  Although there are many ideas for the formulations of statistical hypothesis
testing, we consider that the likelihood ratio test is the most reasonable and
orthodox. However, it is not handy, and thus, it is not usual in elementary
books. That is, the statistical hypothesis testing written in elementary books
is different from the likelihood ratio test. Thus, from the theoretical point
of view, we have the following question: "What is the statistical hypothesis
testing written in elementary books?" For example, we consider that even the
difference between "one sided test" and "two sided test" is not clear yet. In
this paper, we give an answer to this question. That is, we propose a new
formulation of statistical hypothesis testing, which is contrary to the
confidence interval methods. In other words, they are two sides of the same
coin. This will be done in quantum language (or, measurement theory), which is
characterized as the linguistic turn of the Copenhagen interpretation of
quantum mechanics, and also, a kind of system theory such that it is applicable
to both classical and quantum systems. Since quantum language is suited for
theoretical arguments, we believe that our results are essentially final as a
general theory.
