In this paper, we study the problems in the discrete fourier transform (DFT) test included in the National Institute of Standards and Technology (NIST) SP 800-22 released by the NIST, which is a collection of tests for evaluating both physical and pseudo-random number generators for cryptographic applications. The most crucial problem in the DFT test is that its reference distribution of the test statistic is not derived mathematically but rather numerically estimated; the DFT test for randomness is based on a pseudo-random number generator (PRNG). Therefore, the present DFT test should not be used unless the reference distribution is mathematically derived. Here, we prove that a power spectrum, which is a component of the test statistic, follows a chi-squared distribution with two degrees of freedom. Based on this fact, we propose a test, whose reference distribution of the test statistic is mathematically derived. Furthermore, the results of testing non-random sequences and several PRNGs showed that the proposed test is more reliable and definitely more sensitive than the present DFT test.