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What is a One-Sample T-Test?

A one-sample t-test is a parametric statistical test used to determine whether the mean of a single sample differs significantly from a known or hypothesized population value. Unlike the independent or paired samples t-test, which compare two groups against each other, the one-sample t-test compares one group against a fixed reference point. This makes it one of the simplest yet most powerful tools in inferential statistics, frequently used in quality control, clinical research, psychology, and education.

When to Use a One-Sample T-Test

Use a one-sample t-test whenever you have a single group of continuous measurements and want to test whether the group's average is statistically different from a specific value. Common scenarios include:

• Quality control: A factory produces bolts that should weigh exactly 10 grams. You sample 30 bolts and test whether their mean weight differs from 10 g.
• Clinical research: A new drug is expected to lower blood pressure to a target of 120 mmHg. You measure 25 patients after treatment and test whether their mean differs from 120.
• Education: A class takes a standardized exam with a national average of 500. You test whether the class's mean score is significantly different from 500.
• Psychology: A scale is normed at a midpoint of 3.0. You survey a sample and test whether their mean attitude score deviates from 3.0.

Assumptions of the One-Sample T-Test

Before interpreting your results, verify that these assumptions are reasonably met:

How to Report One-Sample T-Test Results in APA Format

According to APA 7th edition guidelines, report the sample descriptives, t-statistic, degrees of freedom, p-value, effect size, and confidence interval. Here is a template:

Note: Report t-values to two decimal places, p-values to three decimal places (use p < .001 when below .001), and always include a measure of effect size such as Cohen's d.

Understanding Cohen's d for One-Sample Tests

In a one-sample t-test, Cohen's d is calculated as the absolute difference between the sample mean and the test value, divided by the sample standard deviation. It quantifies how many standard deviations the sample mean lies from the hypothesized value, making it independent of sample size.

One-Sample T-Test vs. Other Tests

Common Mistakes to Avoid

• Choosing the wrong test value: The test value (population mean) should come from theory, prior research, or a known standard -- not from the same data set you are testing.
• Ignoring non-normality in small samples: With fewer than 30 observations, skewed or heavy-tailed distributions can lead to incorrect p-values. Always check normality for small samples.
• Reporting p = .000: Statistical software sometimes displays p = .000. Report this as p < .001 because a p-value is never exactly zero.
• Ignoring effect size: A statistically significant result does not mean the difference is practically important. Always report and interpret Cohen's d alongside the p-value.
• Treating a non-significant result as proof of equality: Failing to reject the null hypothesis does not prove the sample mean equals the test value. It only indicates insufficient evidence of a difference at the chosen alpha level.