Test statistics, critical value "What is the critical value of the test statistic?" Does this question even make sense?
From my knowledge test statistics is one thing and critical value is another thing. If test statistic is larger than critical value or smaller than negative of the test statistic we reject the null hypothesis at question.
Nevertheless I have that as my quiz question, and if it makes sense can someone explain it to me, perhaps give an example?
Thank you.
 A: In a one-sided hypothesis test, for example $H_0: \mu = 0$ and $H_1: \mu > 0$, one approach is to choose a significance level, for example $\alpha = 0.05$, and then define a test statistic $T(\mathbf{x})$ ie. a function of your observed data. You would then choose a critical value $t$ such that, if the null hypothesis were true, $Pr[T > t] = \alpha$.
So, talking about a critical value only makes sense if you have a predefined significance level $\alpha$.
In the above example typically you'd take $T(\mathbf{x}) = \mathbf{\bar{x}}$ ie. the mean of your observed data, and reject the null hypothesis if $\mathbf{\bar{x}} > t$.
In a two-sided test you would need two critical values $t_1$ and $t_2$ such that $Pr[T < t_1] = \alpha/2$ and $Pr[T > t_2] = \alpha/2$ and then reject the null if either $T < t_1$ or $T > t_2$. In some cases, for example, if your assumed distribution is symmetric around the mean, you may have $t_1 = -t_2$. I think this is what you're referring to, but it will not always be the case.
On the other hand, it's quite common to do away with using critical values and just make your own judgement by looking at $p = Pr[T > T(\mathbf{x})|H_0 \text{ is true} ]$.
A: The critical value is the value you choose as reasonably probable "enough". Often this will be 99% .
A: Let's say, for example, that the rejection region has the form $\{S>c\}$, where $S$ is the test statistic (random variable) and $c$ is the so-called critical value. If the observed value of S is greater than c, you reject the null hypothesis.
You want to have less than 5% chance to reject under the null hypothesis. One way to achieve this is to choose $c$ as a value such that $P(S>c) \leq 5\%$. For example, you can use the quantile of order 0.95 of $S$.
