Sampling

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Sampling

• Population any well-defined set of units of
  analysis the group to which our theories apply
• Sample any subset of units collected in some
  manner from the population the data we use to
  test our theories

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Types of Samples

• Probability sample each element of the
  population has a known probability of being
  included in the sample
• Nonprobability sample - each element of the
  population has an unknown probability of being
  included in the sample

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Types of Nonprobability Samples

• Convenience sample elements are included
  because they are convenient or easy for the
  researcher to select
• Problem may not be representative of the
  population to which we want to generalize

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Famous Example of Convenience Sampling

• Literary Digest used automobile registration
  lists and telephone directories as sampling frame
  for presidential polls
• 1928 - 18 million postcards to accurately predict
  outcome of 1928 election (Hoover-R)
• 1932 20 million postcards to accurately predict
  1932 election (Roosevelt-D)

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Famous Example of Convenience Sampling

• Literary Digest used automobile registration
  lists and telephone directories as sampling frame
  for presidential polls
• 1928 - predicted Hoover-R
• 1932 predicted Roosevelt-D
• 1936 predicted Landon (R) 54
• What happened?

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Famous Example of Convenience Sampling

• Before 1936
• Upper class/Working Class more or less
  representative partisan distribution

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Famous Example of Convenience Sampling

• Before 1936
• Upper class/Working Class more or less
  representative partisan distribution
• 1936 and beyond
• Upper class disproportionately Republican
• Working class disproportionately Democrat

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Types of Nonprobability Samples

• Quota samples elements are chosen based on
  selected characteristics and the representation
  of these characteristics in the population
• Insures accurate representation of selected
  characteristics
• Elements with selected characteristics chosen in
  convenience fashion

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Famous Examples of Quota Samples

• 1936 George Gallup used quota sampling to
  accurately predict
• The (inaccurate) Literary Digest prediction
• The winner of the 1936 election
• 1948 quota sampling incorrectly predicts Dewey
  to defeat Truman

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Types of Probability Samples

• Simple random sample each element of the
  population has an equal chance of being selected
• Systematic sample elements selected from a list
  at predetermined intervals

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Types of Probability Samples

• Stratified sample elements in population are
  grouped into strata, and each strata is randomly
  sampled

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Types of Probability Samples

• Cluster sample elements are grouped into
  clusters, and sampling proceeds in two stages
• A random sample of clusters is chosen
• Elements within selected clusters are then
  randomly selected and aggregated to form final
  sample
• This is the sampling method used in many national
  surveys (e.g. clustersmetropolitan areas, zip
  codes, area codes)

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Sampling and Statistical Inference

• Frequency distributions
• Mean and Standard deviation
• Proportions
• Sampling distribution
• Standard error
• Sampling error
• Confidence interval

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Frequency Distributions

• For any variable X, a tabular or graphical
  display of the number of observations per
  value/category of that variable

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Mean and Standard Deviation

• For any collection of observations measured at
  the interval or ratio level
• Mean the simple average
• Standard deviation the average distance of
  each observation from the mean (intuitive
  definition)

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Mean and Standard Deviation

• Weekly Income Mean Deviation
• 100 100 200 -100
• 150 150 200 -50
• 200 200 200 0
• 300 300 200 100
• 250 250 200 50
• Mean 200 Standard Deviation 79.06

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Proportion

• The ratio of the number of observations taking a
  specific value, to the total number of
  observations
• P / N
• Example 4/10.40

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Sampling Distribution (of sample proportions)

• Population
• Draw Random Sample of Size N
• Calculate sample proportion
• Repeat until all possible random samples of size
  N are exhausted
• The resulting collecting of sample proportions is
  the sampling distribution of sample proportions

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Sampling Distribution of Sample Proportions

• Def A frequency distribution of all possible
  sample proportions for a given sample size (N)
• The mean of the sampling distribution will be
  equal to the population proportion.
  

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Standard Error

• How the sample proportions vary from sample to
  sample (i.e. within the sampling distribution) is
  expressed statistically by the value of the
  standard deviation of the sampling distribution.
• (Standard deviation the average distance of
  each observation from the mean)

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Standard Error, cont.

• The standard error for a sample proportion is
  equal to the square root of P(1-P) / N

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The Standard Error and the Margin of Error in
Surveys

• The standard error is a measure of sampling
  variability for a sample statistic
• If we know the standard error, we can calculate a
  margin of error for our sample proportion

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Calculating the Margin of Error

• First, we must choose a level of certainty
  (confidence level) usually 95
• The margin of error (with a 95 confidence level)
  is equal to 1.96SE

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Example Presidential Approval

• Newsweek Poll conducted by Princeton Survey
  Research Associates International. Sept. 29-30,
  2005. N1,004 adults nationwide. MoE 3..
• "Do you approve or disapprove of the way George
  W. Bush is handling his job as president? (40
  approve)

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Calculating the margin of error for a 40
approval rating

• The standard error for a sample proportion is
  equal to the square root of P(1-P) / N
• The margin of error (with a 95 confidence level)
  is equal to 1.96SE
• .40.60/1004.000239
• v.000239 .01546
• 1.96.01546.03

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Example of a Sampling Distribution
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Example of a Sampling Distribution
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Sample Size and Sampling Error