Chapter 7.2, Problem 6P

Basic Computation: z Score and Raw Score A normal distribution has $\mu =10\text{ and }\sigma =2$.

$\begin{array}{l}\left(\text{a}\right)\text{ Find the }z\text{ score corresponding to }x=12.\hfill \\ \left(\text{b}\right)\text{ Find the }z\text{ score corresponding to }x=4.\hfill \\ \left(\text{c}\right)\text{ Find the raw score corresponding to }z=\mathrm{1.5.}\hfill \\ \left(\text{d}\right)\text{ Find the raw score corresponding to }z=–\mathrm{1.2.}\hfill \end{array}$

To determine

The z score corresponding to $x=12$ for normal distribution with $\mu =10,\sigma =2$.

Answer to Problem 6P

Solution: The z score corresponding to $x=12$ for normal distribution with $\mu =10,\sigma =2$ is 1.0.

Explanation of Solution

$\begin{array}{l}\mu =10,\sigma =2\\ x=12\end{array}$

We use the formula for normal distribution:

$\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ z=\frac{12-10}{2}\\ z=1.0\end{array}$

The z score corresponding to $x=12$ for normal distribution with $\mu =10,\sigma =2$ is 1.0.

To determine

The z score corresponding to $x=4$ for normal distribution with $\mu =10,\sigma =2$.

Answer to Problem 6P

Solution: The z score corresponding to $x=4$ for normal distribution with $\mu =10,\sigma =2$ is -3.0.

Explanation of Solution

$\begin{array}{l}\mu =10,\sigma =2\\ x=4\end{array}$

We use the formula for normal distribution:

$\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ z=\frac{4-10}{2}\\ z=-3.0\end{array}$

The z score corresponding to $x=4$ for normal distribution with $\mu =10,\sigma =2$ is -3.0.

To determine

The raw score corresponding to $z=1.5$ for normal distribution with $\mu =10,\sigma =2$.

Answer to Problem 6P

Solution: The raw score corresponding to $z=1.5$ for normal distribution with $\mu =10,\sigma =2$ is 13.

Explanation of Solution

$\begin{array}{l}\mu =10,\sigma =2\\ z=1.5\end{array}$

We use the formula for normal distribution:

$\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ x=\sigma z+\mu \\ \\ x=2(1.5)+10\\ \\ x=3+10\\ \\ x=13\end{array}$

The raw score corresponding to $z=1.5$ for normal distribution with $\mu =10,\sigma =2$ is 13.

To determine

The raw score corresponding to $z=-1.2$ for normal distribution with $\mu =10,\sigma =2$.

Answer to Problem 6P

Solution: The raw score corresponding to $z=-1.2$ for normal distribution with $\mu =10,\sigma =2$ is 7.6.

Explanation of Solution

$\begin{array}{l}\mu =10,\sigma =2\\ z=-1.2\end{array}$

We use the formula for normal distribution:

$\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ x=\sigma z+\mu \\ \\ x=2(-1.2)+10\\ \\ x=-2.4+10\\ \\ x=7.6\end{array}$

The raw score corresponding to $z=-1.2$ for normal distribution with $\mu =10,\sigma =2$ is 7.6.