Sampling Distribution of the Mean

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Sampling Distribution of the Mean
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Samples and Sampling Error

• With inferential statistics, we use sample
  statistics (e.g., M) to estimate population
  parameters (e.g. µ).
• Different samples from the same population will
  produce different sample means

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Samples and Sampling Error

• Sampling Error The discrepancy (amount of
  error) between a sample statistic and its
  corresponding population parameter.
• An estimate of sampling error tells us how good
  of a job were doing estimating population
  parameters from sample statistics.

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The Distribution of Sample Means

• We can use the DSM to tell us how well a sample
  describes a population
• The Standard Error of the Mean (sM) tells us how
  much samples means (Ms) deviate around the
  population mean (µ).
• We can use this information to estimate the
  amount of sampling error to expect for a given
  population and to determine if a sample mean (M)
  is common or uncommon.

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The Distribution of Sample Means

• We will use the DSM to determine the exact
  probability of getting a certain M, from a normal
  population with a known µ and s.

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The Distribution of Sample Means

• The Distribution of Sample Means (or DSM) A
  collection of sample means for all possible
  random samples of a particular size (n) that can
  be obtained from a population.
• All possible samples are needed in order to
  properly compute the probabilities
• This is a distribution of a statistic (e.g.,
  sample means or Ms) not individual scores (Xs)

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The Distribution of Sample Means
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Constructing a sampling distribution
Population 2, 4, 6, 8
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Distribution of Sample Means

• Sample means pile up around population mean.
• Normal in shape.
• Can use to answer probability questions about
  sample means.

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Central Limit Theorem
For any population with a mean ? and standard
deviation s, the distribution of sample means for
sample size n will have a mean of ? and a
standard deviation of s/sqrt(n), and will
approach a normal distribution as n approaches
infinity.
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Central Limit Theorem

• Shape of sample mean distribution is normal if
• 1. Population from which they were selected is
  normal.
• 2. Number of scores in each sample is large (n
  gt 30)
• The mean of the sample mean distribution will be
  equal to ? and is called the expected value of M
  (?M).
• The standard deviation of the distribution of
  sample means is called the standard error of the
  mean (sM). The standard error measures the
  standard amount of difference between M and ?
  that is reasonable to expect by chance.

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Properties of the Standard Error of the Mean

• The Standard Error of the Mean (SEM or sM) is
  influenced by two things
• Sample size
• Variability in the population (s)

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Properties of the Standard Error of the Mean
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Using Sampling Distributions to test Hypotheses
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Determining Probabilities for Sample Means

• Z-score formula for samples

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Determining Probabilities for Sample Means
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Determining Probabilities for Sample Means
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Determining Probabilities for Sample Means
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Determining Probabilities for Sample Means

• p(z) 3.00 .0013
• We would conclude sample means greater than 86
  not very likely.

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Determining Probabilities for Sample Means

• p gt .05 common
• We can expect to get a sample mean of this size
  by chance.
• p lt .05 extreme
• It is unlikely that we would get a sample mean of
  this size by chance.