Statistics II: Inferential Statistics - Simple Random Sampling and Sample Statistics, Apuntes de Administración de Empresas

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Mikel Esnaola Universitat Autònoma de Barcelona

slides by Xavier Vilà

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Course Agenda

• Activities 1-4: Delivered after the end of the corresponding chapter.
• Midterm Exam: October 16th
• Final exam: January th
• Re-evaluation exam: January 23rd^ (final grade at least 4)

Course Contact

Mikel Esnaola

Office B3-164 (Unitat de Fonaments de l’Anàlisi Econòmica - Dept. d’Economia i d’Història Econòmica)

Office Hours: Mondays 3pm - 6pm & Tuesdays 3pm - 6pm (send email before)

Statistics becomes extremely important for the first of these three items.

In order to study a real problem, the researcher must observe the real world

• Nevertheless, it is also clear that no researcher can observe the whole reality
  • Newton can not observe all the falling apples,
  • An economist can not interview the whole population of a country
• It is hence necessary to somehow summarize the reality,
  • This task has to be done so that such summary closely represents the whole reality.
  • It has to be done so that the conclusions drawn from the summary can be reliably applied to the whole population.

Chapter 1 We will study in detail how the reality is rigorously summarized and what are the main features of the results obtained in this process.

Chapter 2 We will see the first approach on how to generate conclusions about some real issues based on what we observe in the summary.

Chapters 3 and 4 introduce more sophisticated techniques to make inferences about the reality using some of the more elemental results seen in Chapter 2.

Chapter 5 Introduces the linear regression analysis, a technique widely used in the economic analysis (and other sciences) to study the relationship between variables.

Chapter 1 is very important in order to easily understand what other chapters deal with, and also to get an global idea of the whole process of statistical inference.









A careful study and deep understanding of the topics in Chapter 1 is necessary before undertaking the study of other chapters

It is important to understand that

• statistics is based on probabilistic techniques.
• any statistical conclusion drawn from this kind of summary will not be true for sure when applied to the whole reality, but only with a certain probability.

1.1 Inferential Statistics: Definition and Inference

Methods

Statistical inference is mainly built upon four main concepts, which will be defined and described below.

Population Is the set of elements that are the object of study. The goal will be to draw some conclusion regarding some specific feature of this population.

Example 2 All the apples in the world. The feature at study is whether an apple falls down or not. Example 3 Labor force in the European Union. The feature at study is whether a worker is unemployed or not. Example 4 Production of Intel chips in a given day. The feature at study is whether a chip is faulty or not.

Sample Subset of the Population used to draw conclusions about the population

Example 5 50 apples in Newton’s garden. Example 6 Unemployment statistics at the European Union. Example 7 25 Intel chips manufactured in a given day.

Statistic Computation made using the elements in the sample and used to get an approximation to the true value of the parameter. It is important to notice that this value will be known (since we will compute it) and will be used to draw conclusions on the true value of the parameter, which is unknown and is what is of interest to us.

Example 11 Proportion of falling apples among the 50 sampled apples in Newton’s garden. Example 12 Unemployment rate among the workers interviewed in the un- employment statistics in the European Union. Example 13 Proportion of faulty chips among the 25 selected chips produced in a given day.

From this four main concepts, the process of statistical inference works as follows:

1. Using sampling techniques that will be explained below, a sample is selected from the population that is going to be studied.
2. From this sample, the proper computations are done in order to obtain a statistic.
3. From this statistic, using some statistical inference technique that we will see in other chapters, some conclusions are drawn regarding the unknown population parameter that represents the feature of the population that is to be studied.

We can now provide a definition for Statistics (or Statistical Inference, to be more precise) which is more formal than the one offered in the introduction.

Definition 14 Statistical Inference is a subject whose main objective is to draw conclusions regarding a population through the study of one sample by means of probabilistic techniques.

1.2 Definition and properties of Simple Random

Sampling

• We will see what a sample is, that is, how a sample can be selected out of a population
• Since we want to study this sample to produce conclusions about the population, it can not be selected arbitrarily
• There exist rigorous techniques specially tailored for this purpose
• In what follows, the more basic techniques will be introduced

Definition 15 Sampling is a systematic technique to select a sample out of a population in such a way that it is representative of the population