This paper deals with a new Bayesian approach to the standard one-sample $z$- and $t$- tests. More specifically, let $x_1,ldots,x_n$ be an independent random sample from a normal distribution with mean $mu$ and variance $sigma^2$. The goal is to test the null hypothesis $mathcal{H}_0: mu=mu_1$ against all possible alternatives. The approach is based on using the well-known formula of the Kullbak-Leibler divergence between two normal distributions (sampling and hypothesized distributions selected in an appropriate way). The change of the distance from a priori to a posteriori is compared through the relative belief ratio (a measure of evidence). Eliciting the prior, checking for prior-data conflict and bias are also considered. Many theoretical properties of the procedure have been developed. Besides its simplicity, and unlike the classical approach, the new approach possesses attractive and distinctive features such as giving evidence in favor of the null hypothesis. It also avoids several undesirable paradoxes, such as Lindleys paradox that may be encountered by some existing Bayesian methods. The use of the approach has been illustrated through several examples.
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