How do I know which distribution to use

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How do I know which distribution to use?
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Three discrete distributions
Proportional
Poisson
Binomial
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Given a number of categories Probability
proportional to number of opportunities Days of
the week, months of the year
Proportional
Number of successes in n trials Have to know n, p
under the null hypothesis Punnett square, many
p0.5 examples
Binomial
Number of events in interval of space or time n
not fixed, not given p Car wrecks, flowers in a
field
Poisson
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Proportional
Binomial
Binomial with large n, small p converges to the
Poisson distribution
Poisson
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Examples name that distribution

• Asteroids hitting the moon per year
• Babies born at night vs. during the day
• Number of males in classes with 25 students
• Number of snails in 1x1 m quadrats
• Number of wins out of 50 in rock-paper-scissors

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Proportional
Generate expected values
Calculate ?2 test statistic
Binomial
Poisson
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Sample
Null hypothesis
Test statistic
Null distribution
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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Chi-squared goodness of fit test
Null hypothesis Data fit a particular Discrete
distribution
Sample
Calculate expected values
Chi-squared Test statistic

• Null distribution
• 2 With
• N-1-p d.f.

compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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The Normal Distribution
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Babies are normal
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Babies are normal
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Normal distribution
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Normal distribution

• A continuous probability distribution
• Describes a bell-shaped curve
• Good approximation for many biological variables

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Continuous Probability Distribution
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The normal distribution is very common in nature
Human body temperature
Human birth weight
Number of bristles on a Drosophila abdomen
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Discrete probability distribution
Continuous probability distribution
Normal
Poisson
Probability
Probability density
PrX2 0.22
PrX2 ?
Probability that X is EXACTLY 2 is very very small
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Discrete probability distribution
Continuous probability distribution
Normal
Poisson
Probability
Probability density
Pr1.5X2.5 Area under the curve
PrX2 0.22
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Discrete probability distribution
Continuous probability distribution
Normal
Poisson
Probability
Probability density
Pr1.5X2.5 0.06
PrX2 0.22
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Normal distribution
Probability density
x
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Normal distribution
Probability density
x
Area under the curve
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But dont worry about this for now!
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A normal distribution is fully described by its
mean and variance
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A normal distribution is symmetric around its mean
Probability Density
Y
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About 2/3 of random draws from a normal
distribution are within one standard deviation of
the mean
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About 95 of random draws from a normal
distribution are within two standard deviations
of the mean
(Really, its 1.96 SD.)
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Properties of a Normal Distribution

• Fully described by its mean and variance
• Symmetric around its mean
• Mean median mode
• 2/3 of randomly-drawn observations fall between
  ?-? and ??
• 95 of randomly-drawn observations fall between
  ?-2? and ?2?

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Standard normal distribution

• A normal distribution with
• Mean of zero. (m 0)
• Standard deviation of one. (s 1)

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Standard normal table

• Gives the probability of getting a random draw
  from a standard normal distribution greater than
  a given value

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Standard normal table Z 1.96
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Standard normal table Z 1.96
PrZgt1.960.025
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Normal Rules

• PrX lt x 1- PrX gt x

PrXltx PrXgtx1
1
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Standard normal is symmetric, so...

• PrX gt x PrX lt -x

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Normal Rules

• PrX gt x PrX lt -x
• PrX lt x 1- PrX gt x

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Sample standard normal calculations

• PrZ gt 1.09
• PrZ lt -1.09
• PrZ gt -1.75
• Pr0.34 lt Z lt 2.52
• Pr-1.00 lt Z lt 1.00

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What about other normal distributions?

• All normal distributions are shaped alike, just
  with different means and variances

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What about other normal distributions?

• All normal distributions are shaped alike, just
  with different means and variances
• Any normal distribution can be converted to a
  standard normal distribution, by

Z-score
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Z tells us how many standard deviations Y is from
the mean
The probability of getting a value greater than Y
is the same as the probability of getting a value
greater than Z from a standard normal
distribution.
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Z tells us how many standard deviations Y is from
the mean
PrZ gt z PrY gt y
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Example British spies
MI5 says a man has to be shorter than 180.3 cm
tall to be a spy. Mean height of British men is
177.0cm, with standard deviation 7.1cm, with a
normal distribution. What proportion of British
men are excluded from a career as a spy by this
height criteria?
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Draw a rough sketch of the question
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m 177.0cm s 7.1cm y 180.3 PrY gt y
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Part of the standard normal table
PrZ gt 0.46 0.32276, so PrY gt 180.3
0.32276
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Sample problem

• MI5 says a woman has to be shorter than 172.7 cm
  tall to be a spy.
• The mean height of women in Britain is 163.3 cm,
  with standard deviation 6.4 cm. What fraction of
  women meet the height standard for application to
  MI5?