• A count X of successes has the binomial distribution B(n, p) in the binomial setting: there are n trials, all independent, each resulting in a success or a failure, and each having the same probability p of a success.
• The binomial distribution B(n, p) is a good approximation to the sampling distribution of the count of successes in an SRS of size n from a large population containing proportion p of successes. We will use this approximation when the population is at least 20 times larger than the sample.
• The sample proportion of successes is an estimator of the population proportion p. It does not have a binomial distribution, but we can do probability calculations about by restating them in terms of X.
• Binomial probabilities are most easily found by software. There is an exact formula that is practical for calculations when n is small. Table C contains binomial probabilities for some values of n and p. For large n, you can use the Normal approximation.
• The mean and standard deviation of a binomial count X and a sample proportion are
μX = np
The sample proportion is, therefore, an unbiased estimator of the population proportion p.
• The Normal approximation to the binomial distribution says that if X is a count having the distribution, then when n is large,
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We will use this approximation when and . It allows us to approximate probability calculations about X and using the Normal distribution.
• The continuity correction improves the accuracy of the Normal approximations.
• The exact binomial probability formula is
where the possible values of X are k = 0,1, . . . , n. The binomial probability formula uses the binomial coefficient
• Here the factorial n! is
for positive whole numbers n and 0! = 1. The binomial coefficient counts the number of ways of distributing k successes among n trials.
• A count X of successes has a Poisson distribution in the Poisson setting: the number of successes that occur in two nonoverlapping units of measure are independent; the probability that a success will occur in a unit of measure is the same for all units of equal size and is proportional to the size of the unit; the probability that more than one event occurs in a unit of measure is negligible for very small-sized units. In other words, the events occur one at a time.
• If X has the Poisson distribution with mean μ, then the standard deviation of X is , and the possible values of X are the whole numbers 0, 1, 2, 3, and so on.
• The Poisson probability that X takes any of these values is
Sums of independent Poisson random variables also have the Poisson distribution. For example, in a Poisson model with mean μ per unit of measure, the count of successes in a units is a Poisson random variable with mean aμ.