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 $$.14$$.

Example of Evaluating a $$t$$ Score

Assuming the null hypothesis is true, which of the following best describes the distribution of $$t$$ scores?

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.

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A test statistic is a specialized numerical value computed from sample data specifically to aid in determining the $$p$$-value during hypothesis testing. It serves as an intermediate step to evaluate the probability of obtaining the observed results if the null hypothesis were true.

Test Statistic

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The $$t$$ distribution illustrates the spread of possible $$t$$ scores that researchers would expect to observe if the null hypothesis were entirely true. This distribution forms a unimodal and symmetrical curve centered precisely around a mean of $$0$$. By utilizing this known distribution curve, researchers can evaluate any computed $$t$$ score to find its exact $$p$$-value, which represents the proportion of expected scores that are at least as extreme as the observed sample result.

Distribution of t Scores

Degrees of freedom is a core statistical concept that dictates the precise mathematical shape of a test statistic's distribution when the null hypothesis is true. By determining this exact structural shape, the degrees of freedom make it possible for researchers to accurately calculate the probability, or $$p$$ value, associated with any given test score.

Degrees of Freedom

When evaluating the relationship between quantitative variables using Pearson's $$r$$, the appropriate statistical procedure is a hypothesis test of the correlation coefficient. This test shares the fundamental logic of other null hypothesis tests but specifically assesses whether a true, non-zero correlation exists within the broader population.

Hypothesis Test of the Correlation Coefficient

Distribution of $$t$$ Scores

During hypothesis testing, what is the term for the specialized numerical value computed from sample data that serves as an intermediate step to evaluate the probability of obtaining the observed results if the null hypothesis were true?

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