Formalizing the Concepts: Simple Random Sampling

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Formalizing the ConceptsSimple Random Sampling
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Purpose of sampling

• To study a sample of the population to acquire
  knowledge by observing the units selected
  typified by households, persons, institutions, or
  physical objects and making quantitative
  statements about the entire population

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Purpose of sampling

• Why sampling?
• Saves cost compared to full enumeration
• Easier to control quality of sample
• More timely results from sample data
• Measurement can be destructive

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Unit of analysis
Some concepts used in Sampling

• An object on which a measurement is taken
• Most common units of analysis are persons,
  households, farms, and economic establishments

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Target population or universe
Some concepts used in Sampling

• The complete collection of all the units of
  analysis to study.
• Examples population living in households in a
  country students in primary schools

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Sampling frame
Some concepts used in Sampling

• List of all the units of analysis whose
  characteristics are to be measured
• Comprehensive, non-overlapping and must not
  contain irrelevant elements
• Should be updated to ensure complete coverage
• Examples list of establishments census civil
  registration

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Parameter
Some concepts used in Sampling

• Quantity computed from all N values in a
  population set
• Typically, a descriptive measure of a population,
  such as mean, variance
• Poverty rate, average income, etc.
• Objective of sampling is to estimate parameters
  of a population

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Some concepts used in Sampling
Estimation

• Estimator - mathematical formula or function
  using sample results to produce an estimate for
  the entire population
• Estimate - numerical quantity computed from
  sample observations of a characteristic and
  intended to provide information about an unknown
  population value (parameter).
• Examples mean (average), total, proportion,
  ratio

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Some concepts used in Sampling
Unbiased estimator

• When the mean of individual sample estimates
  equals the population parameter, then the
  estimator is unbiased
• Formally, an estimator is unbiased if the
  expected value of the (sample) estimates is equal
  to the (population) parameter being estimated

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Random sampling

• Also known as scientific sampling or probability
  sampling
• Each unit has a non-zero and known probability of
  selection
• Mathematical theory is available to assess the
  sampling error (the error caused by observing a
  sample instead of the whole population).

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Random sampling techniques

• Single stage, equal probability sampling
• Simple Random Sampling (SRS)
• Systematic sampling with equal probability
• Stratified sampling
• Multi-stages sampling
• In real life those techniques are usually
  combined in various ways most sampling designs
  are complex

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Single stage, equal probability sampling
Random sampling techniques

• Random selection of n units from a population
  of N units, so that each unit has an equal
  probability of selection
• N (population ) ? n (sample)
• Probability of selection (sampling fraction) f
  n/N
• Is the most basic form of probability sampling
  and provides the theoretical basis for more
  complicated techniques

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Single stage, equal probability sampling
(continued)
Random sampling techniques

1. Simple Random Sampling. The investigator mixes up
   the whole target population before grabbing n
   units.
2. Systematic Random Sampling. The N units in the
   population are ranked 1 to N in some order (e.g.,
   alphabetic). To select a sample of n units,
   calculate the step k ( k N/n) and take a unit at
   random, from the 1st k units and then take every
   kth unit.

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Random sampling techniques
Single stage, equal probability sampling
(continued)

• Advantage
• self-weighting (simplifies the calculation of
  estimates and variances)
• Disadvantages
• Sample frame may not be available
• May entail high transportation costs

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Stratified sampling
Random sampling techniques

• The population is divided into mutually exclusive
  subgroups called strata.
• Then a random sample is selected from each
  stratum.

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Two-stage sampling
Random sampling techniques

• Units of analysis are divided into groups called
  Primary Sampling Units (PSUs)
• A sample of PSUs is selected first
• Then a sample of units is chosen in each of the
  selected PSUs

This technique can be generalized (multi-stage
sampling)
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Random sampling

• Estimates obtained from random samples can be
  accompanied by measures of the uncertainty
  associated with the estimate.
• The uncertainty is measured by the standard
  error. Confidence intervals around the estimate
  can be calculated taking advantage of the Central
  Limit Theorem.

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Central limit theorem

• The central limit theorem states that given a
  parameter with mean µ and variance s², the
  sampling distribution of the mean approaches a
  normal distribution with mean µ and variance s²/n
• This is true even when the distribution of the
  parameter is not normal.
• The normal distribution is widely used. Part of
  its appeal is that it is well behaved and
  mathematically tractable.

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Sample variance and standard error

• Variance of the sample mean of an SRS of n
  units for a population of size N
• e standard error
• Measure of sampling error. Depends on 3 factors
• ( 1 - n/N ) Finite Population Correction (fpc)
• n sample size
• Var(X) Population variance. Unknown, but can be
  estimated without bias by

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Proportions

• A proportion P (or prevalence) is equal to the
  mean of a dummy variable.
• In this case Var(P) P(1-P), and

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Confidence intervals

• It is not sufficient to simple report the sample
  proportion obtained by Mr Green in the sample
  survey, we also need to give an indication of how
  accurate the estimate is.
• Confidence intervals are used to indicate the
  accuracy of an estimate.
• In other words, instead of estimating the
  parameter of interest by a single value, an
  interval of likely estimates is given.

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Confidence intervals (continued)

• where
• ta 1.28 for confidence level a 80
• ta 1.64 for confidence level a 90
• ta 1.96 for confidence level a 95
• ta 2.58 for confidence level a 99

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Confidence intervals
In a sample of 1,000 electors, 280 of them (28
percent) say they will vote Green.
Standard error is 1.42 percent.
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Confidence intervals
In a sample of 1,000 electors, 280 of them (28
percent) say they will vote Green. Standard error
is 1.42 percent.
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• The required sample size n is determined by
• The variability of the parameter Var(X)
• But we dont know it!
• The maximum margin of error E we are willing to
  accept
• How confident we want to be in that the error of
  our estimation will not exceed that maximum
• For each confidence level a there is a
  coefficient ta
• The size of the population
• But this is not very important!

For a proportion