T - statistic - wikipedia T - statistic This article does not cite any sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. ( February 2011 ) It has been suggested that this article be merged into Studentization. ( Discuss ) Proposed since May 2015. In statistics, the t - statistic is the ratio of the departure of the estimated value of a parameter from its hypothesized value to its standard error. It is used in hypothesis testing. For example, it is used in estimating the population mean from a sampling distribution of sample means if the population standard deviation is unknown. Contents 1 Definition and features 2 Use 2.1 Prediction 3 History 4 Related concepts 5 See also 6 References 7 External links Definition and features Let β ^ ( displaystyle scriptstyle ( hat ( \beta ) ) ) be an estimator of parameter β in some statistical model. Then a t - statistic for this parameter is any quantity of the form t β ^ = β ^ − β 0 s. e. ( β ^ ) ( displaystyle t_ ( hat ( \beta ) ) = ( \frac ( ( hat ( \beta ) ) - \beta _ ( 0 ) ) ( mathrm ( s.e. ) ( ( hat ( \beta ) ) ) ) ) ) where β is a non-random, known constant which may or may not match the actual unknown parameter value β, and s. e. ( β ^ ) ( displaystyle s.e. ( ( hat ( \beta ) ) ) ) is the standard error of the estimator β ^ ( displaystyle scriptstyle ( hat ( \beta ) ) ) for β. By default, statistical packages report t - statistic with β = 0 ( these t - statistics are used to test the significance of corresponding regressor ). However, when t - statistic is needed to test the hypothesis of the form H : β = β, then a non-zero β may be used. If β ^ ( displaystyle scriptstyle ( hat ( \beta ) ) ) 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 parameter β is equal to β, 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 ). In the majority of models the estimator β ^ ( displaystyle scriptstyle ( hat ( \beta ) ) ) is consistent for β and distributed asymptotically normally. If the true value of parameter β is equal to β and the quantity s. e. ( β ^ ) ( displaystyle scriptstyle s.e. ( ( hat ( \beta ) ) ) ) correctly estimates the asymptotic variance of this estimator, then the t - statistic will have asymptotically the standard normal distribution. 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 Main article : t - test 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 population parameters, and thus it can be used regardless of what these may be. One can also divide a residual by the sample standard deviation : g ( x, X ) = x − X  ̄ s ( displaystyle g ( x, X ) = ( \frac ( x - ( overline ( X ) ) ) ( s ) ) ) 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 For more details on this topic, see Prediction interval § Unknown mean, unknown variance. Given a normal distribution N ( μ, σ 2 ) ( displaystyle N ( mu, sigma ^ ( 2 ) ) ) with unknown mean and variance, the t - statistic of a future observation X n + 1, ( displaystyle X_ ( n + 1 ), ) after one has made n observations, is an ancillary statistic -- a pivotal quantity ( does not depend on the values of μ and σ ) that is a statistic ( computed from observations ). This allows one to compute a frequentist prediction interval ( a predictive confidence interval ), via the following t - distribution : X n + 1 − X  ̄ n s n 1 + n − 1 ∼ T n − 1 ( displaystyle ( \frac ( X_ ( n + 1 ) - ( overline ( X ) ) _ ( n ) ) ( s_ ( n ) ( sqrt ( 1 + n ^ ( - 1 ) ) ) ) ) sim T ^ ( n - 1 ) ) Solving for X n + 1 ( displaystyle X_ ( n + 1 ) ) yields the prediction distribution X  ̄ n + s n 1 + n − 1 ⋅ T n − 1 ( displaystyle ( overline ( X ) ) _ ( n ) + s_ ( n ) ( sqrt ( 1 + n ^ ( - 1 ) ) ) cdot T ^ ( n - 1 ) ) 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 X n + 1 ( displaystyle X_ ( n + 1 ) ) will fall in that interval. History For more details on this topic, see Student 's t - test. 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. Related concepts z - score ( standardization ) : If the population parameters are known, then rather than computing the t - statistic, one can compute the z - score ; analogously, rather than using a t - test, one uses a z - test. This is rare outside of standardized testing. Studentized residual : In regression analysis, the standard errors of the estimators at different data points vary ( compare the middle versus endpoints of a simple linear regression ), and thus one must divide the different residuals by different estimates for the error, yielding what are called studentized residuals. See also Statistics portal F - test t - statistic References External links Retrieved from `` https://en.wikipedia.org/w/index.php?title=T-statistic&oldid=808038455 '' Categories : Statistical ratios Parametric statistics Normal distribution Hidden categories : Articles lacking sources from February 2011 All articles lacking sources Articles to be merged from May 2015 All articles to be merged All articles with unsourced statements Articles with unsourced statements from February 2011 Use dmy dates from November 2010 Talk Contents About Wikipedia Add links This page was last edited on 31 October 2017, at 14 : 22. Text is available under the Creative Commons Attribution - ShareAlike License ; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia ® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. About Wikipedia

Question: how many degrees of freedom does the​ t-statistic have
Answer: n − k) degrees of freedom, where n is the number of observations, and k is the number of regressors (including the intercept).