Question

A variable x is normally distributed with mean 17 and standard deviation 7.Round your answers to the nearest hundredth as needed.a) Determine the z-score for x = 20.2b) Determine the z-score for x = 15.zc) What value of has a z-score of 1?d) What value of x has a z-score of -0.1?e) What value of x has a z-score of 0?=

Answer 1

Solution

Given

$\begin{gathered} \text{ mean }\mu=17 \\ \\ \text{ standard deviation }\sigma=7 \end{gathered}$

The Z score is given by the formula;

$Z=\frac{X-\mu}{\sigma}$

(a)

$Z=\frac{20-17}{7}\approx0.43\text{ \lparen nearest hundredth\rparen}$

(b)

$Z=\frac{15-17}{7}\approx-0.29\text{ \lparen nearest hundredth\rparen}$

(c)

when Z = 1

$\begin{gathered} \Rightarrow1=\frac{X-17}{7} \\ \\ \Rightarrow7=X-17 \\ \\ \Rightarrow X=7+17 \\ \\ \Rightarrow X=24 \end{gathered}$

when Z = -0.1

$\begin{gathered} \Rightarrow-0.1=\frac{X-17}{7} \\ \\ \Rightarrow-\frac{7}{10}=X-17 \\ \\ \Rightarrow X=17-\frac{7}{10} \\ \\ \Rightarrow X=16.3 \end{gathered}$

When Z = 0

$\begin{gathered} \Rightarrow0=\frac{X-17}{7} \\ \\ \Rightarrow0=X-17 \\ \\ \Rightarrow X=17 \end{gathered}$