Data analysts are often confronted with the problem of large dimensional data. In some of these situations, it is customary to reduce the dimensionality of the problem by usign principal component analysis and to perform statistical analysis of the data using the new variables (principal components). For example, the new variables are used in the area of classification. Chestnut and Floyd (1981) used the principal components as variables in identification of underwater targets. However, the statistical data analysis using the principal components is adhoc since the distributions of the test statistics based upon the principal components are complicated when the covariance matrix is unknown. Very little work was done in the literature on deriving the distributions of these test statistics even in the asymptotic case. In this paper, we derive the asymptotic distribution of the t statistic based upon the new variable (the most important principal component) instead of using any of the original variables. The above asymptotic distribution is shown to be Student's t distribution. The accuracy of the above approximation is studied by comparing the simulated values using the asymptotic expression with the standard Student's t table. It is found that the accuracy of the above approximation is sufficient for many practical situations.