Psychology 9

1
Psychology 9

• Quantitative Methods in Psychology
• Jack Wright
• Brown University
• Section 15

Note. These lecture materials are intended
solely for the private use of Brown University
students enrolled in Psychology 9, Spring
Semester, 2002-03. All other uses, including
duplication and redistribution, are unauthorized.

2
Agenda

• From binomials to the Normal Distribution
• Announcements
• Assignment Chapter 9

3
Illustrative problem 2

• The probability of a certain trait occurring in a
  population is .40. An investigator intends to
  draw a random sample of 20 people and count the
  number with the trait.
• What is the binomial distribution?

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Binomial Distribution

• For n 20, p .4, and r people with trait

Middle?
Probability
Spread?
Shape?
R ( with trait)
5
Defining mbinomial and sbinomial

• You flip a fair coin 2 times and count heads.
  What is the mean heads (r)?
• Our possible outcomes are TT TH HT HH
• Rs ( heads) for each are 0 1 1
  2
• What is the mean of the rs?
• S(r)/N (0 1 1 2)/4 4/4 1.0
• More simply mbinomial N p 2 .5 1

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Defining mbinomial and sbinomial

• What is the standard deviation of the heads
  (r)?
• Our possible outcomes are TT TH HT HH
• Rs ( heads) for each are 0 1 1
  2
• deviations from mean -1 0 0
  1
• sum of squares 2
• s sqrt(SS/N) sqrt(2/4) sqrt(.5) .71
• Or, more simply
• sbinomial ?(Npq) ?(2.5.5) .71

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Implications of mbinomial and sbinomial

• We now have measures of center and spread of
  binomial
• What about shape?
• Why do we care?
• suppose many binomials take on just one shape
• then, solve properties of this one shape
• then, use this knowledge to solve all binomial
  situations with this shape, no matter what
  mbinomial and sbinomial are

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p.5, n5, 10
height gets lower
Shape appears
area per bar gets smaller
Bars get thinner
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p.5, n20, 50
Shape better defined
10
p.5, n100, 500
Most of area in here
Lets focus on this part...
Approaches, but does not reach 0
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p.5, n100, 500 (truncated)
Shape becomes very well defined
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Effects of increasing n on binomial distributions

• 1. The measure r ( successes) evolves from a
  discrete to a continuous variable
• 2. Histogram evolves into continuous function
• each bar narrower and height lower
• probability in each bar gets smaller
• probability polygon becomes smooth curve
• 3. Tails extend infinitely in both directions
  approach but do not reach 0
• 4. Distribution becomes symmetric and bell
  shaped

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The normal approximation of the binomial

• Our binomial distributions appear to become
  normal
• But are they in fact normal?
• Method
• repeat, but superimpose normal (Gaussian)
  function
• two parameters (as always)
• mbinomial and sbinomial
• Why is this important?

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p.5, n5, 10 w/ Gaussian
Poor fit
Better fit
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p.5, n20, 50 w/ Gaussian
Good fit
Still better fit
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p.5, n100, 500 w/ Gaussian
Nearly perfect fit
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Boundary conditions

• So far, considered only p .5
• We know binomial distribution is skewed when p !
  .5, at least for small n
• What happens when n increases?
• Why important?
• So, consider extreme case

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p.1, n5, 10
Poor fit
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p.1, n20, 50
Better fit
20
p.1, n100, 500
Good fit, even when p .1
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Summary

• With n large, the binomial distribution becomes a
  continuous normal (Gaussian) function
• occurs even when p ! .5
• Therefore, can solve large binomial problems with
  only 3 pieces of information
• the middle or mbinomial
• the width of the bell around the middle or
  sbinomial
• the Gaussian shape

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Summary

• How large is large enough?
• rule of thumb when Np and Nq gt 10
• eg p .5, q .5, N must be about 20
• eg p .1, q .9, N must be about 100

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From binomials to sample sums

• So far, dealt only with binomials ( successes)
• Now examine how our normal approximation applies
  to other things
• Suppose you roll a die. What is the probability
  of getting each number?

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Probability Distribution

• We have 6 outcomes, each with p 1/6 .17

Probability
Outcome
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From binomials to sample sums

• Now suppose you roll the die twice and take the
  SUM or the mean of the outcomes? What will the
  distribution be?
• sum mean ways to get this result p
• 2 1 11 1/36 .027
• 3 1.5 12 21 .055
• 4 2 13 31 22 .083
• 5 2.5 23 32 14 41 .111
• 6 3 15 51 24 42 33 .138
• 7 3.5 16 61 25 52 34 43 .167
• 8 4 26 62 35 53 44 .138
• 9 4.5 36 63 45 54 .111
• 10 5 46 64 55 .083
• 11 5.5 56 65 .055
• 12 6 66 .027

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Probability distribution of sums

• We have 11 possible outcomes...

Probability
Sum
Mean
1 2 3 4 5
6
Outcome
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Extension to random samples

• For previous problem, could work out all possible
  combinations
• For bigger problems, this would be laborious
• Instead of all possible samples, imagine we take
  random samples.
• Example
• we roll the die 10 times, sum up the dots
• take the average number of dots
• repeat this many times
• draw a probability histogram
• try to fit a Gaussian function

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n2 rolls
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n3 rolls
30
n5 rolls
31
n10 rolls
32
n50 rolls
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Final extension

• In previous case, distribution was flat
• Now consider cases in which the population is
  an asymmetric or irregular distribution

Frequency
Value
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Final extension

• Using this population...
• Suppose we draw random samples of size n
• for each sample, compute the mean and record it
• repeat this 1000 times
• draw a probability histogram of the distribution
  of sample means
• try to fit a Gaussian function

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n1
36
n2
37
n3
38
n5
39
n10
40
n50
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Significance

• 1. Binomial situation provides account of how
  chance events combine
• Small n exact solutions through binomial formula
  and tables
• Large n Gaussian distribution provides good
  approximation, even when p ! .5
• For infinite n, the binomial distribution
  converges with the Gaussian function

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Significance

• 2. Many phenomena can be understood as the result
  of adding together several independent events
• eg multiple genes and other determinants of
  physical characteristics
• eg multiple genes and other determinants of
  cognitive capacities and behavioral traits

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Significance

• 3. Phenomena that are the result of adding
  independent random elements will often have an
  approximately normal distribution.
• This is known as the Central Limit Theorem.
• Central refers to fundamental
• Limit indicates that the normal distribution
  emerges as we reach the limit of the binomial
  (infinite n).

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Significance

• 4. For random samples from a population, the
  sample sum and mean, by definition, result from
  adding independent elements.
• Therefore, sample means necessarily satisfy the
  central limit theorem and will often be normally
  distributed
• Even when the parent population is not normally
  distributed.

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Significance

• 5. For these reasons, the normal or Gaussian
  distribution becomes an indispensable tool in
  studying a wide range of cases in which the
  Central Limit Theorem is satisfied.

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Significance

• The Gaussian or normal distribution is the most
  widely used in Statistics There are two major
  reasons for its dominance. The first is that
  the mathematics tend to be relatively simple, and
  the second is that often the random mechanism can
  be specified as approximately Gaussian due to the
  Central Limit Theorem.
• Chambers Hastie (1992)

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Significance

• Thus, you see, it is no accident that the normal
  distribution is the workhorse of inferential
  statistics. The assumption of normal population
  distributions or the use of the normal
  distribution as an approximation device is not as
  arbitrary as it sometimes appears this
  distribution is part of the very fabric of
  inferential statistics.
• William Hays (1963)