Transcript Aim: How do we use binominal distribution for sample counts?

Aim: How do we use binominal
distribution for sample counts?
Binomial Setting
• The distribution of X depends on how the data
are produced
1. There a fixed number n of observations
2. The n observations are all independent
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
Examples of a binomial setting
• Tossing a coin n times because each toss gives
you either heads or tails
• The outcome of successive tosses are
independent
Binomial Distribution
• The distribution of the count X of successes in
binomial setting is called the binomial
distribution with parameters n and p
• The parameter n is the number of
observations and p is the probability of a
success on any observation
• The possible values of X are the whole
numbers from 0 to n
• As an abbreviation, we say that X is B(n, p)
Example
• Genetics says that children receive genes from
their parents independently. Each child of a
particular pair of parents has probability 0.25 of
having type O blood.
• If these parents have 3 children, the number who
have type O blood is the count X of successes in 3
independent trials with probability 0.25 of a
success on each trial.
– So X (the distribution) has the B(3,0.25) distribution.
Binomial Distribution in Statistical
Sampling
• Binomial distributions are important in
statistics when we wish to make inferences
about the proportion p of “successes” in a
populations
Sampling distribution of a count
• A population contains proportion p of success.
If the population is much larger than the
sample, the count X of successes in an SRS of
size n has approximately the binomial
distribution B(n, p)
Class Work #3
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Toss a fair coin 20 times. Give the distribution of X, and the number of heads that you observe.
Genetics says that children receive genes from their parents independently. Suppose each child
of a particular pair of parents has probability 0.25 of having type O blood. If these parents have 4
children, what is the distribution of the number who have type O blood? Explain your answer.
A educational research team wanted to examine the relationship between faculty participation in
decision making and job satisfaction in Mongolian public universities. They are planning to
randomly select 300 faculty members from a list of 2500 faculty members in these universities.
The Job Descriptive Index (JDI) will be used to measure job satisfaction, and the Conway
Adaptation of the Alutto-Belasco Decisional Participation Scale will be used to measure decision
participation. Describe the population and the sample for this study. Can you determine the
response rate?
A study was designed to assess the impact of taxes on forestland usage in part of the Upper
Wabash River Watershed in Indiana.29 A survey was sent to 772 forest owners from this region
and 348 were returned. Consider the population, the sample, and the response rate for this
study. Describe these based on the information given and indicate any additional information
that you would need to give a complete answer.