Finite normal mixture models are used in a wide range of applications. Hypothesis testing on the order of the normal mixture is an important yet unsolved problem. Existing procedures often lack a rigorous theoretical foundation. Many are also hard to implement numerically. In this article, we develop a new method to fill the void in this important area. An effective expectation-maximization (EM) test is invented for testing the null hypothesis of arbitrary order <italic>m</italic> <sub>0</sub> under a finite normal mixture model. For any positive integer <italic>m</italic> <sub>0</sub> ⩾ 2, the limiting distribution of the proposed test statistic is <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="uasa_a_695668_o_ilm0001.gif"/>. We also use a novel computer experiment to provide empirical formulas for the tuning parameter selection. The finite sample performance of the test is examined through simulation studies. Real-data examples are provided. The procedure has been implemented in R code. The <italic>p</italic>-values for testing the null order of <italic>m</italic> <sub>0</sub> = 2 or <italic>m</italic> <sub>0</sub> = 3 can be calculated with a single command. This article has supplementary materials available online.
