Chapter 2, Problem 1CRE

(a)

To determine

To find: mean and standard deviation.

Answer to Problem 1CRE

The standard deviation is $7.5\text{cm}$ and the mean is $170\text{cm}$

Explanation of Solution

Given:

A z-score of 0 corresponds to a height of 170 centimeters

A z-score of 1 corresponds to a height of 177.5cm

Calculation:

To find the mean, use the fact that a z-score of 0 corresponds to a height of $170

Then theresult is shown below.

  $\begin{array}{l}z=\frac{x-\mu }{\sigma }\hfill \\ \Rightarrow 0=\frac{170-\mu }{\sigma }\hfill \\ \Rightarrow 0=170-\mu \hfill \\ \Rightarrow \mu =170\text{cm}\hfill \end{array}$

Thus, the mean of the height distribution for 15 -year-old males is $170\text{cm}$

To find the standard deviation, use the fact that a $z$ -score of 1 corresponds to a height of 177.5 Then the result is shown below.

  $\begin{array}{l}z=\frac{x-\mu }{\sigma }\hfill \\ \Rightarrow 1=\frac{177.5-170}{\sigma }\hfill \\ \Rightarrow 1=\frac{7.5}{\sigma }\hfill \\ \Rightarrow \sigma =7.5\text{cm}\hfill \end{array}$

Thus, the standard deviation of the height distribution for 15 -year-old males is $7.5\text{cm}$

Conclusion:

Thus, the standard deviation is $7.5\text{cm}$ and the mean is $170\text{cm}$

(b)

To determine

To find:the height corresponding to a z-score of 2.5.

Answer to Problem 1CRE

A height of $188.75\text{cm}$ has a z-score of 2.5

Explanation of Solution

Calculation:

To find height correspond to a z-score of 2.5 we use the equation.

  $z=\frac{x-\mu }{\sigma }$

  $\Rightarrow 2.5=\frac{x-170}{7.5}$

  $\begin{array}{l}\Rightarrow x=170+(2.5\times 7.5)\hfill \\ \Rightarrow x=188.75\text{cm}\hfill \end{array}$

A height of $188.75\text{cm}$ has a z-score of 2.5

Conclusion:

Therefore, a height of $188.75\text{cm}$ has a z-score of 2.5