# Confidence Region and Not-reject Region

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