Randomness provided by pseudo-random number generators is the one of the most vital parts of cryptographic applications. There are two gaps in the cryptographic randomness test procedures used to evaluate the degree of randomness. Firstly, although there are more accurate alternatives, the usual chi-square test is directly applied regardless of the predictive power of the tests. Secondly, although there are more than 100 cryptographic randomness tests available in the literature of cryptography, the statistical characteristics and accuracy of those hypothesis tests have not been figured out by an extensive simulation study. To conduct appropriate and reliable hypothesis tests, the main statistical characteristics of the tests should be studied. In this article, the usage of alternatives to the chi-square test, such as Anderson-Darling, Kolmogorov-Smirnov, and Jarque-Bera tests, in testing the cryptographic randomness is proposed to get better statistical properties. Also, the effects of type-I error, sensitivity, specificity, power, negative predictive value, and goodness-of-fit tests on the accuracy of recently proposed and existing cryptographic randomness tests are evaluated by an extensive Monte Carlo simulation study. The results are beneficial for practitioners wishing to choose the most appropriate cryptographic randomness test procedure and for the evaluation of accuracy of the cryptographic randomness tests in the detection of non-randomness for cryptographic applications.