2023 NJC P2 Q6

The random variable \(X\) is normally distributed with mean \(\mu \) and variance \({{\sigma }^{2}}\).

1. Find \(\mathrm{P}\left( X>\mu +0.05\sigma \right)\).[2]
2. The mean of a random sample of \(n\) independent observations of \(X\) is denoted by \(\bar{X}\).
   Find the least value of \(n\) such that the probability that \(\bar{X}\) differs from \(\mu \) by at least \(0.05\sigma \) is less than \(0.4\).[3]
3. In another random sample of \(m\) independent observations of \(X\), the probability that at least \(4\) of these observations exceeds \(\mu +0.05\sigma \) each is at most \(\mathrm{0}\mathrm{.95}\). Find the greatest value of \(m\).[3]