One-Sample t-Test

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.

• AKA: t-Value Test, Single-Sample t-Test, One-Sample Location Test.
• Context:
  • It is based on an One-Sample t-Statistic.
  • It is described by an One-Sample t-Test Task and solved by an One-Sample t-Test System.
  • It is used when population mean is unknown. A Z-Test replaces an one-sample t-test when the population mean is known.
  • It is based on numerical data (i. e. interval data and ratio data) from a single sample drawn from a single population.
  • It is a parametric statistical test for which the population mean ([math]\displaystyle{ \mu }[/math]) and population variance ([math]\displaystyle{ \sigma^2 }[/math]) are parameter estimations based on the sample data.
  • It can range being One-Tailed Hypothesis Test to being a Two-Tailed Hypothesis Test.
• Example(s)
  • An One Tailed One-Sample t-Test.
  • A Two-Tailed One-Sample t-Test.
• Counter-Example(s):
  • Z-Value Test.
  • Welch's t-Test.
  • Linear Regression t-Test.
  • Two-Sample t-Test.
  • Matched-pair t-Test.
• See: Statistical Significance, Sample Average, Sample Variance.

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References

2017

• (Stat Trek, 2017) ⇒ http://stattrek.com/statistics/dictionary.aspx?definition=One-sample%20t-test
  • A one-sample t-test is used to test whether a population mean is significantly different from some hypothesized value.

    (...) The test statistic is a t statistic (t) defined by the following equation.

[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

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Student%27s_t-test#One-sample_t-test
  • In testing the null hypothesis that the population mean is equal to a specified value μ₀, one uses the statistic

[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. (The degree of approximation will depend on how close the parent population is to a normal distribution and the sample size, n.)

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