Step 1 :Given that $X$ is a normal random variable with mean $\mu=46$ and standard deviation $\sigma=11$.
Step 2 :To compute the $z$-value corresponding to $X=31$, we use the formula $z = \frac{X - \mu}{\sigma}$. Substituting the given values, we get $z = \frac{31 - 46}{11} = -1.36$.
Step 3 :The area under the standard normal curve to the left of the $Z$-value found in part (a) is given as 0.0863. This is equivalent to the cumulative distribution function (CDF) at $X=31$.
Step 4 :To find the area under the normal curve to the right of $X=31$, we subtract the CDF at $X=31$ from 1. This gives us $1 - 0.0863 = 0.9137$.
Step 5 :Final Answer: (a) The z-value corresponding to $X=31$ is \(\boxed{-1.36}\). (b) The area under the normal curve to the left of $X=31$ is \(\boxed{0.0863}\). (c) The area under the normal curve to the right of $X=31$ is \(\boxed{0.9137}\).