The t-statistic is used in a t-test to determine whether to support or reject the null hypothesis.
It is very similar to the z-score but with the difference that t-statistic is used when the sample size is small or the population standard deviation is unknown.
For example, the t-statistic is used in estimating the population mean from a sampling distribution of sample means if the population standard deviation is unknown.
However, when t-statistic is needed to test the hypothesis of the form H0: β = β0, then a non-zero β0 may be used.
is an ordinary least squares estimator in the classical linear regression model (that is, with normally distributed and homoscedastic error terms), and if the true value of the parameter β is equal to β0, then the sampling distribution of the t-statistic is the Student's t-distribution with (n − k) degrees of freedom, where n is the number of observations, and k is the number of regressors (including the intercept)[citation needed].
If the true value of the parameter β is equal to β0, and the quantity
The key property of the t statistic is that it is a pivotal quantity – while defined in terms of the sample mean, its sampling distribution does not depend on the population parameters, and thus it can be used regardless of what these may be.
with unknown mean and variance, the t-statistic of a future observation
yields the prediction distribution from which one may compute predictive confidence intervals – given a probability p, one may compute intervals such that 100p% of the time, the next observation
The term "t-statistic" is abbreviated from "hypothesis test statistic".
[1][citation needed] In statistics, the t-distribution was first derived as a posterior distribution in 1876 by Helmert[2][3][4] and Lüroth.
[8] However, the T-Distribution, also known as Student's T Distribution gets its name from William Sealy Gosset who was first to publish the result in English in his 1908 paper titled "The Probable Error of a Mean" (in Biometrika) using his pseudonym "Student"[9][10] because his employer preferred their staff to use pen names when publishing scientific papers instead of their real name, so he used the name "Student" to hide his identity.
[11] Gosset worked at the Guinness Brewery in Dublin, Ireland, and was interested in the problems of small samples – for example, the chemical properties of barley where sample sizes might be as few as 3.
Hence a second version of the etymology of the term Student is that Guinness did not want their competitors to know that they were using the t-test to determine the quality of raw material.