Data Handling Lecture 2 The normal distribution

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Data HandlingLecture 2The normal distribution

• Andrew Jackson

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Computer Lab Recap

• How do I know I can trust the user uploaded R
  packages?
• Good ones will have an accompanying peer reviewed
  published paper
• Always have to be aware of potential bugs
• Why did we pull random numbers from the binomial
  distribution?

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The statistical method

• Think about your data and what question you are
  asking.
• Form a hypothesis and null hypothesis
• Decide on the appropriate model
• Observe some data
• Can you reject the null hypothesis?

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1 - Think about our system

• Repeated coin tosses
• There is some probability of getting a headP(H)
• With a corresponding probability of getting a
  tailP(T) 1 P(H)

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2 - Form your Hypotheses

• Hypothesis is that the coin is unfair
• What is the null hypothesis H0?
• That the coin has 0.5 chance of heads and 0.5
  chance of tails
• i.e. the coin is fair

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Decide on the model

• You just have to know what this is going to be
• Comes with familiarity and exposure to lots of
  different kinds
• In this case its a Binomial distribution
• Observation Binom(p,Ntrials)

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Observe some data

• Toss coins like we did 6 times
• For each observation write down the number of
  heads
• Or
• Simulate some data using a computer
• u lt- rbinom(23,6,0.5)
• 4 3 1 2 3 4 4 3 4 5 3 2 3 3 6 5 5 1 5 4 4 2 3

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Accept or Reject H0?

• Say we get 5 heads
• What is the probability of getting 5 or more
  heads given the model?
• P(5) 0.094
• P(6) 0.016
• p 0.11 (1-tailed test)
• p 0.22 (2-tailed test)

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You can check this in R

• Use the binomial test (http//en.wikipedia.org/wik
  i/Binomial_test)
• In R, type ?binom.test to bring up the help files
• binom.test(x,n,p0.5, alternativec("two.sided",
  "less", "greater"), )
• For one-sidedbinom.test(5,6,p0.5,alternativegr
  eater)
• For two-sidedbinom.test(5,6,p0.5,alternativetw
  o.sided)

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The normal distribution

• Also known as the Gaussian distribution
• First introduced by Abraham de Moivre in 1733 to
  approximate the Binomial for very large Ntrials

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Characteristics

• Continuous
• unlike the discrete binomial
• Can take any value
• unlike binomial which must be positive integer
  less than Ntrials

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Where does it come from?

• Binomial from repeated trials
• yes/no, head/tail, present/absent, true/false
• All things being equal we would all be the same
  height
• But lots of reasons why we are not
• Some effects will make us taller, some shorter
• Sum these additive effects and you get a normal
  distribution!

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Demonstration

• Draw lots of uniform random numbers between -0.5
  and 0.5
• i.e. some additive some subtractive

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The normal appears

• Sum all these numbers
• Do this again and again

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The normal distribution

• Represents an addition of an infinite number of
  random additive errors that are uniformly
  distributed

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Getting a p-value
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Cumulative Distribution Function CDF

• Obtained by integration
• Say we 0bserve a value of 2
• CDF at 2 0.98
• One-sidedp-value 0.02
• Two-sidedp-value 0.04

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Populations and sampling

• Populations are often very large
• We cant measure everyone
• So
• We have to sample
• and hope that our samples bear some relation to
  the population!

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Sampling Error
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Comparison of Populations Based on Samples
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Summary

• Statistical method
• Form a hypothesis and appropriate null H0
• Pick an appropriate model
• Collect data test H0
• Normal distribution represents random additive
  error
• Sampling Error
• Estimating the population values