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\sigma_{e}^{2}=\sqrt{\sigma_{y}^{2}-\hat{\beta}_{1}^{2} \sigma_{x}^{2}}=\sqrt{\left(\frac{\mathrm{USL}_{y}-\mathrm{LSL}_{y}}{\Phi^{-1}(0.9973)}\right)^{2}-\hat{\beta}_{1}^{2}\left(\frac{\mathrm{USL}_{x}-\mathrm{LSL}_{x}}{\Phi^{-1}(0.99865)}\right)^{2}}