Question

Suppose the random variable x is best described by a normal distribution with $\mathbf{\mu }\mathbf{=}\mathbf{30}$ and $\mathbf{\sigma }\mathbf{=}\mathbf{4}$. Find the z-score that corresponds to each of the following x values:

$$\begin{gathered} \mathbf{a}\mathbf{.}\mathbf{x}\mathbf{=}\mathbf{20}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{b}\mathbf{.}\mathbf{x}\mathbf{=}\mathbf{30} \\ \mathbf{c}\mathbf{.}\mathbf{x}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{75}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{d}\mathbf{.}\mathbf{x}\mathbf{=}\mathbf{15} \\ \mathbf{e}\mathbf{.}\mathbf{x}\mathbf{=}\mathbf{35}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{f}\mathbf{.}\mathbf{x}\mathbf{=}\mathbf{25} \end{gathered}$$

Short answer

1. z = -2.5
2. z = 0
3. z = -6.1825
4. z = -3.75
5. z = 1.25
6. z = -1.25

Step-by-step solution

Given information

Here, x is a random variable that is normally distributed with$\mu =30$ and$\sigma =4$

Finding the z-score when x=20

a.

When x = 20, the z score will be -

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ =\frac{20-30}{14} \\ =\frac{-10}{14} \\ =-2.5 \end{gathered}$$

Therefore, the z-score forx = 20is -2.5.

Finding the z-score when x=30

b.

When x=30, the z score will be -

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ =\frac{30-30}{4} \\ =\frac{0}{4} \\ =0 \end{gathered}$$

Therefore, the z-score for x = 30 is 0.

Finding the z-score when x=2.75

c.

When x=2.75, the z score will be -

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ =\frac{2.75-30}{4} \\ =\frac{-27.25}{4} \\ =-6.8125 \end{gathered}$$

Therefore, the z-score for x=2.75 is -6.8125.

Finding the z-score when x=15

d.

When x=15, the z score will be -

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ =\frac{15-30}{4} \\ =\frac{-15}{4} \\ =-3.75 \end{gathered}$$

Therefore, the z-score for x=15 is -3.75.

Finding the z-score when x=35

e.

When x=35, the z score will be -

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ =\frac{35-30}{4} \\ =\frac{5}{4} \\ =1.25 \end{gathered}$$

Therefore, the z-score for x=35 is 1.25.

Finding the z-score when x=25

f.

When x=25, the z score will be -

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ =\frac{25-30}{4} \\ =\frac{-5}{4} \\ =-1.25 \end{gathered}$$

Therefore, the z-score for x=25 is -1.25.