Proportions

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Proportions
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First, some more probability!

• Why were there so many birthday pairs?

14 people 91 pairs
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Permutation
An ordered set of events
Girl, Boy
Boy, Girl
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Combination
An unordered set of events
Girl, Boy
or
Boy, Girl
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Permutations - Example

• If you have five children, three boys and two
  girls, how many possible birth orders are there?
• Example BGBGB

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Checking...
BBBGG BBGBG BBGGB BGBBG BGBGB BGGBB GBBBG GBBGB G
BGBB GGBBB
Answer 10 permutations One combination (three
boys, two girls)
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0!11!1
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Multiplication rule
If two events A and B are independent,
then PrA and B PrA x PrB
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Permutations and probabilities

• The multiplication rule also holds for
  permutations
• As long as the events are independent!

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PrBBBGGPrBPrBPrBPrGPrG 0.51
20.5120.5120.4880.488 (0.512)3(0.488)2
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Probability of a combination

• Consider the previous example
• What is the probability of the combination three
  boys, two girls?

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Probability of a combination
(Number of permutations giving that
combination) ? (Probability of one of those
permutations)
Assumes that events are independent
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Probability of a combination
10 ? 0.031
0.31
Assumes that events are independent
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Generalizing
? px (1-p)n-x
Assumes that events are independent
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The probability of a combination of independent
events
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The binomial distribution
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Binomial distribution

• The probability distribution for the number of
  successes in a fixed number of independent
  trials, when the probability of success is the
  same in each trial

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The binomial distribution
P(x) probability of a total of x successes p
probability of success in each trial n total
number of trials
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Sheep null distribution from computer simulation
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Binomial distribution, n 20, p 0.5
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Remember
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Example Paradise flycatchers
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Example Paradise flycatchers
A population of paradise flycatchers has 80
brown males And 20 white. You capture 5 male
flycatchers at random. What is the probability
that 3 of these are brown and 2 are white?
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Example Paradise flycatchers
A population of paradise flycatchers has 80
brown males And 20 white. You capture 5 male
flycatchers at random. What is the probability
that 3 of these are brown and 2 are white?
n 5 x 3 p 0.8
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Example Paradise flycatchers
A population of paradise flycatchers has 80
brown males And 20 white. You capture 5 male
flycatchers at random. What is the probability
that 3 of these are brown and 2 are white?
n 5 x 3 p 0.8
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Example Paradise flycatchers
A population of paradise flycatchers has 80
brown males And 20 white. You capture 5 male
flycatchers at random. What is the probability
that 3 of these are brown and 2 are white?
n 5 x 3 p 0.8
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Properties of the binomial distribution

• Mean
• Variance

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Proportion of successes in a sample
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Example A coin is flipped 8 times 5 were heads
Proportion of heads 5/8
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Properties of proportions

• Mean p
• Variance
• Standard error

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Sampling distribution of a proportion
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The law of large numbers
The greater the sample size, the closer an
estimate of a proportion is likely to be to its
true value.
Sample size
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Estimating a proportion
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Confidence interval for a proportion
where Z 1.96 for a 95 confidence interval
The Agresti-Couli method
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Example The daughters of radiologists
30 out of 87 offspring of radiologists are males,
and the rest female. What is the best estimate of
the proportion of sons among radiologists?
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Hypothesis testing on proportions
The binomial test
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Binomial test
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Example
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Hypotheses
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N 10, p0 0.2
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P 0.12
The is greater than the common a value of 0.05,
so we would not reject the null hypothesis.
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Summary