1. **Stating the problem:** We need to evaluate the expression: $$N' = \left[ \frac{k}{s} \sqrt{N \frac{\Sigma (Y^2) - (\Sigma x)^2}{\Sigma X}} \right] = \left[ \frac{2}{0.05} \sqrt{5 (17986.5) - 90000} \div 300 \right]^2$$ 2. **Identify given values:** - $k = 2$ - $s = 0.05$ - $N = 5$ - $\Sigma (Y^2) = 17986.5$ - $(\Sigma x)^2 = 90000$ - $\Sigma X = 300$ 3. **Calculate the expression inside the square root:** $$5 \times 17986.5 - 90000 = 89932.5 - 90000 = -67.5$$ 4. **Evaluate the square root:** Since the value inside the square root is negative ($-67.5$), the expression involves an imaginary number: $$\sqrt{-67.5} = i \sqrt{67.5}$$ 5. **Calculate the fraction outside the root:** $$\frac{k}{s} = \frac{2}{0.05} = 40$$ 6. **Calculate the entire bracket before squaring:** $$40 \times \frac{i \sqrt{67.5}}{300} = \frac{40}{300} i \sqrt{67.5} = \frac{2}{15} i \sqrt{67.5}$$ 7. **Square the entire expression:** $$\left( \frac{2}{15} i \sqrt{67.5} \right)^2 = \left( \frac{2}{15} \right)^2 \times (i)^2 \times (\sqrt{67.5})^2 = \frac{4}{225} \times (-1) \times 67.5 = -\frac{270}{225} = -1.2$$ **Final answer:** $$N' = -1.2$$ Since the value inside the square root was negative, the result is a negative real number after squaring the imaginary term.