What is a 'critical value' in statistics?

Question

Here's where I encountered this word:

The raw material needed for the manufacture of medicine has to be at least $97\%$ pure. A buyer analyzes the nullhypothesis, that the proportion is $\mu_0=97\%$, with the alternative hypothesis that the proportion is higher than $97\%$. He decides to buy the raw material if the nulhypothesis gets rejected with $\alpha = 0.05$. So if the calculated critical value is equal to $t_{\alpha} = 98 \%$, he'll only buy if he finds a proportion of $98\%$ or higher with his analysis. The risk that he buys a raw material with a proportion of $97\%$ (nullhypothesis is true) is $100 \times \alpha = 5 \%$

I don't really understand what is meant by 'critical value'

Answer 1

A critical value is the point (or points) on the scale of the test statistic beyond which we reject the null hypothesis, and is derived from the level of significance $\alpha$ of the test.

You may be used to doing hypothesis tests like this:

1. Calculate test statistics
2. Calculate p-value of test statistic.
3. Compare p-value to the significance level $\alpha$.

However, you can also do hypothesis tests in a slightly different way:

1. Calculate test statistic
2. Calculate critical value(s) based on the significance level $\alpha$.
3. Compare test statistic to critical value.

Basically, rather than mapping the test statistic onto the scale of the significance level with a p-value, we're mapping the significance level onto the scale of the test statistic with one or more critical values. The two methods are completely equivalent.

In the theoretical underpinnings, hypothesis tests are based on the notion of critical regions: the null hypothesis is rejected if the test statistic falls in the critical region. The critical values are the boundaries of the critical region. If the test is one-sided (like a $\chi^2$ test or a one-sided $t$-test) then there will be just one critical value, but in other cases (like a two-sided $t$-test) there will be two.