SECTION 5.1 Summary

• A count of successes has the binomial distribution in the binomial setting: there are trials, all independent, each resulting in a success or a failure, and each having the same probability of a success.

• If has the binomial distribution with parameters and , the possible values of are the whole numbers . The binomial probability that takes any value is

  Binomial probabilities are most easily found by software. This formula is practical for calculations when is small. Table C contains binomial probabilities for some values of and .

• The binomial coefficient

  counts the number of ways successes can be arranged among observations. Here, the factorial is

  for positive whole numbers , and .

• The mean and standard deviation of a binomial count and a sample proportion are
  

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• The Normal approximation to the binomial distribution says that if is a count having the distribution, then when is large,

  We will use this approximation when and . It allows us to approximate probability calculations about and using the Normal distribution. The continuity correction improves the accuracy of the Normal approximations.

• A simple check for the adequacy of the binomial model is to compare the binomial-based standard deviation (or variance) with the observed count standard deviation (or variance). In addition, some software packages provide fitting of the binomial model on the observed histogram to assess compatibility.