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I am fairly happy with the concept of confidence interval. In a very basic guise we have a certain level of confidence that a statistic, say the population mean of a random variable, lies in an interval: $$\bar{x}\pm\Delta x.$$

On the other hand, I understand how hypothesis testing works. One guise, more or less, assumes some properties of a random variable, we compute an associated $p$ value of a sample, and we compare $p$ with some significance level.

I understand that there is an alternative approach, which says based on the assumptions on the random variable, the probability that a sample falls into a certain critical region is small, and if we see this the 'chances' of the assumptions being correct AND the sample falling into the critical region is so small that we reject the null hypothesis assumptions.

For example, in a simple two sided test for a population mean, the critical region for a sample mean will be of the form, where $\mu_0$ is the assumed population mean, and $\Delta x$ based on $\mu_0$ not $\bar{x}$:

$$(-\infty,\mu_0-\Delta x)\cup (\mu_0+\Delta x,\infty).$$

Speaking at the moment about a two-sided test for a population mean, what I like to do instead is:

  • Let $H_0$ be the null hypothesis that $\mu=\mu_0$
  • Let $H_A$ be the alternative hypothesis that $\mu\neq \mu_0$
  • Assuming $H_0$, construct the interval $\mathcal{I}=(\mu_0- \Delta x,\mu_0+\Delta x)$ (for an appropriate $\Delta x$)
  • Take a sample yielding $\bar{x}$.
  • If $\bar{x}\in\mathcal{I}$, this is consistent with $H_0$, and so $H_0$ cannot be rejected
  • If $\bar{x}\not\in\mathcal{I}$, this is inconsistent with $H_0$, and so we reject $H_0$.

I know this is equivalent to using $p$ values and critical regions, the interval $\mathcal{I}$ is the complement of the critical region (save for a set of measure zero), and I think it is a better way to think about hypothesis testing.

Question: is there a commonly-used name for the interval $\mathcal{I}$?

I have in the past used "Confidence Interval" with inverted commas but I don't think this is appropriate.

Thanks for your help.

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  • $\begingroup$ Complement of critical region is often called acceptance region, 'inverting' which you get a confidence interval. $\endgroup$ Jun 12, 2019 at 10:55
  • $\begingroup$ @StubbornAtom any issue with using the word "Acceptance" in conjunction with "cannot reject"? $\endgroup$ Jun 12, 2019 at 11:11
  • $\begingroup$ Otherwise this is a good answer that I would accept. $\endgroup$ Jun 12, 2019 at 11:13

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