Step 1: Analyzing Statement I.
The given order for radii is \( \text{Al}>\text{Mg}>\text{Mg}^{2+}>\text{Al}^{3+} \). This order is correct because:
- \( \text{Al} \) (atomic radius of aluminum) is larger than \( \text{Mg} \) (atomic radius of magnesium) due to fewer protons in Al's nucleus compared to Mg.
- \( \text{Mg}^{2+} \) (Mg ion) has a smaller radius than Mg due to the loss of electrons and increased nuclear attraction.
- \( \text{Al}^{3+} \) (Al ion) has the smallest radius due to the even higher nuclear charge and fewer electrons.
So, Statement I is correct.
Step 2: Analyzing Statement II.
Statement II says atomic size always depends on electronegativity. However, atomic size primarily depends on factors like nuclear charge, electron shielding, and quantum levels, rather than just electronegativity. Therefore, Statement II is incorrect.
Thus, the correct answer is (3).