(本题满分 11 分) 设 $X_{1}, X_{2}, \dots, X_{n}$ 是总体 $N(\mu, \sigma^{2})$ 的简单随机样本，记 $$ \bar {X} = \frac {1}{n} \sum_ {i = 1} ^ {n} X _ {i}, \quad S ^ {2} = \frac {1}{n - 1} \sum_ {i = 1} ^ {n} \left(X _ {i} - \bar {X}\right) ^ {2}, \quad T = \bar {X} ^ {2} - \frac {1}{n} S ^ {2}. $$ (I) 证明 $T$ 是 $\mu^2$ 的无偏估计量； （Ⅱ）当 $\mu = 0, \sigma = 1$ 时，求 $D(T)$