools for the understanding and handling of random phenomena related to the economy. Therefore, it has a practical profile so the student can analyze, solve and interpret economic realities with the aim of making decisions with scientific rigor.

The first units are dedicated to the study of the basic random model, the most frequently used to explain real phenomena mainly linked to economic variables. Afterwards the general scope of an economic problem is presented, in which the theoretical model is not completely known and some empirical research is required in order to have a grasp of it.

The next lesson addresses the issue of how to select a sample which allows to infer the unknown facets of the proposed method. Particular emphasis is placed in the random sampling method, which is the basis of the inferential methods that will be discussed later. A key aspect addressing this problem is to determine the size of the sample for our conclusions to be reliable and offer probabilistic guarantees.

The next lessons go into detail about the inferential methods, from the perspective of both parameter estimation and hypotheses testing. The student will learn the different approaches and conclusions, as well as the interpretation of the achieved results. The last lesson addresses the comparison of two phenomena, a very common practice when studying two different economic realities or groups –either geographic or temporary.

The contents of the module have a practical goal, namely, for the student to get the tools and the skills required to apply them to different situations, obtaining the more outstanding conclusions and providing the interpretations needed for their understanding.

3.2. Competences

Specific competences:

• Assess the situation and the development expected for companies and organizations, making decisions and extracting the relevant knowledge with reference to social responsibility.
• Understand and apply professional criteria and scientific rigor to solve economic, business and organizational problems.
• Develop and draft a project.

Transversal competences:

• Ability to solve problems
• Ability for self-organization and strategic planning
• Ability to analyze and extract information from a variety of sources
• Ability to make decisions
• Motivation for quality and excellence
• Capacity for adaptation to new situations
• Capacity to apply theoretical knowledge to real situations
• Ability to use the technological tools needed in the exercise of the profession

4.1. Assessment tasks (description of tasks, marking system and assessment criteria)

The student will demonstrate the achievement of the desired learning outcomes through the following assessment activities.

The assessment of the subject is global, based on two different tests: a theoretical-practical test and a computer-based test.

The theoretical-practical test involves solving problems and theoretical questions similar to those taught in the practical sessions and in the classroom lectures. The test consists of three questions with several items each. Tests from previous years are available for the students in the learning platform, which can help them to know the nature of the test.

The computer-based test consists of solving problems based on real data with the spreadsheet. The proposed problems are similar to those solved in the practical sessions. Alternatively, this test may be passed by doing two intermediate tests, PT1 and PT2, during the semester.

The theoretical-practical test (WT) is worth 60% of the overall mark and the computer-based test (PT) is worth 40%. There are two alternatives for the first sitting:

• Students who take only the theoretical-practical test (WT): those students who have obtained a minimum of three points in each of the intermediate test PT1 and PT2. Their grade in the PT test will be 0.5 PT1 + 0.5 PT2.
• Students who take both tests (WT and PT): those students who either have not taken the intermediate tests, PT1 and PT2, have failed one of them or both with not less than three points, or who want to improve their mark.

The final assessment mark for the course will be computed as follows:

Final mark = 0.6 WT + 0.4 PT.

The written test (WT), as well as the intermediate tests (PT1 and PT2) and the computer-based test (PT), will be marked on a scale from 0 to 10. The student should obtain at least 3 points in each (WT, PT1 and PT2; or WT and PT) to apply the formula and, consequently, obtain the final mark. If this condition is not fulfilled the final mark will be the

Final mark = min{WT,PT}.

In order to pass the course the student should obtain at least 5 points as the final mark. If the final mark is under 5 or the student has less than 3 points in any of the two parts WT and PT the student is required to re-sit. Reassessment will consist of a written test and a computer-based test, and the assessment mark will be similar to the global test in the first sitting. Students will not have to re-sit one test if the mark obtained in the first sitting is not less than three points. If the student do not take the re-sit of one test the mark of this test will be the mark that obtained in the first sitting.

Assessment criteria

In the theoretical-practical test, the student will demonstrate his/her ability to model and develop the suggested problems, obtain their solutions and explain them in the context of the proposed situation.

In the computer-based tests, the student will demonstrate his/her ability to obtain accurate numerical results and analyze their adequacy and correspondence with the real situation under study.

