Chapter 7: Sampling Distribution of the Means

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Chapter 7 Sampling Distribution of the Means

• Examining the relationship between samples and
  populations

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Sampling Distributions Example

• Consider a small, imaginary population with 4
  values
• 10, 11, 12, 13
• Lets take a sample of size 2 from this
  population
• Furthermore, lets take all possible samples of
  size 2

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Sampling Distribution of the Mean

• A distribution whose scores are MEANS of samples
  drawn from some population
• A distribution of all possible sample means from
  samples of a given size drawn from some
  population
• Thus, for previous example, we should find the
  MEANS of all of the possible samples.

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Sampling Distributions and Probabilities

• What are the probabilities of each of the sample
  means?
• Does each sample mean have an equal probability
  of occurrence?

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The Shape of the Sampling Distribution of the Mean

• How does the shape of the sampling distribution
  compare to the shape of the population?

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Some Features of the Sampling Distribution

• It will approximate a normal curve even if the
  population you started with does NOT look normal
• Sampling distribution serves as a bridge between
  the sample and the population

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Third Property Sample Size and the Standard Error

• The larger the sample size, the smaller the
  standard error of the mean
• Or
• As n increases, the standard error of the mean
  decreases

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• See Fig 7.6

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Why?

• The larger the size of the sample taken from the
  population, the closer the sample mean will be to
  the population mean
• With large samples, you are more likely to obtain
  a distribution that accurately reflects the
  population

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Fourth Property Central Limit Theorem

• Small samples Shape of sampling distribution is
  less normal
• Large sample Shape of sampling distribution is
  more normal.

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Why?

• If you take a mean for a large sample, extreme
  values are diluted and the mean tends to be an
  intermediate value
• Therefore, all the means themselves will tend to
  cluster around an intermediate value.
• See Fig 7.5
• Thus, the Central Limit Theorem holds regardless
  of the shape of the parent population
• The population does not have to be normal for the
  Central Limit Theorem to hold

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Summary

• The sampling distribution of the mean is a
  hypothetical distribution of sample means that
  has four important properties that allow us to
  make inferences or generalizations.

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Properties (cont.)

• Third Property Sample Size and the Standard
  Error
• Fourth Property Central Limit Theorem