The binomial distribution is a probability distribution that summarizes the likelihood of a value taking one of two independent states and is determined by the number of trials and the probability of success in each trial.
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A binomial distribution is characterized by two parameters: $n$ (number of trials) and $p$ (probability of success).
The mean of a binomial distribution is given by $np$, where $n$ is the number of trials and $p$ is the probability of success.
The standard deviation for a binomial distribution can be calculated using $\sqrt{np(1-p)}$.
In hypothesis testing, the binomial distribution can be used to test claims about a population proportion when sample sizes are sufficiently large.
For constructing confidence intervals for a population proportion, the normal approximation to the binomial distribution is often used when both $np \geq 5$ and $n(1-p) \geq 5$.
Review Questions
What are the key parameters that define a binomial distribution?
How do you calculate the mean and standard deviation of a binomial distribution?
When can normal approximation be applied to a binomial distribution?
Related terms
Population Proportion: The fraction or percentage that represents part of a total population with a particular characteristic.
Hypothesis Testing: A statistical method used to make decisions about population parameters based on sample data.
Normal Approximation: An approach where the binomial distribution is approximated by the normal distribution under certain conditions, typically when sample size is large.