This paper presents likelihood-based statistical tests for high-energy physics experiments, focusing on methods to account for systematic uncertainties. It derives explicit formulae for the asymptotic distributions of test statistics using results from Wilks and Wald. The paper introduces the "Asimov data set," a representative data set that provides a simple method to obtain the median experimental sensitivity and fluctuations around this expectation. The formalism outlines the statistical test procedure, defining statistical significance and sensitivity. Test statistics based on the profile likelihood ratio are defined, and their sampling distributions are derived using the approximations from Wilks and Wald. The paper discusses how to determine the median significance for assumed signal strengths and provides numerical implementation in the RooStats package. It also explores the limitations and additional aspects of the method, including the distribution of test statistics for discovery and upper limits. The paper extends previous work by providing more accurate formulas for exclusion significance and a quantitative measure of statistical fluctuations in discovery significance and exclusion limits.This paper presents likelihood-based statistical tests for high-energy physics experiments, focusing on methods to account for systematic uncertainties. It derives explicit formulae for the asymptotic distributions of test statistics using results from Wilks and Wald. The paper introduces the "Asimov data set," a representative data set that provides a simple method to obtain the median experimental sensitivity and fluctuations around this expectation. The formalism outlines the statistical test procedure, defining statistical significance and sensitivity. Test statistics based on the profile likelihood ratio are defined, and their sampling distributions are derived using the approximations from Wilks and Wald. The paper discusses how to determine the median significance for assumed signal strengths and provides numerical implementation in the RooStats package. It also explores the limitations and additional aspects of the method, including the distribution of test statistics for discovery and upper limits. The paper extends previous work by providing more accurate formulas for exclusion significance and a quantitative measure of statistical fluctuations in discovery significance and exclusion limits.