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Mathematically, the first vector is the oblique projection of the data vector onto the subspace spanned by the vector of 1's. The 1 degree of freedom is the dimension of this subspace. The second residual vector is the least-squares projection onto the (n − 1)-dimensional orthogonal complement of this subspace, and has n − 1 degrees of freedom.
Under the null hypothesis that model 2 does not provide a significantly better fit than model 1, F will have an F distribution, with (p 2 −p 1, n−p 2) degrees of freedom. The null hypothesis is rejected if the F calculated from the data is greater than the critical value of the F-distribution for some desired false-rejection probability (e ...
the number of degrees of freedom for each mean ( df = N − k ) where N is the total number of observations.) The distribution of q has been tabulated and appears in many textbooks on statistics.
In these formulae, n i − 1 is the number of degrees of freedom for each group, and the total sample size minus two (that is, n 1 + n 2 − 2) is the total number of degrees of freedom, which is used in significance testing. The minimum detectable effect (MDE) is: [25]
If is a -dimensional Gaussian random vector with mean vector and rank covariance matrix , then = () is chi-squared distributed with degrees of freedom. The sum of squares of statistically independent unit-variance Gaussian variables which do not have mean zero yields a generalization of the chi-squared distribution called the noncentral chi ...
The test statistic T in this instance could be set to be the sum of squares about the sample mean, divided by the nominal value for the variance (i.e. the value to be tested as holding). Then T has a chi-squared distribution with n − 1 degrees of freedom.
has the Studentized range distribution for n groups and ν degrees of freedom. In applications, the x i are typically the means of samples each of size m, s 2 is the pooled variance, and the degrees of freedom are ν = n(m − 1). The critical value of q is based on three factors: α (the probability of rejecting a true null hypothesis)
In statistics, Welch's t-test, or unequal variances t-test, is a two-sample location test which is used to test the (null) hypothesis that two populations have equal means. It is named for its creator, Bernard Lewis Welch , and is an adaptation of Student's t -test , [ 1 ] and is more reliable when the two samples have unequal variances and ...