Statistical Sciences 2244A/B Chapter Notes - Chapter 12: Random Variable, Binomial Distribution, Simple Random Sample
22 May 2018
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Stats 2244
Chapter 12
CHAPTER 12.1
The Binomial Setting and Binomial Distributions
- The distribution of a count depends on how the data is produced
- The Binomial Setting
1. There are a fixed number n of observations
2. The n observations are all independent. That is, knowing the result of one observation
does not change the probabilities we assign to other observations
3. Each observation falls into one of just two categories, which for convenience we call
success and failure
4. The probability of a success, call it p, is the same for each observation
- Example: Think of an obstetrician overseeing n single-birth deliveries as an example of the
binomial setting
o Each single-birth delivery is either a baby girl or a baby boy
o Knowing the outcome of one birth doesnt change the probability of a girl on any other
birth, so the births are independent
o If we call a girl a success, then p is the probability of a girl and remains the same over
all the births overseen by that obstetrician
o The number of girls we count is a discrete random variable, X
▪ Discrete random variable: a random variable that has a countable (typically
finite) list of possible outcomes
o The distribution of X is called a binomial distribution
- binomial distribution:
o the count X of successes in the binomial setting has the binomial distribution with
parameters n and p
o the parameter n is the number of observations,
o the parameter p is the probability of a success on any one observation
o the possible values of x are the whole numbers from 0 to n
o the binomial distributions are an important class of probability distributions
- note: pay attention to the binomial setting, bc not all counts have binomial distributions
- example:
- example:
VIDEO: binomial distributions in statistical sampling
- population proportion: the fraction of the population having the characteristic of interest
o ex: fraction of patients in hospital with type O blood
o ex: fraction of computers at a repair shop with failed mother boards
o ex: fraction of registered voters who actually voted in the 2008 presidential election
- if each individual in a population is equally likely to be sampled:
o P(randomly selected individual has characteristic of interest) = population proportion
having characteristic of interest
- So probability = population proportion
find more resources at oneclass.com
find more resources at oneclass.com
- Sampling distribution of a count
o Begin with a very large population with proportion p of successes
o We call that success the characteristic of interest
o So p is the proportion or percentage of the population with that characteristic
1. Take an SRS of size n
2. Count the # of individuals in the sample with the characteristic of interest
▪ X= count of successes in sample has approx. binomial distribution
o Why?
▪ Each individual either has the characteristic of interest or doesnt →these are
the 2 possible outcomes
▪ The sample n is fixed before we take our SRS
▪ The population is much larger than sample →this implies that the probability of
success is almost constant on each individual
• Ex: if you have a large bowl of candy and take a handful, no one will
notice but if you have a small bowl of candy and take a handful,
someone will notice
• So if we take a sample from a very large population, we will not sample
enough to change the probability of success
▪ Sample is simple random sample →Individuals are independent
▪ These 4 properties satisfy the binomial setting so we can say the counts of
successes from a random sample has an approx. binomial distribution
- Examples of sampling distribution of count
o Count of: number of plants that die after applying herbicide to 50 plants is approx.
binomial
o Number of Hispanics in a SRS of 1000 ppl is approx. binomial
o Number of parts that are defective in a random sample of 20 from a shipment of 14000
parts is an approx. binomial
o Number of students who carried an umbrella on a rainy day in an SRS of 150 students is
an approx. binomial
o Number of voters who factor a school bond levy in a random sample of 500 registered
voters is an approx. binomial
CHAPTER 12.2
Binomial Distributions in Statistical Sampling
- The binomial distributions are important in stats when we want to make inferences about the
proportion p of successes in a population
- Example:
- This shows that when the population is much larger than the sample, a cound of successes in an
SRS of size n has approx. the binomial distribution with n = sample size and p = proportion of
successes in the population
- Sampling distribution of a count:
o Choose an SRS of size n from a population with proportion p of successes
find more resources at oneclass.com
find more resources at oneclass.com
Document Summary
The distribution of a count depends on how the data is produced. The binomial setting: there are a fixed number n of observations, the n observations are all independent. That is, knowing the result of one observation does not change the probabilities we assign to other observations. Note: pay attention to the binomial setting, bc not all counts have binomial distributions example: example: Binomial distributions in statistical sampling proportion p of (cid:494)successes(cid:495) in a population. The binomial distributions are important in stats when we want to make inferences about the. This shows that when the population is much larger than the sample, a cound of successes in an. Srs of size n has approx. the binomial distribution with n = sample size and p = proportion of successes in the population. The pattern of this calculation works for any binomial probability. To use it, we must count the number of arrangements of k successes in n observations.
