In statistics, the t-statistic is a ratio of the departure of an estimated parameter from its notional value and its standard error. It is used in hypothesis testing. For example, it is used in determining the population mean from a sampling distribution of sample means if the population standard deviation is unknown.
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Definition
Let
where β0 is a non-random, known constant which may or may not match the actual unknown parameter value β, and
If
In the majority of models the estimator
In some models the distribution of t-statistic is different from normal, even asymptotically. For example, when a time series with unit root is regressed in the augmented Dickey–Fuller test, the test t-statistic will asymptotically have one of the Dickey–Fuller distributions (depending on the test setting).
Use
Most frequently, t statistics are used in Student's t-tests, a form of statistical hypothesis testing, and in the computation of certain confidence intervals.
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 sample parameters, and thus it can be used regardless of what these may be.
One can also divide a residual by the sample standard deviation:
to compute an estimate for the number of standard deviations a given sample is from the mean, as a sample version of a z-score, the z-score requiring the population parameters.
Prediction
Given a normal distribution
Solving for
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
History
The term "t-statistic" is abbreviated from "hypothesis test statistic", while "Student" was the pen name of William Sealy Gosset, who introduced the t-statistic and t-test in 1908, while working for the Guinness brewery in Dublin, Ireland.