Critical value (statistics)

In test theory, a critical value is the threshold value that separates the critical area from the non-rejection area. Further details are given in the Lemma Statistical Test .

example

Since the test statistic is a random variable , its realization can either be before or after the critical value. If it is after the critical value, the null hypothesis is rejected.

For a two- tailed one- sample t-test with a reject area

${\displaystyle A:=\left(-\infty ,-t_{1-{\frac {\alpha }{2}}}(n-1)\right)\cup \left(t_{1-{\frac {\alpha }{2}}}(n-1),\infty \right)}$.

the critical value is the - quantile of the t-distribution with degrees of freedom . The null hypothesis of this test is rejected if the critical value is less than the calculated test statistic . This is equivalent to saying that the p-value less than the significance level is ${\displaystyle t_{1-{\frac {\alpha }{2}}}(n-1)}$ ${\displaystyle (n-1)}$ ${\displaystyle \alpha }$

${\displaystyle p<\alpha \Leftrightarrow t>k}$.

The figure opposite shows the rejection area for a one-sided test. He surrenders through

${\displaystyle A:=\left(t_{1-\alpha ;n-1},\infty \right)}$