Psychology 2113

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Psychology 2113

• Sampling Distributions

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Sampling Distributions Introduction

• You need sampling distributions to make
  inferences
• To get probabilities of statistics for decision
  making about parameters
• To get information necessary to estimate
  parameters
• Def A sampling distribution is a distribution of
  a statistic across all samples of a given size N
  drawn from a specified population.
• Every statistic has a sampling distribution

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Types of Distributions

• Population distribution
• Distribution of all possible scores, Xs
• Usually large, unobtainable, and hypothetical
• Parameters ? and ?2
• Unknown shape
• We want to infer to one of the parameters or to
  the distribution itself

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Types of Distributions, cont.

• Sample distribution
• Distribution of the N scores that we actually
  have, Xs
• Usually a manageable size, already obtained, and
  real
• In our real world
• Has known statistics like X and s2
• Known shape
• We want to infer from one of the statistics to a
  parameter

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Types of Distributions, cont.

• Sampling distribution
• Distribution of a statistic over all possible
  samples, for example, Xs
• Shows the variability of the statistic
• Theoretical
• Has parameters and usually a known shape
• Bridge infer from sample to population, from
  statistic to parameter
• Where we get the probabilities of the statistic
  so we can make decisions about the parameter

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Types of Distributions, cont.
Sampling Distribution
Population ? ?2
Real World
Sample X
s2
N59
? X ?2/N
88.07
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Sampling Distribution of X
Sampling Distribution

• Sampling distribution of X
• Purpose to obtain probabilities
• Has specific characteristics
• Mean ?X ?
• Variance ?2X ?2/N
• Shape is normal if
• Population is normal
• N is large (Central Limit Theorem)

? X ?2/N
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Sampling Distribution of X (Review)
Sampling Distribution

• Sampling distribution of X
• Purpose is ___________________
• Definition is _________________
• Has specific characteristics
• Mean ?X _____
• Variance ?2X ______
• Shape is _______ if
• Population is _________
• N is ______ (_____________ Theorem)

? X ?2/N
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Sampling Distribution of x Use of ZX

• IQ of deaf children
• What is the mean of this population distribution?
  Is it 100, like for the population of all IQ
  scores (? 100 and ?2 225)?
• What is the probability of getting X 88.07 or
  less if ? 100 (and ?2 225)?
• To get this probability, we need a new statistic,
  ZX (X-?)/?(?2/N)
• ZX (88.07-100)/ ?(225/59) -6.11
• p (X lt 88.07) p(z lt -6.11) lt .00003

Sampling Distribution
? X ?2/N
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Use of ZX

• IQ of deaf children
• So, what does this look like and how does it help
  us decide about ? 100? Is the mean of the IQ of
  deaf children 100?
• Because the probability of getting X 88.07 or
  less if ? 100 is so small, less than .00003, we
  reject the idea that ? 100
• It is very unlikely to get the data
• that led to X88.07 from a
• population with ? 100

Sampling Distribution
?100 X ?2/N
X88.07
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Other Sampling Distributions

• The sampling distribution of X is the first
  sampling distribution we learn, but it is not the
  only one (all statistics have sampling
  distributions)
• All sampling distributions have in common
• Purpose to obtain probabilities
• Definition the distribution of a statistic
• But each sampling distribution has specific
  characteristics like mean, variance, and shape

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Other Sampling Distributions, cont.

• Sampling distributions of s2 and s2
• Both have shapes that are positively skewed
• The mean of s2 is (N-1)/N?2, always smaller
  than ?2
• The mean of s2 is ?2

Population ? ?2
Real World
Sampling Distributions
Sample X s2
N59
s2 (N-1)/N ?2
positive skew

s2 ?2 positive skew
? X ?2/N
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Other Sampling Distributions, cont.

• Sampling distributions of r, s, and s
• r the mean is ? (rho) if ? 0, and shape is
  symmetric but not normal
• s and s neither has a mean equal to ?

Population ? ?2
Real World
Sampling Distributions
Sample X s2
N59
s Mean is not ?
s Mean is not ?
r
? symmetric
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Estimation

• You need sampling distributions to make
  inferences
• To get probabilities of statistics for decision
  making about parameters
• To get information necessary to estimate
  parameters
• Estimation is the calculation of an approximate
  value of a parameter
• Point estimation is the use of a statistic as a
  single value (point) to estimate a parameter
• Any statistic can be used to estimate any
  parameter
• Some statistics are good, and logical, estimates
  of particular parameters, such as X as an
  estimate of ?
• Unbiased estimate is one definition of good
  estimate

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Estimation Unbiased

• Unbiased estimate A statistic is an unbiased
  estimate of a parameter if the mean of its
  sampling distribution is equal to the parameter
  ?statistic desired parameter
• The following statistics are unbiased estimates
  of their corresponding parameters
• X is an unbiased estimate of ? because ?X ?
• s2 is an unbiased estimate of ?2 because ?s2 ?2
• r is an unbiased estimate of ? because ?r ? if
  ? 0
• Note that the statistic and parameter can change,
  but the definition of unbiased is ?statistic
  desired parameter

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Estimation Unbiased, cont.

• The following statistics are not unbiased
  estimates of their corresponding parameters (each
  is a biased estimate)
• s2 is a biased estimate of ?2 because ?s2 ? ?2
• s is a biased estimate of ? because ?s ? ?
• s is a biased estimate of ? because ?s ? ?