Question

Compare the z-scores to decide which of the following x values lie the greatest distance above the mean and the greatest distance below the mean.

$$\begin{gathered} \mathbf{a}\mathbf{.}\mathbf{ }\mathbf{x}\mathbf{=}\mathbf{100}\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{\mu }\mathbf{=}\mathbf{50}\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{\sigma }\mathbf{=}\mathbf{25} \\ \mathbf{b}\mathbf{.}\mathbf{ }\mathbf{x}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{\mu }\mathbf{=}\mathbf{4}\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{\sigma }\mathbf{=}\mathbf{1} \\ \mathbf{c}\mathbf{.}\mathbf{ }\mathbf{x}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{\mu }\mathbf{=}\mathbf{200}\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{\sigma }\mathbf{=}\mathbf{100} \\ \mathbf{d}\mathbf{.}\mathbf{x}\mathbf{=}\mathbf{10}\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{\mu }\mathbf{=}\mathbf{5}\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{\sigma }\mathbf{=}\mathbf{3} \end{gathered}$$

Short answer

a.Z=2

b.Z=-3

c.Z=-2

d.Z=1.67

The x values lie the most significant distance above the mean in part (a), and they lie the most significant distance below the mean in part (c).

Step-by-step solution

Computing the z-score when x=100,  μ=50,  σ=25

Z-score can be computed using the formula:$\mathbf{Z}\mathbf{=}\frac{\mathbf{x}\mathbf{-}\mathbf{\mu }}{\mathbf{\sigma }}$

Substituting the values:

$$\begin{gathered} \mathbf{Z}\mathbf{=}\frac{\mathbf{100}\mathbf{-}\mathbf{50}}{\mathbf{25}} \\ \mathbf{Z}\mathbf{=}\frac{\mathbf{50}}{\mathbf{25}} \\ \mathbf{Z}\mathbf{=}\mathbf{2} \end{gathered}$$

Thus, the z-score is 2.

Calculating the z-score when x=1,  μ=4,  σ=1

Using the values in the z-score formula, one gets:

$$\begin{gathered} \mathbf{Z}\mathbf{=}\frac{\mathbf{1}\mathbf{-}\mathbf{4}}{\mathbf{1}} \\ \mathbf{Z}\mathbf{=}\frac{\mathbf{-}\mathbf{3}}{\mathbf{1}} \\ \mathbf{Z}\mathbf{=}\mathbf{-}\mathbf{3} \end{gathered}$$

Therefore, the z-score is -3.

Finding the z-score when x=0,  μ=200,  σ=100

$$\begin{gathered} \mathbf{Z}\mathbf{=}\frac{\mathbf{0}\mathbf{-}\mathbf{200}}{\mathbf{100}} \\ \mathbf{Z}\mathbf{=}\frac{\mathbf{-}\mathbf{200}}{\mathbf{100}} \\ \mathbf{Z}\mathbf{=}\mathbf{-}\mathbf{2} \end{gathered}$$

Hence, the z-score is -2.

Determining the z-score when x=10,  μ=5,  σ=3

$$\begin{gathered} \mathbf{Z}\mathbf{=}\frac{\mathbf{10}\mathbf{-}\mathbf{5}}{\mathbf{3}} \\ \mathbf{Z}\mathbf{=}\frac{\mathbf{5}}{\mathbf{3}} \\ \mathbf{Z}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{67} \end{gathered}$$

Here, the z-score is 1.67

Comparing the z-scores

From the z-scores computed above, the x values that lie the greatest distance above the mean isx=100 (part a), and the x values that lie the greatest distance below the mean isx=0 (part c).