One-sample t-Statistic

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A One-sample t-Statistic is a t-statistic used in one-sample t-test to measure the difference between sample mean and the population mean.



References

2016


Assumptions:
The sample values are independent.
The sample values are all identically normally distributed (same mean and variance).

2014

where x is the sample mean, μ is the population mean, s is the standard deviation of the sample, and n is the sample size.

2007

  • (Goldsman, 2007) ⇒ David Goldsman (2007). Chapter 6 - Sampling Distributions, Course Notes: "ISyE 3770 - Probability and Statistics" [1], PDF file
    • QUOTE: Let [math]\displaystyle{ X \sim N(\mu, \sigma^2) }[/math]. Then [math]\displaystyle{ X \sim N(\mu, \sigma^2/n) }[/math] or, equivalently, [math]\displaystyle{ Z =(X − \mu)/(\sigma/\sqrt{n}) \sim N(0, 1) }[/math]. In most cases, the value of [math]\displaystyle{ \sigma^2 }[/math] is not available. Thus, we will use [math]\displaystyle{ S^2 }[/math] to estimate [math]\displaystyle{ \sigma^2 }[/math]. The t-distribution deals with the distribution about the statistic T defined by
[math]\displaystyle{ T =\frac{X-\mu}{S/\sqrt{n}} }[/math]
[...] Let [math]\displaystyle{ Z \sim N(0, 1) }[/math] and [math]\displaystyle{ W \sim \chi^{2\nu} }[/math] be two independent random variables. The random variable [math]\displaystyle{ T =Z/\sqrt{W/\nu} }[/math] is said to possess a t-distribution with [math]\displaystyle{ \nu }[/math] degrees of freedom and is denoted by [math]\displaystyle{ T \sim t_\nu }[/math]