In a one-sample $$t$$-test, the null hypothesis assumes that the true population mean ($$\mu$$) is exactly equal to the hypothetical comparison mean ($$\mu_0$$), written as $$\mu = \mu_0$$. The alternative hypothesis states that the population mean differs from the hypothetical mean, denoted as $$\mu 
eq \mu_0$$.

Hypotheses in a One-Sample $$t$$-Test

To illustrate a one-sample $$t$$-test, suppose a researcher investigates whether students accurately estimate a cookie's calories, known to be $$250$$ (a hypothetical population mean $$\mu_0 = 250$$). The null hypothesis states the population mean estimate $$\mu$$ is $$250$$. From a sample of $$n = 10$$ students, the mean estimate ($$M$$) is $$212.00$$ with a standard deviation ($$SD$$) of $$39.17$$. Using the $$t$$-statistic formula, the score is calculated as $$t = \frac{212 - 250}{39.17 / \sqrt{10}} = -3.07$$.

Example of a One-Sample $$t$$-Test

The $$t$$ statistic for a one-sample $$t$$-test is calculated by subtracting the hypothetical population mean ($$\mu_0$$) from the sample mean ($$M$$), and dividing this difference by the sample standard deviation ($$SD$$) divided by the square root of the sample size ($$N$$). The formula is $$t = \frac{M - \mu_0}{\frac{SD}{\sqrt{N}}}$$.

One-Sample $$t$$-Test Formula

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

The one-sample $$t$$-test is a statistical procedure used to compare a sample mean ($$M$$) against a hypothetical population mean ($$\mu_0$$). It is used to evaluate whether the sample likely comes from a population with that specific mean.

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There are three primary variations of the $$t$$-test, each suited for slightly different research designs: the one-sample $$t$$-test, the dependent-samples $$t$$-test, and the independent-samples $$t$$-test.

Types of $$t$$-Tests

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