Chapter 11 Sampling and Sampling Distributions

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Chapter 11Sampling and Sampling Distributions

• Aims of Sampling
• Basic Principles of Probability
• Types of Random Samples
• Sampling Distributions
• Sampling Distribution of the Mean
• Standard Error of the Mean
• The Central Limit Theorem

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Sampling

• Population A group that includes all the cases
  (individuals, objects, or groups) in which the
  researcher is interested.
• Sample A relatively small subset from a
  population.

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Notation
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Sampling

• Parameter A measure (for example, mean or
  standard deviation) used to describe a population
  distribution.
• Statistic A measure (for example, mean or
  standard deviation) used to describe a sample
  distribution.

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Sampling Parameter Statistic
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Probability Sampling

• Probability sampling A method of sampling that
  enables the researcher to specify for each case
  in the population the probability of its
  inclusion in the sample.

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Random Sampling

• Simple Random Sample A sample designed in such
  a way as to ensure that (1) every member of the
  population has an equal chance of being chosen
  and (2) every combination of N members has an
  equal chance of being chosen.
• This can be done using a computer, calculator, or
  a table of random numbers

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Population inferences can be made...
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...by selecting a representative sample from the
population
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Random Sampling

• Systematic random sampling A method of sampling
  in which every Kth member (K is a ration obtained
  by dividing the population size by the desired
  sample size) in the total population is chosen
  for inclusion in the sample after the first
  member of the sample is selected at random from
  among the first K members of the population.

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Systematic Random Sampling
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Stratified Random Sampling

• Stratified random sample A method of sampling
  obtained by (1) dividing the population into
  subgroups based on one or more variables central
  to our analysis and (2) then drawing a simple
  random sample from each of the subgroups

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Stratified Random Sampling

• Proportionate stratified sample The size of the
  sample selected from each subgroup is
  proportional to the size of that subgroup in the
  entire population.
• Disproportionate stratified sample The size of
  the sample selected from each subgroup is
  disproportional to the size of that subgroup in
  the population.

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Disproportionate Stratified Sample
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Sampling Distributions

• Sampling error The discrepancy between a sample
  estimate of a population parameter and the real
  population parameter.
• Sampling distribution A theoretical
  distribution of all possible sample values for
  the statistic in which we are interested.

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Sampling Distributions

• Sampling distribution of the mean A theoretical
  probability distribution of sample means that
  would be obtained by drawing from the population
  all possible samples of the same size.
• If we repeatedly drew samples from a population
  and calculated the sample means, those sample
  means would be normally distributed (as the
  number of samples drawn increases.) The next
  several slides demonstrate this.
• Standard error of the mean The standard
  deviation of the sampling distribution of the
  mean. It describes how much dispersion there is
  in the sampling distribution of the mean.

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Distribution of Sample Means with 21 Samples
10 8 6 4 2 0
S.D. 2.02 Mean of means 41.0 Number of Means
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Frequency
37 38 39 40 41 42 43 44 45 46
Sample Means
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Distribution of Sample Means with 96 Samples
14 12 10 8 6 4 2 0
S.D. 1.80 Mean of Means 41.12 Number of Means
96
Frequency
37 38 39 40 41 42 43 44 45 46
Sample Means
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Distribution of Sample Means with 170 Samples
30 20 10 0
S.D. 1.71 Mean of Means 41.12 Number of Means
170
Frequency
37 38 39 40 41 42 43 44 45 46
Sample Means
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The Central Limit Theorem

• If all possible random samples of size N are
  drawn from a population with mean ?y and a
  standard deviation , then as N becomes
  larger, the sampling distribution of sample means
  becomes approximately normal, with mean ?y and
  standard deviation .