The contents of this dissertation can be divided into two parts. In the first part, after defining the random variable I_n, the number of times that two empirical distribution functions intersect, the distribution of I_n is derived. The mean and variance of I_n are obtained; and asymptotic expressions for these parameters are derived.

In the second part, a distribution-free test is formulated using the variable I_n. Then, using Monte Carlo methods, the power of this test is compared to three other tests: t-test, Smirnov, and Wilcoxon. The Monte Carlo simulation was carried out for certain normal and non-normal distributions with samples sizes of 10, 20, and 40.