1. **State the problem:** We need to evaluate the expression $$N' = \left[ \frac{k}{s} \sqrt{N \sum X^2 - (\sum X)^2} \over \sum X \right]^2$$ with the given values: $$k=2, \quad s=0.05, \quad N=5, \quad \sum X^2=16788.5, \quad (\sum X)^2=83621, \quad \sum X=289.$$ 2. **Write the formula clearly:** $$N' = \left[ \frac{k}{s} \cdot \frac{\sqrt{N \sum X^2 - (\sum X)^2}}{\sum X} \right]^2$$ 3. **Calculate the numerator inside the square root:** $$N \sum X^2 - (\sum X)^2 = 5 \times 16788.5 - 83621 = 83942.5 - 83621 = 321.5$$ 4. **Calculate the square root:** $$\sqrt{321.5} \approx 17.93$$ 5. **Calculate the fraction inside the brackets:** $$\frac{k}{s} = \frac{2}{0.05} = 40$$ 6. **Calculate the entire fraction inside the brackets:** $$\frac{40 \times 17.93}{289} = \frac{717.2}{289} \approx 2.48$$ 7. **Square the result:** $$N' = (2.48)^2 = 6.15$$ **Final answer:** $$\boxed{6.15}$$