When conducting a one-sample $t$-test, what does the null hypothesis assume about the relationship between the true population mean ($\mu$) and the hypothetical comparison mean ($\mu_0$)?

In a one-sample t-test, the alternative hypothesis (μ ≠ μ₀) allows for the possibility that the true population mean is either higher or lower than the hypothetical comparison mean.

A researcher wants to determine if the average IQ of a group of chess players differs from the general population mean of 100. Match each statistical component with its corresponding description for this study.

A clinical researcher is investigating whether the average depression score of individuals who practice daily meditation differs from the known general population mean of 18. Arrange the following steps in the correct logical sequence to construct the statistical hypotheses for this research, starting with the establishment of the comparison standard.

A social psychologist wants to design a study to test whether the average self-reported loneliness score of university students who live in single-occupancy dorms differs from the national average loneliness score of 42 (on a standardized 0-80 scale). The researcher plans to use a one-sample $t$-test.

Which of the following correctly formulates the null and alternative hypotheses for this study?

A researcher is investigating whether the average sleep duration of psychology graduate students differs from the general population mean of 7 hours. Arrange the following logical steps in the correct order to construct and analyze the hypotheses for a one-sample $t$-test.

A clinical psychologist is investigating whether the average number of therapy sessions required for recovery at her clinic differs from the national average of 12 sessions. Match each component of the one-sample $t$-test hypotheses to its specific meaning or notation in this study.

A researcher uses a one-sample t-test to determine whether the average anxiety score of her sample differs from a known population average of 50. In this test, the null hypothesis predicts that any difference between her sample mean and the value of 50 is due to sampling error alone, rather than reflecting a true difference in the population mean.

In a one-sample $t$-test, what does the alternative hypothesis state regarding the population mean ($\mu$) and the hypothetical comparison mean ($\mu_0$)?

You are creating a research plan to investigate whether the average level of job satisfaction among remote workers ($\mu$) differs from the national pre-pandemic average of $7.2$ on a $10$-point scale ($\mu_0$). As you develop the statistical framework for this one-sample $t$-test, which set of hypotheses should you formulate to accurately represent this research question?

A researcher comparing a sample mean ($\mu$) to a hypothetical comparison mean ($\mu_0$) is criticized for assuming that the effect will only be an increase ($\mu > \mu_0$). To perform a more objective evaluation that remains open to any difference from the comparison standard, the researcher should instead define the _____ hypothesis as $\mu \neq \mu_0$.

A researcher is reviewing a draft of a study where a one-sample $t$-test is used to determine if the mean IQ of a specific community ($\mu$) differs from the national average ($\mu_0$) of $100$. The draft incorrectly lists the alternative hypothesis as $H_1: \mu = 100$. To properly evaluate the researcher's claim that a difference exists, the operator in this alternative hypothesis must be changed to _____.

A cognitive psychologist wants to test whether undergraduate students at their university sleep a different number of hours per night than the national student average of $7.4$ hours ($\mu_0$). The researcher plans to use a one-sample $t$-test. Apply the concepts of the null and alternative hypotheses to this study. State both hypotheses using proper notation and describe in words what each hypothesis predicts regarding the relationship between the university students' true population mean ($\mu$) and the national average.

Analyze the psychologist's proposed null and alternative hypotheses. Identify the errors in their formulation of hypotheses for a standard one-sample $t$-test, justify why these statements are incorrect based on the definition of a one-sample $t$-test, and write the correct null and alternative hypotheses using mathematical notation.

A researcher states that because they found their sample mean ($M = 7.1$) is numerically different from the hypothetical comparison mean ($\mu_0 = 7.0$), they can automatically reject the null hypothesis ($\mu = \mu_0$) without running a statistical test. Evaluate this statement. Explain whether the researcher's decision is justified and why a one-sample $t$-test is necessary.