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\begin{aligned} p o w e r & = {\begin{cases} \int_{0}^{\infty} P (F (1, N - 2, λ) > \\ h (u) F_{1 - α} (1, v (u)) | u) f (u) d u, & two-sided \\ \int_{0}^{\infty} P (t (N - 2, λ^{\frac{1}{2}}) > \\ {[h (u)]}^{\frac{1}{2}} t_{1 - α} (v (u)) | u) f (u) d u, & upper one-sided \\ \int_{0}^{\infty} P (t (N - 2, λ^{\frac{1}{2}}) < \\ {[h (u)]}^{\frac{1}{2}} t_{α} (v (u)) | u) f (u) d u, & lower one-sided \end{cases} \\ where \\ h (u) & = \frac{(\frac{1}{n_{1}} + \frac{u}{n_{2}}) (n_{1} + n_{2} - 2)}{[(n_{1} - 1) + (n_{2} - 1) \frac{u σ_{1}^{2}}{σ_{2}^{2}}] (\frac{1}{n_{1}} + \frac{σ_{2}^{2}}{σ_{1}^{2} n_{2}})} \\ v (u) & = \frac{{(\frac{1}{n_{1}} + \frac{u}{n_{2}})}^{2}}{\frac{1}{n_{1}^{2} (n_{1} - 1)} + \frac{u^{2}}{n_{2}^{2} (n_{2} - 1)}} \\ λ & = \frac{(μ_{d i f f} - μ_{0})^{2}}{\frac{σ_{1}^{2}}{n_{1}} + \frac{σ_{2}^{2}}{n_{2}}} \\ f (u) & = \frac{Γ (\frac{n_{1} + n_{2} - 2}{2})}{Γ (\frac{n_{1} - 1}{2}) Γ (\frac{n_{2} - 1}{2})} {[\frac{σ_{1}^{2} (n_{2} - 1)}{σ_{2}^{2} (n_{1} - 1)}]}^{\frac{n_{2} - 1}{2}} u^{\frac{n_{2} - 3}{2}} {[1 + (\frac{n_{2} - 1}{n_{1} - 1}) \frac{u σ_{1}^{2}}{σ_{2}^{2}}]}^{- (\frac{n_{1} + n_{2} - 2}{2})} \end{aligned}

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\begin{align*} \mr{power} & = \left\{ \begin{array}{ll} \int _0^\infty P\left(F(1,N-2, \lambda ) > \right. \\ \quad \left. h(u) F_{1-\alpha }(1, v(u)) | u\right) f(u) \mr{d}u, & \mbox{two-sided} \\ \int _0^\infty P\left(t(N-2, \lambda ^\frac {1}{2}) > \right. \\ \quad \left. \left[h(u)\right]^\frac {1}{2} t_{1-\alpha }(v(u)) | u\right) f(u) \mr{d}u, & \mbox{upper one-sided} \\ \int _0^\infty P\left(t(N-2, \lambda ^\frac {1}{2}) < \right. \\ \quad \left. \left[h(u)\right]^\frac {1}{2} t_{\alpha }(v(u)) | u\right) f(u) \mr{d}u, & \mbox{lower one-sided} \\ \end{array} \right. \\ \mbox{where} & \\ h(u) & = \frac{\left(\frac{1}{n_1} + \frac{u}{n_2}\right) (n_1+n_2-2)}{\left[(n_1-1) + (n_2-1)\frac{u\sigma _1^2}{\sigma _2^2}\right] \left(\frac{1}{n_1} + \frac{\sigma _2^2}{\sigma _1^2n_2}\right)} \\ v(u) & = \frac{\left(\frac{1}{n_1} + \frac{u}{n_2}\right)^2}{\frac{1}{n_1^2(n_1-1)} + \frac{u^2}{n_2^2(n_2-1)}} \\ \lambda & = \frac{(\mu _\mr {diff}-\mu _0)^2}{\frac{\sigma _1^2}{n_1} + \frac{\sigma _2^2}{n_2}} \\ f(u) & = \frac{\Gamma \left(\frac{n_1+n_2-2}{2}\right)}{\Gamma \left(\frac{n_1-1}{2}\right) \Gamma \left(\frac{n_2-1}{2}\right)} \left[ \frac{\sigma _1^2(n_2-1)}{\sigma _2^2(n_1-1)}\right]^\frac {n_2-1}{2} u^\frac {n_2-3}{2} \left[1+\left(\frac{n_2-1}{n_1-1}\right) \frac{u\sigma _1^2}{\sigma _2^2}\right]^{-\left(\frac{n_1+n_2-2}{2}\right)} \end{align*}