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Chapter 13

# Chapter 13.odt

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Simon Fraser University

Statistics

STAT 101

Qian( Michelle) Zhou

Fall

Description

Chapter 13 – Binomial Distributions
The Binomial Setting and Binomial Distribution
• Distribution of a count depends on how data is produced
• The Binomial Setting
1. There are fixed number of “n” observations
◦ “count observations until the first success” is NOT FIXED, so it won't be binomial
distribution
2. The “n”observations are all independent; result of one observation does not change the
probabilities we assign to other observations
3. Each observations falls into one of just two categories, “success” and “failure”
4. Probability of success, let's call it p, is the same for each observation
• Ex) coin toss – knowing outcome of one toss doesn't change probability of a head on any other
toss – it's independent
◦ number of heads we count is a discrete random variable X
◦ distribution of X is called binomial distribution
◦ ex) P(H) = 0.5
▪ P(HH) = 0.5 x 0.5 = 0.25
▪ P(T) P(H) will happen = P(T) x P(H) = 0.5 x 0.5 = 0.25
• The count X of successes in a binomial setting has the binomial distribution w/ parameters n and
p
◦ n = number of trials
◦ p = probability of success on any trial
• Warning: not all counts have binomial distributions
Ex 13.1) Blood Types
Each child has probability of 0.25 of having type blood O. If a pair of parents have 5 children, the number
who have blood type O is the count X of successes in 5 independent observations w/ 0.25 probability of
success. So X has the binomial distribution, n = 5 and p = 0.25
Binomial Distributions in Statistical Sampling
Example 13.3) Choosing an SRS of Cds
Music inspector inspects SRS of 10 Cds from shipment of 10,000 Cds. Suppose 10% of the Cds in
shipment are defective. Count the number of X of bad CDS in sample.
This is not quite a binomial setting. Removing 1 CD changes the proportion of bad CDS remaining in
shipment.
• When population is much larger than the sample, a count of successes in SRS of size n has
approximately the binomial distribution with n equal to the sample size and p equal to the
proportion of successes in the population
• Sampling Distribution of a Count
◦ Choose an SRS of size n from a population w/ proportion p of successes. When population
much larger than sample, the count X of successes in sample has approx the binomial
distribution w/ parameters n and p
Ex 13.1) Opinion poll calls residential telephon numbers @ random, only 20% of calls reach a live
person. You watch random sampling machine make 15 calls. X is number that reach a live person.
This is binomial distribution. P = 0.25; n is fixed, so it's n = 15. X (success) = success of reaching
a live person. And each call is independent of another.
Binomial Probabilities
• formula for probability that a binomial random variable takes any value by adding probabilities
for the different ways of getting exactly that many successes in n observations Example 13.4) Inheriting blood type
Blood types of successive children born to the same parents are independent and have fix probabilities
that depend on genetic makeup of parents. Each child born to certain set of parents have 0.25 probability
of having “O”. If parents have 5 children

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