  The Full Bayesian Significance Test (FBST) for precise hypotheses was
presented by Pereira and Stern (1999) as a Bayesian alternative instead of the
traditional significance test based on p-value. The FBST uses the evidence in
favor of the null hypothesis ($H_0$) calculated as the complement of the
posterior probability of the highest posterior density region, which is tangent
to the set defined by $H_0$. An important practical issue for the
implementation of the FBST is the determination of how large the evidence must
be in order to decide for its rejection. In the Classical significance tests,
the most used measure for rejecting a hypothesis is p-value. It is known that
p-value decreases as sample size increases, so by setting a single significance
level, it usually leads $H_0$ rejection. In the FBST procedure, the evidence in
favor of $H_0$ exhibits the same behavior as the p-value when the sample size
increases. This suggests that the cut-off point to define the rejection of
$H_0$ in the FBST should be a sample size function. In this work, we focus on
the case of two-sided normal mean hypothesis testing and present a method to
find a cut-off value for the evidence in the FBST by minimizing the linear
combination of the type I error probability and the expected type II error
probability for a given sample size.
