## 4.4 Main sampling techniques

### 4.4.1 Simple Random Sample

Simple Random Sampling (SRS) is the most basic sample selection method, being a reference for all other sampling plans. From a complete list of \(N\) elementary units of the population, each sampling unit is selected with equal probability, such that in each draw the elements have the same chance of being chosen. The need for a complete list of the population to carry out an SRS may eventually be a limitation in the application of this type of methodology, as in practice it is not always possible to obtain such a list. The sample plans A and B discussed in Examples 4.16, 4.19 and 4.20 are characterized as SRS.

#### SRS without replacement - SRSwo

If the drawn unit is removed from the population and a new draw is carried out, it is said that a *restricted* or *without replacement* SRS was carried out, indicated by SRSwo.

**Example 4.26 **(SRSwo) From a box with \(N\) cards numbered from \(1\) to \(N\), one is drawn at random, with probability \(1/N\). The drawn card is left out of the ballot box and a new draw takes place, where each of the remaining \(N-1\) cards has a \(1/(N-1)\) probability of being drawn. This procedure is repeated until all \(n\) desired draws are completed. This is an SRS process *without* replacement. \(\\\)

**Exercise 4.8 **Define the draw probabilities of Example 4.26 considering \(n=3\) draws. \(\\\)

**Exercise 4.9 **Calculate the draw probabilities of Example 4.26 considering \(n=3\) draws and \(N=10\). \(\\\)

**Example 4.27 **(Caixa Econômica Federal Mega-Sena) In R you can draw a sample without replacement to try your luck in the Caixa Econômica Federal game. Note that \(N=60\), \(n=6\).

```
set.seed(1234) # Fixing pseudo-random generation
sort(sample(1:60, size = 6, replace = F)) # Betting on the CEF Mega-Sena via SRSwo
```

`## [1] 16 22 28 37 44 58`

**Exercise 4.10 **Read the documentation of the functions used in the Example 4.27 by doing `?set.seed`

,

`?sort`

e `?sample`

.

#### SRS with replacement - SRSwi

If the selected unit has the chance to participate in the sample again, the procedure is called SRS *unrestricted* or *with replacement*, indicated by SRSwi.

**Example 4.28 **(SRSwi) From a box with \(N\) cards numbered from \(1\) to \(N\), one is drawn at random, with probability \(1/N\). The drawn card is placed back in the urn and a new draw takes place. This procedure is repeated until all \(n\) desired draws are completed. This is an SRS process *with* replacement. \(\\\)

**Exercise 4.11 **Define the draw probabilities of Example 4.28 considering \(n=3\) draws. \(\\\)

**Exercise 4.12 **Calculate the draw probabilities of Example 4.28 considering \(n=3\) draws with \(N=10\). \(\\\)

### 4.4.2 Systematic Sampling

Consider a population of \(N\) elementary units numbered from 1 to \(N\). To select a sample of \(n\) observations, one of the first \(k=N/n\) units, say \(a\), is randomly selected, with probability \(1/k\), systematically selecting the next observations every \(k\) units. Mathematically, \[a, a+k, a+2k, \ldots, a+(n-1)k.\]

**Example 4.29 **Let there be a population with \(N=100\), from which you want to take a systematic sample of size \(n=5\). In this case \(k=100/5=20\), then we randomly draw a number between 1 and 20 with probability \(1/20\). If the number drawn is \(a=4\), the systematic sample is then defined as \[4, 4+20, 4+2 \times 20, 4+3 \times 20, 4+4 \times 20 = 4,24,44,64,84.\]

`## [1] 20`

```
set.seed(1) # Fixing pseudo-random generation
(a <- sample(1:k, size = 1)) # Sample size 1 with probability 1/k
```

`## [1] 4`

```
## [1] 4
## [1] 24
## [1] 44
## [1] 64
## [1] 84
```

**Exercise 4.13 **Consider the code in Example 4.29.

(a) Run the code repeatedly removing the `set.seed(1)`

line. What do you observe?

(b) Redo the exercise considering other values of \(N\) and \(n\), such that \(n<N\).

### 4.4.3 Stratified Sampling

According (Bolfarine and Bussab 2005, 93),

Stratified sampling consists of dividing a population into groups (strata) according to some characteristic(s) known in the population under study, and from each of these strata samples are selected in convenient proportions.

The objective of this type of sampling is that the researcher can use pre-existing structures in order to improve inferences, reducing their variability.

### References

*Elementos de Amostragem*. Editora Blucher. https://www.blucher.com.br/livro/detalhes/elementos-de-amostragem-331.