5.1. Methodological overview

The learning process proposed for this module is based on the following premises:

Several teaching methods will be used in the learning process, based on the objectives set and the competences to develop. Explanatory techniques will be used in the lectures, aiming to analyze and develop the basic concepts of the subject, and collaborative training techniques will be used to get the student involved in order to develop her/his ability to organize, plan and make decisions.

Furthermore, computer tools and solving case studies will be used to tackle the competences related to the use of technological tools, problem solving and ability to analyze and extract information from external sources. Moreover, the classroom practicals will enable the student to develop the capacity to adapt to new situations and apply the knowledge acquired in professional practice.

The learning platform Moodle (http://moodle2.unizar.es) will provide the educational support. All the documentation and material needed for the lectures and the classroom practicals and the associated information, including this teaching guide will be published in this platform. In particular, documents such as conceptual maps and video tutorials will be of great interest for the student when preparing and revising the subject.

5.2. Learning tasks

The programme will help the student to achieve the expected results by means of the following activities:

Lectures (30 classroom hours and 45 autonomous working hours): Will be used mainly to introduce the basic concepts and the theoretical developments of each lesson. Explanatory techniques will be used, always promoting participation and discussion in the classroom. The teacher’s explanations will be supported by a presentation and by the development of the corresponding conceptual map. Class attendance, participation and note-taking are highly recommended. The presentation, its complementary theoretical developments and the conceptual map will be published in advance.

Classroom practicals (12 classroom hours and 15 autonomous working hours): This activity aims to show the student how to deal with problems. The teacher will announce the cases that will be analyzed in each session in advance, so that the student can study them and attempt to solve them. In the classroom, the cases will be analyzed and explained by the teacher with the help of the blackboard and the computer. The session is intended to be participative and to encourage the students to discuss and come to an agreement on both the analysis or the problem and its solution. With this aim, some type problems will be published in advance so the students can use them as a basis for their adaptation to more complex situations and contexts.

Computer practicals (14 classroom hours and 18 autonomous working hours): This activity will take place in computer rooms. As a general norm there will be two students per computer. The goal is solving problems that are more complex and real than those studied in the classroom practicals. The session will consist of two parts: in the first, the students will learn the basic techniques, under the teacher’s guidance. In the second, the student will be asked to develop the rest of the task until the problem is completely solved. Some video tutorials will be published in advance in order to prepare the student for the practicals, as well as a template of the practical, that will speed up the process of drawing conclusions.

Small group classes (4 classroom hours and 4 autonomous working hours): Will take place in the computer room within the established hours. The goal of these sessions is to help the student to acquire skill and fluency in the resolution of statistical problems with the computer. The teacher will supervise the individual or pair work of the students and will resolve any doubts that may arise.

5.3. Syllabus

Unit 1: Discrete random variable.

Random variables. Probability distribution. Discrete and continuous random variable. Discrete random variable: Probability distribution or mass function.  Expected value and its properties. Binomial, Hypergeometric and Poisson distributions.

Unit 2: Continuous random variable.

Continuous random variable: density and probability density functions. Uniform, Exponential and Gamma distributions. Continuous approximations for discrete distributions.

Unit 3: Introduction to sampling theory.

Basic concepts: population, sample, parameter and statistics. Sampling methods. Sampling distribution of statistics: Exact, Monte Carlo and asymptotic methods. Sample-size determination.

Unit 4: Point and interval estimation.

Estimation. Building estimators. Properties of estimators. Confidence interval. Methods of finding interval estimators: the pivot method. Confidence intervals for estimation of means, proportions and variances.

Unit 5: Parametric hypotheses.

Basic concepts: Simple, compound, null and alternative hypotheses, significance level, power of a test. Maximum power tests: Neyman-pearson lemma. Likelihood-ratio test. Tests of the mean and variance of a normal distribution, tests of the population proportion. Goodness of fit tests.

Unit 6: Two-population hypothesis tests.

Independent and paired samples. Comparing proportions, means and variances: confidence intervals and tests of statistical hypotheses.

5.4. Course planning and calendar

The module consists of 6 ECTS credits (150 hours of study), distributed between classroom hours and individual homework. This study load is distributed as follows:

The detailed schedule of the course is as follows (the four small group classes are not included because they can be taught at any time during the semester, depending on classroom availability:

5.5. Bibliography and recommended resources

[BB: Bibliografía básica / BC: Bibliografía complementaria]