### Abstract:

In many statistical applications an interval is needed that will contain the values of all J future observations with some preassigned probability. For example, suppose twenty rockets have been fired in a test program and three have failed. If two more test programs are to be conducted, an interval that will, with probability 1-α, contain the maximum number of failures in either of the two programs is called an a confidence level prediction interval. In this thesis a general procedure is given for predicting future observations when there is one unknown parameter and other conditions are satisfied. The normal and the gamma distributions are used as examples to illustrate the procedure in the continuous case. It is shown that Poisson random variables can be predicted using the negative multinomial distribution. Tables of negative multinomial probabilities are provided and approximation procedures are suggested. It is also shown that negative binomial random variables can be predicted using the multivariate beta negative binomial and binomial random variables can be predicted using the multivariate negative hypergeometric distribution. The prediction intervals given in this thesis can also be used for simultaneous hypothesis testing for the Poisson, negative binomial and binomial distributions.