Suppose $X$ is a normal random variable with mean $\mu=46$ and standard deviation $\sigma=11$
(a) Compute the $z$-value corresponding to $X=31$.
(b) Suppose the area under the standard normal curve to the left of the $Z$-value found in part (a) is 0.0863 . What is the area under the normal curve to the left of $X=31$ ?
(c) What is the area under the normal curve to the right of $X=31$ ?
(a) $z=1.9757$
(Round to two decimal places as needed.)
(b) The area is 0.0155
(Round to four decimal places as needed.)
(c) The area is 57
(Round to four decimal places as needed.)

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 \(\boxed{0.9137}\).