Abstract:

Binomial distributions widely exist in nature and human society, which is classified as discrete by probability theory. In theoretical studies in mathematical statistics, a random variable of binomial distribution B(n,p） is equivalent to the sum of a number of n independent and identical variables of Bernoulli distribution B(1,p）. Estimation and testing on parameter p of binomial distribution B(n,p） is therefore equivalent to that of Bernoulli distribution B(1,p）. Three corrections were made in this article, relevant to the calculation of two-tailed probability, and the construction of hypothesis test statistics. (1) Assume p_k(k=0,1,...,n) is the probability list of binomial distribution B(n,p）, and the probability by ascending order is given by p_(k). The two-tailed exact probability is equal to , given the value of the observed k. (2) When testing the difference between parameter p of B(n,p） against a given value p₀, the test statistic was corrected by , which asymptotically approaches to normal distribution N(p-p₀,1) under the condition of large samples. (3) When testing the difference between two parameters of binomial distributions B(n₁,p₁）and B(n₂,p₂）, the test statistic was corrected by , which asymptotically approaches to normal distribution N(p₁-p₂,1) under the condition of large samples. By the correction, the two-tailed probability has the exact value, and avoids the embarrassing situation of a probability exceeding one. Under either the null or alternative hypothesis conditions, the asymptotical normal distributions always have the variance at one, and therefore are more suitable to study the statistical power in testing the alternative hypothesis. Exact test on binomial distributions under the condition of small samples was also introduced, together with the comparison between exact and approximate tests. Probability theory underlying the corrections was provided. Comparison was made between the tests on parameter of Bernoulli distribution and mean of normal distribution. The general rule in determining the small probability and large sample was present as well. By doing so, the author wishes to provide the readers with a perspective picture on hypothesis testing and statistical inference, consisting of the core content of modern statistics. 

Key words: binomial distribution, normal distribution, hypothesis test, testing statistic, correction, testing power