The purposes of the thesis were to review some of the existing methods for testing normality and to investigate the use of generated data combined with observed to test for normality. The approach to testing for normality is in contrast to the existing methods which are derived from observed data only. The test of normality proposed follows a characterization theorem by Bernstein (1941) and uses a test statistic D*, which is the average of the Hoeffding’s D-Statistic between linear combinations of the observed and generated data to test for normality. Overall, the proposed method showed considerable potential and achieved adequate power for many of the alternative distributions investigated. The simulation results revealed that the power of the test was comparable to some of the most commonly used methods of testing for normality. The test is performed with the use of a computer-based statistical package and in general takes a longer time to run than some of the existing methods of testing for normality.