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Critical values of $$t$$ are specific scores that mark the boundaries of the most extreme portions of the $$t$$ distribution for a given alpha level and degrees of freedom. For example, the two-tailed critical values when there are $$24$$ degrees of freedom and $$\alpha$$ is $$.05$$ are $$2.064$$ and $$-2.064$$. If a calculated $$t$$ score falls beyond a critical value in either direction, it is in the extreme tail of the distribution, leading to the rejection of the null hypothesis.

Critical Values of $$t$$

A two-tailed $$t$$-test evaluates whether a sample mean differs from a hypothetical population mean in either direction (greater than or less than). In this test, the extreme $$5\%$$ of the distribution is split into two critical regions of $$2.5\%$$ each, one at the positive end and one at the negative end of the $$t$$ distribution.

Two-Tailed $$t$$-Test

A one-tailed $$t$$-test evaluates differences in only one specific, pre-determined direction. It places the entire extreme $$5\%$$ of the probability distribution into a single tail. This test is only appropriate when researchers have a strong theoretical justification to expect the difference to occur in that particular direction prior to data collection.

One-Tailed $$t$$-Test

What do critical values of $$t$$ represent in a $$t$$ distribution?

In a hypothesis test, if a calculated $$t$$ score falls beyond the established critical values of $$t$$, the score is considered to be in the extreme tail of the distribution, which leads the researcher to reject the null hypothesis.

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