  We describe a method based on the CL$_s$ approach to present results in
searches of new physics, under the condition that the relevant parameter space
is continuous. Our method relies on a class of test statistics developed for
non-nested hypotheses testing problems, denoted by $\Delta T$, which has a
Gaussian approximation to its parent distribution when the sample size is
large. This leads to a simple procedure of forming exclusion sets for the
parameters of interest, which we call the Gaussian CL$_s$ method. Our work
provides a self-contained mathematical proof for the Gaussian CL$_s$ method,
that explicitly outlines the required conditions. These conditions are milder
than that required by the Wilks' theorem to set confidence intervals (CIs). We
illustrate the Gaussian CL$_s$ method in an example of searching for a sterile
neutrino, where the CL$_s$ approach was rarely used before. We also compare
data analysis results produced by the Gaussian CL$_s$ method and various CI
methods to showcase their differences.
