High-dimensional data often pose challenges in comparing mean values of two sets with different covariance matrices. A new L2-norm-based test was developed to address this issue, showing that the test statistic and a chi-square-type mixture have similar distributions under certain conditions. This allows for approximating the null distribution of the test using the chi-square-type mixture, leading to a more reliable test called a normal reference test. The test's power was proven through simulations and real data analysis, demonstrating its effectiveness compared to other methods.