Weighted normal sum and probit moment

Let $X_1,X_2,…,X_n$ be independent and identically distributed random variables having $N(\mu, \sigma^2)$ distribution, where $\mu \in \mathbb{R}$ and $\sigma > 0$. Consider the following statements: (I) If $\frac{\sqrt{5}}{\sigma(2n+1)} \sum_{i=1}^{n} i (X_i - \mu)$ has $N(0,1)$ distribution, then $n = 2$. (II) $E\left(\left(-\log_e\left(\Phi\left(\frac{X_1 - \mu}{\sigma}\right)\right)\right)^3\right) = 6$, where $\Phi(\cdot)$ denotes the cumulative distribution function of $N(0,1)$ random variable. Which of the above statements is/are true?