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

2. Binomial Setting
The distribution of X depends on how the data are produced
There a fixed number n of observations
The n observations are all independent
Each observation falls into one of just two categories, which for convenience we call “Success” and “Failure”
The probability of a success, call it p is the same for each observation

3. 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

4. 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)

5. 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.

6. 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

7. 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)

8. Class Work #3
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.