Statistical Estimation and Confidence Intervals, Diapositivas de Estadística

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Xavier Vilà Universitat Autònoma de Barcelona

1. Introduction to Inferential Statistics and Estimation

Statistical Inference is a collection of techniques by means of which we can draw conclusions with regards to a reality from the study of a sample of such reality

Example 1 When an electoral survey is conducted it is clear that its results do not exactly coincide with the results in the nal election. Nevertheless, if the survey is "well done", that is, if the sample (which in this case is the set of people interviewed) closely represents the whole reality (which in this case is the whole population that has the right to vote), then the survey result will be close to the nal results with a high probability

1.1 Inferential Statistics: Denition and Inference

Methods

Statistical inference is mainly built upon four main concepts, which will be dened 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 specic 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.

Parameter Is the feature of the population that we want to know something about. This feature has to be a numerical one and, obviously, its true value must be unknown

Example 8 What is the proportion of falling apples. Example 9 What is the unemployment rate at the European Union Example 10 What is the proportion of faulty chips among those produced 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.

This process can be represented as in Figure 1

Population

Sample

Parameter (unkonwn)

Statistic (known)

Statistical Sampling Inference

Figure 1: The process of Statistical Inference

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

Denition 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.

Denition 15 A statistic (or estimator) is a formula that uses the values in the sample at hand (observations) in order to produce an approximation to the true value of an unknown population parameter.

Denition 16 An estimate (or estimation) is the particular value of an es- timator that is obtained from a particular sample of data and normally used to indicate the value of an unknown population parameter.

Hence,

• a statistic is not a number but a formula
• an estimate is the number that is obtained when the formula (the estimator) is applied to the observations of the specic sample that we have at hand.

Important

Given that the sample is obtained by means of a random technique, the statistic is a random variable 







The statistic will produce dierent estimates with dierent probabilities (depend- ing on the specic sample that is nally "selected" at random).

In this sense, an estimate is a specic realization of this random variable.

The following example aims to clarify this idea.

For the study, we

• want to obtain a random sample of size 2.
• compute the average number of cars in the sample
• use it to infer some conclusion regarding the true average in the population.

The sample mean (or just mean, for short) will play the role of statistic in this example. We will use it to draw conclusions on the true population parameter that is of interest to us: the average number of cars per family in the whole population, that is, the population mean.

The following Table summarizes:

1. the 6 possible samples than can be the result of a sampling process on this popu- lation,
2. the probability of being selected (all of them will have the same probability as we are assuming SRS)
3. the estimate value that would result from applying the sample average formula to the corresponding sample Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Sample 6 Elements {A, B} {A, C} {A, D} {B, C} {B, D} {C, D} Probability (^161616161616) Estimate 1.5 1.5 2.5 2 3 3

We summarize what are the possible values the statistic can take an what is the probability associated to each of them:

statistic value =

1. 5 p = (^13) 2 p = (^16)
2. 5 p = (^16) 3 p = (^13)

In this example, we have seen how the statistic can take dierent values (4 in this case) with dierent probabilities. Hence, the statistic is a random variable

It will be necessary to know their main properties and, specially, the probability distributions of the statistics that are more frequently used.

The main statistics (or estimators) that are studied are

• the sample mean,
• the sample variance, and
• the sample proportion.

In all cases, we will assume that a sample of size n has been obtained by means of a SRS. The elements of the sample will be denoted by

{x 1 , x 2 , · · · xn}