2018 HCI MYE Q10

The random variable \(X\) has the distribution \(X\sim N\left( \mu ,{{\sigma }^{2}} \right)\). It is given that \({{X}_{1}}\) and \({{X}_{2}}\) are \(2\) independent observations of \(X\) with \(\mathrm{P}\left( 4 < X < 6 \right)=0.5\) and \(\mathrm{P}\left( {{X}_{1}}+{{X}_{2}}>10 \right)=0.5\).

1. Show that \(\mu =5\) and state, with a reason, the value of \(\mathrm{P}\left( 2X>10 \right)\).[3]
2. Find the value of \(\sigma \), giving your answer to 5 significant figures.[3]
3. A random sample of \(n\) observations of \(X\) is obtained. Given that there is a probability of more than 95% such that the sample mean is within 0.1 units of \(\mu \), show that \(\mathrm{P}\left( Z < -\frac{a\sqrt{n}}{b} \right) < c\), where \(a\), \(b\), \(c\) are real values to be determined.
   Hence find the least value of \(n\).[4]