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Systematic sampling

In this sampling of a population of units numbered in some order. To select a sample of units (being ) take a random unity among the first units, and thereafter take every -th unit. called the selection range.

This type of sampling has apparent advantages over simple random sampling, such as:

1. is easier and faster to obtain the sample. No big
2. succession of elements in the list is not represented, because of this sometimes systematic sampling may be more representative sample Simple random.
3. In practice it is easier llervarlo out and therefore are less exposed to selection errors committed by field researchers.
4. can be implemented without knowing in advance the size of the population.

The process for selecting a sample using this method begins with determining the value . This is important because if we take a very large value the sample is very small and if we take a very small sample size becomes larger.
In practice it must follow the following procedure to select the range of selection:

1. If is known, determine the sample size for the survey of approximately and then select as the entire .
2. If population size is unknown can not select the precise value of .

Estimators and their variances.

The inclusion probabilities are given by

and second order as

if and belong to the same sample and is zero otherwise.

Expression of unbiased estimators is the following, obtained estimator from Horvitz - Thompson:

• For Media :

• For Total:

• For Ratio:

Variance Estimation.

In practice to estimate the variances of the estimates for this method, you can use several methods such as interpenetrating samples , successive differences, etc. In this case we will consider the method of random people, which is used when the sample can be considered sufficiently random and the estimates of the variances as the expressions of simple random sampling.

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A random variable is said to follow a normal distribution with mean and variance if its density function is written in the form:

Properties:

Take
• values \u200b\u200baround.
• function is symmetric about .

• Strictly and strictly increasing if decreasing if .
• has a maximum.

• has turning points and .
• In has a horizontal asymptote.

Features:

1. Hope: Hope is distribution.
2. Variance: The variance of the distribution is .
3. moment generating function: The moment generating function is
   


origin and scale change: The variable is not affected by changes of origin and scale, ie if then has the following distribution