The Binomial Distribution

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Chapter 19

• The Binomial Distribution

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What is the binomial distribution?

• The binomial distribution is the probability
  distribution of the number of times a dichotomous
  event event E occurs in N attempts.
• The event is considered to either occur or not.
• Binomial distributions are not normal
  distributions.
• Binomial means 2 numbers.

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Examples of situations generating binomial
distributions

• Coin toss event heads
• Birth event female gender
• Two alternative forced choice for mood event
  happy
• Notice that there is an anti-event for each
  event.
• For example tails, male, not happy.

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Examples of situations generating binomial
distributions

• Either the event or the anti-event happens.
• There are no other possibilities.
• The first 2 examples are inherently dichotomous,
  whereas the third example is made binary by
  experimental design.
• If the probability of the event is P, then the
  probability of the anti-event is _______?

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Experiment

• Flip a coin 10 times.
• How many heads do you get?
• Now flip a coin 20 times.
• How many heads do you get.

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What happens as the number of trials increases?
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Explanation

• More observations means more opportunities for an
  event to occur.
• More observations means there are more possible
  outcomes in terms of the number of events.

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What happens if P ? .5?

• For example, if an event is rolling 1 on a die,
  then P 1/6.

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Explanation

• Small P means the event is less likely, therefore
  the expected number of times the event will occur
  is less.
• Symmetry is lost when P ? (1-P).

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ExampleCola taste test

• The binomial distribution can be used to test
  hypotheses.
• In a pilot study Pepsi tested 12 tasters to see
  if they preferred Pepsi to Coke.
• 9 of the 12 preferred Pepsi over Coke.
• Pepsi executives were convinced that their
  product was better.
• Word leaked out to Coca-Cola.
• The executives there were convinced that the
  results were due to sampling error.
• Who is correct?

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ExampleCola taste test

• Let us look at things from the Coke executives
  point of view.
• They are confident that Coke is at least as good
  as Pepsi.
• Thus, they make a null hypothesis that the
  probability P, that a person would pick Pepsi, is
  .5.
• Coke executives always keep a copy of Cohens
  book close by.
• So, they look in table A13 to find the
  probabilities for 9 or more tasters picking Pepsi
  when P.5.
• The sum of such probabilities is the probability
  that Pepsi has beaten Coke by chance.

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ExampleCola taste test

• Let X be the number of tasters who prefer Pepsi.
• When n 12 and P.5
• p( X9 ) .0537
• p( X10 ) .0161
• p( X11 ) .0029
• p( X12 ) .0002
• So the total probability on that tail of the
  distribution is
• p .0537 .0161 .0029 .0002 .0729
• So, with an ? .05, the probability is not
  significant.

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When the null distribution is not based on P.5

• The Coke executives were lucky that their null
  hypothesis was based on P .5.
• What if it is not?
• Fortunately, we can produce our own distribution
  using the general equation for the probability

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When the null distribution is not based on P.5

• When would P ? .5?
• Suppose a gambler is at a casino and notices that
  3 is rolled less than 1/6th of the time.
• In particular, 3 appears only once in 24 rolls.
• Is this a fair die?
• Left as an exercise.

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Approximating the binomial distribution using the
normal distribution

• Throughout history, statisticians have had an
  obsession with the normal distribution.
• So, naturally they would try to approximate the
  binomial distribution with the normal.
• If N, the number of trials, is large and P is
  close to .5, then the approximation is good.
• ? is approximately NP (why?).
• The standard deviation is
• And, z is then given by
• Or,

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Approximating the binomial distribution using the
normal distribution

• How good is the approximation?
• When P is near .5 and Ngt25, the error is small.
• If P is not near .5, NPQ should be at least 9.
• NPQ can be manipulated experimentally. How?

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Z test for proportions

• Suppose we dont have X but have a proportion
  instead.
• For example say 58 of voters sampled prefer
  candidate A and 42 favor candidate B.
• Our formula for z then becomes

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Z test for proportionsExample

• For example say 58 of voters sampled prefer
  candidate A and 42 favor candidate B.
• 200 voters were surveyed.
• Our formula for z then becomes

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Z test for proportionsExample

• Plugging in the numbers from the survey
• Which is larger than either the 1 or 2 tailed
  zcrit.

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Adjusting the z approximation for bias

• The z distribution ( the approximating
  distribution ) is continuous and the binomial
  distribution is discrete.
• This causes a bias which is corrected by adding a
  correction factor of .5 in the numerator.
• The bias is greatest when N is small, so apply
  the corrected formula in this case.

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General rules for testing hypotheses with the
binomial distribution

• There are 3 approaches.
• You can always use the formula for probabilities
  in the binomial distribution.
• This formula can be used to custom make any
  distribution, distribution table (like A13) or
  part thereof.
• If P .5, you may be able to use table A13.
• If N is large enough and P is close enough to .5,
  you may use a normal approximation.

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One tailed vs. two tailed tests

• http//onlinestatbook.com/chapter9/tails.html
• Note the binomial calculator.

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Exercises

• Page 619
• 1, 2,
• Answer the question about the casino,
• 6, 8

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SPSS

• Suppose a teacher knows from past experience that
  children are equally likely to leave the class
  room from any of 4 doors.
• Suppose she hypothesizes that oppositional
  children will more likely leave by the door in
  the left rear of the room.
• P for children leaving by the left rear door is
  .25 under the null hypothesis.
• Suppose she collects the following data with a
  class of 80 oppositional students.

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SPSS

• Data-gtWeight cases
• Select Weight cases by.
• Put Number of oppositional children in box.
• OK.
• Analyze-gtNonparametric Tests-gtBinomial Test.
• Set Test Proportion to .75, which is the null
  hypothesized portion for the first door listed.
• If the order of the rows was reversed we would
  enter .25.
• Put door in the Test Variable List.
• OK.

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SPSS

• Significance of your test is given in the Asymp.
  Sig. column.
• Notice that it is a 1 tailed test.
• You can convert this to a 2 tailed test by
  multiplying by 2.
• However, due to skewing, this only works if the
  distribution is symmetric.
• The distribution is symmetric if
• P .5

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SPSS

• What if the data is not condensed?
• Open George grades.sav data.
• Suppose we want to know if the the number of
  female students is statistically greater than the
  number of male students.
• Each students gender is tabulated individually.
• Go directly to the analysis.
• Move sex to the Test Variable List.
• Set the test proportion to .5.
• OK
• You should find that p.031

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SPSS Exercises

• Page 626
• 3, 4, 7, 8