Either Confidence Interval (CI) or Null Hypothesis Significance Test (NHST) has the same business, to advise whether some sample X\equiv\left(X_{1},X_{2},\dots,X_{n}\right) is or is not disliked by some hypothesized parameter \vartheta.

NHST.com manages a database. For each Miss \vartheta, NHST spies out all she dislikes. Mr *X* logs in NHST.com and inputs a girl name and his credit card number, to bet his luck whispering-- *Does she dislike me?
*

CI.com manages a database too. For each Mr

*X*, CI only needs his credit card with his name

*X*on it, then serves him a full list of available girls.

NHST.com has been historically monopolizing the market. Nevertheless, somebody prefer visiting CI.com and find that the two may share database in most cases.

Not-reject Region of \vartheta is defined as A\left(\vartheta\right)=\left\{ x:\vartheta\; doesn't\;dislike\;x\right\} .

Confidence Region of

xis defined as S\left(x\right)\equiv\left\{ \vartheta:\vartheta\; doesn't\;dislike\;x\right\} .\theta\in S\left(X\right)\Leftrightarrow \theta\,does\,not\,dislike\,X \Leftrightarrow\,X\in\,A\left(\theta\right)

So, Pr_{\vartheta}\left(\vartheta\in S\left(X\right)\right)\ge1-\alpha,\forall\vartheta\Longleftrightarrow Pr_{\vartheta}\left(X\notin A\left(\vartheta\right)\right)\le\alpha,\forall\vartheta