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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

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

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

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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