I'm trying to analyze a set of 15 paired samples. The elements of each pair are results of a measurement that produces a scalar value. The first measurement of the pair was done in presence of some factor (let's call it $A$), whereas the second was done in its absence. I'm looking for a method to infer whether $A$ has an impact on the results or not.

The only usable method that I know of is the paired sample t-test for mean values, with the null hypothesis $H_0: \mu_1 = \mu_2$. With $H_0$ formulated in this way, the test yields a meaningful result only if there exists a difference that could be attributed to $A$ (i.e. when $H_0$ is rejected). In my case though, it makes more sense to use a test that is meaningful if $A$ has no impact on the result. That is, my $H_0$ should be $\mu_1 \neq \mu_2$, or maybe $|\mu_1 - \mu_2| \leq \delta$, where $\delta$ is the absolute value of some sufficiently small difference.

Does anyone know how can I adapt the paired sample t-test to my needs, or what other methods can I use instead?