Sampling variability

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Sampling variability

• Toward statistical inference

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Parameter vs. statistic

• Number that describes a sample
• Value is known after examining the sample
• Value changes from sample to sample

• Number that describes a population
• Ex mean, variance
• Learn about it by taking sample - statistical
  inference

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Sampling variability

• If you take a simple random sample from a
  population, every sample will likely be
  different.
• Thus, the value of a statistic varies with
  repeated random sampling.
• If we take many such samples, we notice an
  emerging pattern - the sampling distribution

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Sampling distribution

• Portrays how the statistic changes from sample to
  sample
• We can get an idea of it by
• Taking many random samples of the same size, n,
  from the same population
• Calculating the sample proportion for each sample
• Examining a histogram of all the sample
  proportions obtained.

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Estimating population proportion

• What proportion of the population meets a certain
  criterion?
• View this as people/objects marked with 1 (if
  meet criterion) and 0 (if dont meet criterion)
• Take sample, add up the 1s, divide by n -
  yields sample proportion

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Population proportion (real life examples)

• What percentage of California voters voted for
  Arnold?
• What percentage of items coming off the assembly
  line have a certain defect?
• What percentage of seeds (in an agricultural
  field trial) sprouted?

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Simulation

• To understand shape of sampling distribution, we
  can use simulation
• Use computer to take random draws from
  population, graph results
• For population of 0s and 1s, we can use the
  computer to generate from a binomial
  distribution, where n is sample size and p is
  population percentage meeting our criterion

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Mean of a statistic

• The mean is the center of the sampling
  distribution for the statistic.
• It is the average value of the statistic over
  many samples.
• If the mean of the sampling distribution is equal
  to the parameter of interest, the statistic is
  unbiased.

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Variability of a statistic

• What is the spread of the sampling distribution?
• Spread is smaller for larger sample sizes - more
  data helps you understand the parameter better
• Variability of a statistic is unaffected by the
  size of the population as long as the population
  is much bigger than the sample size, n.

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Goldenrod and the gall fly
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Cautions

• The sampling distribution only shows how the
  statistic varies due to chance.
• Sampling distribution doesnt show possible bias
  due to undercoverage, nonresponse, lack of
  realism, etc.

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