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For the statistic t, with ν degrees of freedom, A(t | ν) is the probability that t would be less than the observed value if the two means were the same (provided that the smaller mean is subtracted from the larger, so that t ≥ 0). It can be easily calculated from the cumulative distribution function F ν (t) of the t distribution:
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 ...
From the t-test, the difference between the group means is 6-2=4. From the regression, the slope is also 4 indicating that a 1-unit change in drug dose (from 0 to 1) gives a 4-unit change in mean word recall (from 2 to 6). The t-test p-value for the difference in means, and the regression p-value for the slope, are both 0.00805. The methods ...
If there are no ties – or the ties occur within a particular sample (which does not affect the value of the test statistic) – exact tables of S are available; for example, Jonckheere [1] provided selected tables for values of k from 3 to 6 and equal samples sizes (m) from 2 to 5.
The test is based on the work of Michael E. Sobel, [1] [2] and is an application of the delta method. In mediation, the relationship between the independent variable and the dependent variable is hypothesized to be an indirect effect that exists due to the influence of a third variable (the mediator).
The function T(h, a) gives the probability of the event (X > h and 0 < Y < aX) where X and Y are independent standard normal random variables.. This function can be used to calculate bivariate normal distribution probabilities [2] [3] and, from there, in the calculation of multivariate normal distribution probabilities. [4]
The phrase "T distribution" may refer to Student's t-distribution in univariate probability theory, Hotelling's T-square distribution in multivariate statistics.
These small standard errors will cause the estimated t-statistic to be inflated, suggesting significance where perhaps there is none. The inflated t-statistic, may in turn, lead us to incorrectly reject null hypotheses, about population values of the parameters of the regression model more often than we would if the standard errors were ...