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Assuming the null hypothesis is true, the distribution of $$t$$ scores is symmetrical and unimodal, with a mean of $$0$$. This known distribution allows researchers to find the $$p$$-value associated with any computed $$t$$ score based on its degrees of freedom.

Distribution of $$t$$ Scores

The degrees of freedom ($$df$$) determine the exact shape of a $$t$$ distribution. For a one-sample $$t$$-test, the degrees of freedom are calculated as the sample size minus one ($$N - 1$$).

Degrees of Freedom in a One-Sample $$t$$-Test

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 $$t$$

Consider a $$t$$ score of $$1.50$$ based on a sample of $$25$$. The probability of a $$t$$ score at least this extreme is given by the proportion of $$t$$ scores in the distribution that are at least this extreme. Defining extreme as being far from zero in either direction, the $$p$$-value is the proportion of $$t$$ scores that are $$1.50$$ or above or that are $$-1.50$$ or below, which is $$.14$$.

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