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[tex]

\sup_{\sum a_i = 0} T_a^2 = \sup_{\sum a_i = 0} \left(

\left( S^2_p \sum a_i^2 / n_i \right)^{-1/2} \left( \sum a_i \bar Y_{i \cdot} - \sum a_i \theta_i\right)

\right)^2 = \left( S^2_p \right)^{-1} \sum n_i \left( \bar Y_{i \cdot} - \bar{\bar Y} - \theta_i + \bar{\theta} \right)^2

[/tex]

where all the summations are from 1 to [itex] k [/itex] the no. of treatments and [itex] S^2_p, n_i, \theta_i, \bar Y_{i \cdot}[/itex] are the pooled sample variance, no. of observations of treatment [itex] i [/itex], its mean, and sample mean respectively. The term inside the square between equals signs follows t distribution but for whatever reason the supremum of the square follows [itex] (k-1) F(k-1, n-k)[/itex], as opposed to [itex] t^2 [/itex].