AP STATISTICS LESSON 8

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AP STATISTICSLESSON 8 1 ( DAY 2 )

• THE BINOMIAL DISTRIBUTION
• (BINOMIAL FORMULAS)

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Binomial FormulasExample 8.9page 446

• Each child born to a particular set of parents
  has probability 0.25 of having blood type O.
• If these parents have 5 children, what is the
  probability that exactly 2 of them have type O
  blood?

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ESSENTIAL QUESTION What are the binomial
formulas and how are they used to solve problems
that can be modeled in binomial settings?

• Objectives
• To define and use the binomial formulas.
• To derive and use binomial means and standard
  deviation

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Binomial Coefficient

• The number of ways of arranging k successes
  among n observations is given by the binomial
  coefficient
• for k 0,1,2..n.

nk
n!

k!(n k )!
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Vocabulary

• Factorial - ! The formula for binomial
  coefficients uses the factorial notation. For
  any positive whole number n, its factorial n! is
• n! n x (n-1) x ( n 2 ) x x 3 x 2 x1
• The notation is not related to the
• Fraction . A helpful way to remember its
• meaning is to read it as binomial coefficient
  n choose k.

nk
nk
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Binomial Probability

• If X has the binomial distribution with n
  observations and probability p of success on each
  observation, the possible values of X are 0, 1,
  2, 3, n. If k is any one of these values,
• P(X k ) pk(1 p)n - k

nk
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Example 8.10Defective SwitchesPage 448

• The number X of switches that fail inspection
  in Example 8.3 has approximately the binomial
  distribution with n 10 and
• p 0.1.
• Find the probability that no more than 1 fails.

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Mean and Standard Deviation of a Binomial Random
Variable

• If a count X has the binomial distribution
  with number of observations n and probability p,
  the mean and the standard deviation of X are
• µ np
• s v np(1 p )
• These short formulas are good only for
  binomial distributions. They cant be used for
  other discrete random variables.

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The Normal Approximation to Binomial Distributions

• The formula for binomial probabilities becomes
  awkward as the number of trials n increases.
• As the number of trials n gets larger, the
  binomial distribution gets close to normal
  distribution.
• When n is large, we can use normal probability
  calculations to approximate binomial
  probabilities.

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Example 8.12Attitudes Toward ShoppingPage 452
Are attitudes toward shopping changing?

• A recent survey asked a nationwide random
  sample of 2500 adults if they agreed or disagreed
  that I like buying new clothes, but shopping is
  often frustrating and time-consuming.
• p 60 and X 1520

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Normal Approximation for Binomial Distributions

• Suppose that a count X has the binomial
  distribution with n trials and success
  probability p. When n is large, the distribution
  of X ix approximately normal,
• N(np,vnp(1 p)).
• As a rule of thumb, we will use normal
  approximation when n and p satisfy
• np 10 and n(1-p) 10.