Transcript binomial distribution

Binomial Distributions
Section 8.1
The 4 “Commandments” of
Binomial Distributions
There are n trials.
 Each trial results in a
success or a failure.
 The probability of a success,
p, is constant from trial to
trial.
 The trials are independent.
-Knowing the result of one
observation tells you nothing
about the other observations.
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Sampling Distribution of a Count
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Choose an SRS of size n from a population with
proportion p of successes. When the
population is much larger than the sample, the
count X of successes in the sample has
approximately the binomial distribution with
parameters n and p.
Essentially, it is sometimes sufficient for
outcomes of an event to be “close enough” to
independent to use binomial calculations.
x  np
Key formulas
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x  np
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 x  np(1  p)
If data fits binomial setting, then random
variable X = number of successes is called a
binomial random variable.
And the probability distribution of X is called a
binomial distribution. We represent this
distribution as B(n,p).
P.D.F.
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Given a discrete random variable X, the
probability distribution function assigns a
probability to each value of X. The probabilities
must satisfy the rules for probabilities given in
Chapter 6…
Rules of Probability--Chapter 6
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Rule 1: 0 ≤ P(A) ≤ 1 for any event A.
Rule 2: P(S) = 1.
Rule 3: complement rule; for any event A,
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Rule 4: Addition rule:
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P(AC) = 1 – P(A)
P(A or B) = P(A) + P(B) – P(A and B)
Rule 5: Multiplication rule:
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P(A and B) = P(A)P(B|A)
C.D.F.
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Given a random variable X, the cumulative
distribution function of X calculates the sum
of the probabilities for 0, 1, 2, …, up to the
value X. That is, it calculates the probability of
obtaining at most X successes in n trials.
Calculator Tips
To determine P(X = x)
Use binompdf(n, p, x): where n is the number of
observations, p is the probability of success.
 To determine P(X ≤ x)
Use binomcdf(n, p, x): where n is the number of
observations, p is the probability of success.
 To determine P(X > x)
Use 1-binomcdf(n, p, x): where n is the number of
observations, p is the probability of success.
 To determine P(X < x)
Use binomcdf(n, p, x-1): where n is the number of
observations, p is the probability of success.
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Binomial Setting Example
A baseball pitcher throws 30 pitches in an inning.
The pitcher throws a strike 60% of the time.
A) Is this binomial setting? Let’s check!
1. Can each observation be categorized as a success or failure?
YES: Throwing a strike is a success, throwing a ball (not a strike) is a failure.
2. Are there a fixed number of observations?
YES: The pitcher throws 30 pitches.
3. Are all n of the observations independent?
YES: While it is possible that one pitch impacts another, it is still safe to assume
that they are independent.
4. Is the probability of success the same for each observation?
YES: While a pitcher may get tired as the game wears on, thus changing the
probability of throwing a strike, it is safe to assume that throughout a season,
the probability of throwing a strike is the same.
Binomial Setting Example (cont.)
B) How many strikes does the pitcher expect to
throw? np = (30)(0.6) = 18
C) What is the standard deviation?
  np(1  p)  (30)(0.6)(1  0.6)  2.6833
D) What is the probability that the pitcher throws
exactly 21 strikes in the inning?
binompdf(30, 0.6, 21) ≈ 0.0823
E) What is the probability that he throws 15 or
fewer strikes? binomcdf(30, 0.6, 15) ≈ 0.1754
F) What is the probability that he throws more
than 11 strikes? 1 – binomcdf(30, 0.6, 11) ≈ 0.9917
What if we are between values?
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Consider the pitcher scenario.
What is the probability that he throws between
12 and 20 strikes?
We can’t do binomcdf directly or 1-binomcdf
Try:
binomcdf(30, 0.6, 20) – binomcdf(30, 0.6, 11)
 =0.8154
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Normal Approximation to
the Binomial Distribution
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If X is a count having the binomial distribution with
parameters n and p, then when n is larger, X is
approximately N(np, np(1  p) ).
As a rule of thumb, we can use this approximation when
np ≥ 10 and n(1-p) ≥ 10.
Essentially, we can use this approximation if we expect at
least 10 successes and 10 failures.
The accuracy of the Normal Approximation improves as
the sample size increases
It is most accurate for any fixed n when p is close to ½
and least accurate when p is near 0 or 1 and the
distribution is skewed.
Normal Approximation Example
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Many local polls of public opinion use samples of
size 400 to 800. Consider a poll of 400 adults in
Atlanta that asks the question “Do you approve of
President Bush’s response to the World Trade
Center terrorists attacks in September 2001?”
Suppose we know that President Bush’s approval
rating on this issue nationally is 92% a week after
the incident.
What is the random variable X?
X = the number of polled people that approve of
Bush’s response
Normal Approx. Ex. Continued
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Is X binomial?
n = 400, approve = success & not = failure, if polled
separately should be independent, with random polling
probability should be same for each person polled
Calculate the binomial probability that at most 358 of the 400
adults in the Atlanta poll answer “Yes” to this question.
binomcdf(400, 0.92, 358) ≈ 0.0441
Find the expected number of people in the sample who
indicate approval. Find the standard deviation of X.
We expect  X  (400)(.92)  368 with   (400)(.92)(.08)  5.4259
Perform a normal approximation to the question above if
possible.
np=368≥10, n(1-p)=32≥10, normalcdf(0, 358, 368, 29.44 ) ≈
0.0327
X