One-Sample $t$-Test Formula

What is the primary purpose of a one-sample $t$-test?

A clinical psychologist can use a one-sample $t$-test to evaluate whether a sample of patients with a specific disorder has a different average anxiety score than the established national average for the general population.

A clinical psychologist wants to determine if the average sleep duration of 31 patients with a specific diagnosis differs from the general population mean of 7.0 hours. Match each statistical component of the one-sample $t$-test to its correct value or description from this study.

A neuropsychologist is using one-sample $t$-tests to compare cognitive scores from different patient groups to a known population mean. Assuming the sample mean ($M$) is higher than the population mean ($\mu_0$) in all groups, rank the following scenarios from the highest calculated $t$-value (top) to the lowest calculated $t$-value (bottom).

An experimental psychologist is critiquing a research report that mistakenly uses a one-sample $t$-test to compare the average test scores of two separate and independent student groups. The psychologist points out that this is an improper application of the statistic because a one-sample $t$-test is specifically designed to evaluate a single sample mean ($M$) against a(n) ______ mean ($\mu_0$).

A researcher is evaluating whether the average test score of a sample of students ($M = 82$) is significantly different from the district-wide average of $75$. Given that the researcher is comparing a single sample to a fixed value and lacks the population standard deviation, the most appropriate statistical procedure for this evaluation is a(n) ______.

What is the primary purpose of a one-sample $t$-test?

A researcher would use a one-sample $t$-test to determine whether the average memory score of a specific group of participants ($M$) significantly differs from a known national average score ($\mu_0$).

A social psychologist investigates whether the average level of 'community belonging' among students at a specific college differs from the national average score of 12.0. They survey a sample of 31 students and find a mean ($M$) score of 14.5 with a standard deviation ($s$) of 2.1. Match each numerical value from this study to its correct statistical role in a one-sample $t$-test.

A psychologist is testing whether the average score on a new 'Resilience Scale' for a group of trauma survivors differs from the established general population mean of 50. Arrange the following steps in the logical order required to conduct and interpret a one-sample $t$-test.

Hypotheses in a One-Sample $t$-Test

Example of a One-Sample $t$-Test

Define the one-sample $t$-test and identify the two specific values that are compared in this statistical procedure.

Based on the researcher's goal to compare the sample's performance to the national standard, explain why a one-sample $t$-test is the appropriate statistical procedure for this scenario. Be sure to identify what values in this scenario correspond to $M$ and $\mu_0$.

A developmental psychologist wants to evaluate if the average vocabulary size of 5-year-olds in a specific town differs from the known national average of 2,500 words. How should the psychologist use the sample mean ($M$) and hypothetical population mean ($\mu_0$) in a one-sample $t$-test to achieve this goal?