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

Critical Values of $t$

Example of Evaluating a $t$ Score

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

A researcher is applying the properties of the $t$ distribution to interpret results from a psychological study. Match each specific research observation or requirement to the theoretical property of the $t$ distribution (visualized in the image) that explains it.

Two researchers calculate a $t$ score of 2.25. Study A has a smaller sample size ($df = 5$) than Study B ($df = 50$). Using the provided image of the $t$ distribution as a guide, arrange the following steps to analyze why Study A results in a larger $p$-value than Study B.

A researcher evaluates two different experimental outcomes and concludes that a $t$ score of $t = 2.58$ and a $t$ score of $t = -2.58$ are equally likely to occur if the null hypothesis is true. This evaluation is logically sound because the distribution of $t$ scores is symmetrical and has a mean of $0$.

Imagine you are a software architect designing a new 'Null Hypothesis Visualization' module for a statistical application used in psychology research. You need to create a programmatic blueprint that correctly produces a theoretical $t$ distribution based on its known properties. Which set of design specifications must you synthesize to ensure the resulting model accurately reflects the behavior of $t$ scores?

Assuming the null hypothesis is true, match each term related to the properties of the $t$ distribution with its correct description or value.

Because the distribution of $t$ scores is known to be symmetrical and unimodal under the null hypothesis, researchers can use a computed $t$ score and its degrees of freedom to determine the associated _____.

A researcher runs a one-sample $t$-test and obtains $t = 2.85$. She argues that, because the $t$ distribution has a mean of $0$ when the null hypothesis is true, a score of 2.85 lies far in the upper tail of the distribution and is therefore associated with a small $p$-value. The researcher's reasoning correctly applies the known properties of the $t$ distribution under the null hypothesis.

A researcher computes a $t$ score of 2.10 in Study A ($df = 12$) and the same $t$ score of 2.10 in Study B ($df = 55$). Although the computed $t$ scores are identical, the associated $p$-values differ between the two studies because the exact shape of the $t$ distribution — and therefore the probability of obtaining any given $t$ score by chance — varies according to the _____.

A student claims: 'Since the $t$ distribution is symmetric around zero under the null hypothesis, I can always ignore the negative sign on any $t$ score and look up the $p$-value using only the absolute value.' Arrange the following steps in the correct order to fully evaluate whether this claim is justified.

Describe the characteristics of the distribution of $t$ scores under the assumption that the null hypothesis is true. Identify its shape, its unimodality or multimodality, its mean, and explain what this known distribution enables researchers to determine.

Based on your understanding of the distribution of $t$ scores under the null hypothesis, explain to the student why the degrees of freedom are necessary to find the correct $p$-value for their computed $t$ score.

A researcher conducts a psychological experiment and calculates a test statistic of $t = -3.10$. Explain how the researcher utilizes the symmetry and mean of the null hypothesis $t$ distribution to evaluate the probability of obtaining a score this extreme or more extreme.