# Difference between revisions of "t-Statistic"

A t-Statistic is a test statistic used in a Student's t-Test.

## References

### 2016

• (Changing Minds, 2016) ⇒ http://changingminds.org/explanations/research/analysis/t-test.htm Retrieved 2016-10-16
• QUOTE: The t-test (or student's t-test) gives an indication of the separateness of two sets of measurements, and is thus used to check whether two sets of measures are essentially different (and usually that an experimental effect has been demonstrated). The typical way of doing this is with the null hypothesis that means of the two sets of measures are equal.
The t-test assumes:
• A normal distribution (parametric data)
• Underlying variances are equal (if not, use Welch's test)
It is used when there is random assignment and only two sets of measurement to compare.
There are two main types of t-test:
• Independent-measures t-test: when samples are not matched.
• Matched-pair t-test: When samples appear in pairs (eg. before-and-after).
A single-sample t-test compares a sample against a known figure, for example where measures of a manufactured item are compared against the required standard.

### 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 $X \sim N(\mu, \sigma^2)$. Then $X \sim N(\mu, \sigma^2/n)$ or, equivalently, $Z =(X − \mu)/(\sigma/\sqrt{n}) \sim N(0, 1)$. In most cases, the value of $\sigma^2$ is not available. Thus, we will use $S^2$ to estimate $\sigma^2$. The t-distribution deals with the distribution about the statistic T defined by
$T =\frac{X-\mu}{S/\sqrt{n}}$
[...] Let $Z \sim N(0, 1)$ and $W \sim \chi^{2\nu}$ be two independent random variables. The random variable $T =Z/\sqrt{W/\nu}$ is said to possess a t-distribution with $\nu$ degrees of freedom and is denoted by $T \sim t_\nu$