One-Sample t-Test

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An One-Sample t-Test is a student's t-test based on a single sample and used to test whether a population mean differs from the observed sample mean.



References

2017

[math]\displaystyle{ t = \frac{(x - M )}{s \sqrt{n}} }[/math]
where [math]\displaystyle{ x }[/math] is the observed sample mean, [math]\displaystyle{ M }[/math] is the hypothesized population mean (from the null hypothesis), and [math]\displaystyle{ s }[/math] is the standard deviation of the sample.

2011

[math]\displaystyle{ t = \frac{\overline{x} - \mu_0}{s/\sqrt{n}} }[/math]
where [math]\displaystyle{ \overline{x} }[/math] is the sample mean, s is the sample standard deviation of the sample and n is the sample size. The degrees of freedom used in this test are n − 1. Although the parent population does not need to be normally distributed, the distribution of the population of sample means, [math]\displaystyle{ \overline {x} }[/math], is assumed to be normal. By the central limit theorem, if the sampling of the parent population is independent then the sample means will be approximately normal.[1] (The degree of approximation will depend on how close the parent population is to a normal distribution and the sample size, n.)


  1. George Box, William Hunter, and J. Stuart Hunter, Statistics for Experimenters, ISBN 978-0471093152, pp. 66–67.