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\begin{aligned} Pr (half-width \leq h) & = {\begin{cases} P (χ^{2} (N - 1) \leq \frac{h^{2} N (N - 1)}{σ_{d i f f}^{2} (t_{1 - \frac{α}{2}}^{2} (N - 1))}), & two-sided \\ P (χ^{2} (N - 1) \leq \frac{h^{2} N (N - 1)}{σ_{d i f f}^{2} (t_{1 - α}^{2} (N - 1))}), & one-sided \end{cases} \\ \begin{array}{r} Pr (half-width \leq h | \\ validity) \end{array} & = {\begin{cases} (\frac{1}{1 - α}) 2 [Q_{N - 1} ((t_{1 - \frac{α}{2}} (N - 1)), 0; \\ 0, b_{1}) - Q_{N - 1} (0, 0; 0, b_{1})], & two-sided \\ (\frac{1}{1 - α}) Q_{N - 1} ((t_{1 - α} (N - 1)), 0; 0, b_{1}), & one-sided \end{cases} \end{aligned}

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\begin{align*} \mbox{Pr}(\mbox{half-width} \le h) & = \left\{ \begin{array}{ll} P\left( \chi ^2(N-1) \le \frac{h^2 N(N-1)}{\sigma ^2_\mr {diff}(t^2_{1-\frac{\alpha }{2}}(N-1))} \right), & \mbox{two-sided} \\ P\left( \chi ^2(N-1) \le \frac{h^2 N(N-1)}{\sigma ^2_\mr {diff}(t^2_{1-\alpha }(N-1))} \right), & \mbox{one-sided} \\ \end{array} \right. \\ \begin{array}{r} \mbox{Pr}(\mbox{half-width} \le h | \\ \mbox{validity}) \end{array}& = \left\{ \begin{array}{ll} \left(\frac{1}{1-\alpha }\right) 2 \left[ Q_{N-1}\left((t_{1-\frac{\alpha }{2}}(N-1)),0; \right. \right. \\ \quad \left. \left. 0,b_1\right) - Q_{N-1}(0,0;0,b_1)\right], & \mbox{two-sided} \\ \left(\frac{1}{1-\alpha }\right) Q_{N-1}\left((t_{1-\alpha }(N-1)),0;0,b_1\right), & \mbox{one-sided} \\ \end{array} \right. \\ \end{align*}