"Critical value at 5%" or "critical value at 95%": what is the right way to say it? I have a quite simple (and not particularly intriguing) question about the way to present a test. I was wondering what is the right expression (or more widespread) when I talk about critical values.
In particular, if for example the rejection region is defined by the 95 percentile, is it preferred to say that I am using the critical value of the, say, t "at 95%" or "at 5%"?
Is is correct in both ways? Is one "more correct" than the other?
Do I incur in some ambiguity if I do not specify further? Are there some conventions? Do they vary based on the distribution I am referring?
Thank you very much
 A: 
In particular, if for example the rejection region is defined by the 95 percentile

Consider three hypothesis tests for a parameter, with alternatives respectively:
$$H_{1a}: \theta < \theta_0\\
H_{1b}: \theta > \theta_0\\
H_{1c}: \theta \neq \theta_0\,,$$
all conducted with a significance level of $\alpha = 0.05$.
Further, assume that the test statistic is a simple function of $\hat \theta - \theta_0$ and has a symmetric distribution, so everything is 'nice and simple', but includes the $t$ test you mention as an example.
The 95th percentile comes up in computing only one of those three rejection regions.
The test is a test at the 5% level because that's what you specify when you choose the type I error rate.
If you're talking about the hypothesis test itself, the phrase "95%" need never be mentioned. If for some reason you need to refer to how you calculated the critical value, in one of the above cases you compute the $1-\alpha$ quantile of the distribution of the test statistic, and in the other cases you compute the $\alpha$ and the $1-\alpha/2$ quantiles. You can mention those if you need to for some reason but they're not part of how you describe the hypothesis test itself. You choose your rejection region to be of size $\alpha$.
[In any case, pedagogy aside, these days it would be odd to compute a critical value when packages give $p$ values as a matter of course. Why would one compute a critical value when the $p$ value allows a reader to conclude whether they would reject at any significance level they wish?]
If you read the wikipedia article on Statistical Hypothesis Testing, you'll see that (in spite of some poor wording), even for the case where the critical value is computed, the discussion is in terms of $\alpha$. This is no accident.
A: I generally think of a critical value as indicating ranges in the sampling distribution of the test statistic under the null hypothesis which we consider to be consistent with that null hypothesis. Hence I would say that -1.96 and 1.96 are critical values for the central 95% of the asymptotic distribution of the test statistic under $\mathcal{H}_0$. We would trust that anything falling beyond such a range, leading to a rejection of the null hypothesis, would be a false positive error with probability 1/20. 
In general, there are much more useful values to report than a critical value. Take, for instance, the 95% confidence interval. Rather than indicating an interval which is consistent with an idea (that idea being the null hypothesis), you get an interval which is consistent with that data. The null hypothesis can be reduced to a singular statistic (such as a 0 mean difference or a 1/1 odds ratio), and we simply look whether the CI is consistent with that point or not to make inference.
