Chapter 6.2, Problem 11P

(a)

To determine

Find the z interval for $4.5<x$.

Answer to Problem 11P

The z interval for $4.5<x$ is $z>-1.00$.

Explanation of Solution

Calculation:

Z score:

The number of standard deviations the original measurement x is from the value of mean $\mu$ is measured using the z-score or z value. The formula for z score is,

$z=\frac{x-\mu }{\sigma }$

In the formula, x is the raw score, $\mu$ is the mean and $\sigma$ is the standard deviation.

The variable x is red blood cell (RBC) count in millions per cubic millimetre for women. The healthy females are normally distributed with mean $\mu =4.8$ and standard deviation $\sigma =0.3$. Then the z score is,

$z=\frac{x-4.8}{0.3}$

For the z interval consider,

$\begin{array}{c}4.5<x\\ x>4.5\end{array}$

Subtract 4.8 on both sides of the inequality.

$\begin{array}{c}x-4.8>4.5-4.8\\ x-4.8>-0.3\end{array}$

Divide 0.3 on both sides of the inequality.

$\begin{array}{c}\frac{x-4.8}{0.3}>\frac{-0.3}{0.3}\\ z>-1.00\end{array}$

Hence, the z interval for $4.5<x$ is $z>-1.00$.

(b)

To determine

Find the z interval for $x<4.2$.

Answer to Problem 11P

The z interval for $x<4.2$ is $z<-2.00$.

Explanation of Solution

Calculation:

For the z interval consider,

$x<4.2$

Subtract 4.8 on both sides of the inequality.

$\begin{array}{c}x-4.8<4.2-4.8\\ x-4.8<-0.6\end{array}$

Divide 0.3 on both sides of the inequality.

$\begin{array}{c}\frac{x-4.8}{0.3}<\frac{-0.6}{0.3}\\ z<-2.00\end{array}$

Hence, the z interval for $x<4.2$ is $z<-2.00$.

(c)

To determine

Find the z interval for $4.0<x<5.5$.

Answer to Problem 11P

The z interval for $4.0<x<5.5$ is $-2.67<z<2.33$.

Explanation of Solution

Calculation:

For the z interval consider,

$4.0<x<5.5$

Subtract 4.8 for each part of the inequality.

$\begin{array}{l}4.0-4.8<x-4.8<5.5-4.8\\ -0.8<x-4.8<0.7\end{array}$

Divide 4.3 for each part of the inequality.

$\begin{array}{c}\frac{-0.8}{0.3}<\frac{x-4.8}{0.3}<\frac{0.7}{0.3}\\ -2.67<z<2.33\end{array}$

Hence, the z interval for $4.0<x<5.5$ is $-2.67<z<2.33$.

(d)

To determine

Find the x interval for $z<-1.44$.

Answer to Problem 11P

The x interval for $z<-1.44$ is $x<4.4$.

Explanation of Solution

Calculation:

The z score is,

$\begin{array}{l}z=\frac{x-4.8}{0.3}\\ x=4.8+0.3z\end{array}$

For the x interval consider,

$z<-1.44$

Multiply 0.3 on both sides of the inequality.

$\begin{array}{c}0.3z<-1.44\left(0.3\right)\\ 0.3z<-0.432\end{array}$

Add 4.8 on both sides of the inequality.

$\begin{array}{c}4.8+0.3z<-0.432+4.8\\ x<4.4\end{array}$

Hence, the x interval for $z<-1.44$ is $x<4.4$.

(e)

To determine

Find the x interval for $1.28<z$.

Answer to Problem 11P

The x interval for $1.28<z$ is $x>5.2$.

Explanation of Solution

Calculation:

For the x interval consider,

$\begin{array}{c}1.28<z\\ z>1.28\end{array}$

Multiply 0.3 on both sides of the inequality.

$\begin{array}{c}0.3z>1.28\left(0.3\right)\\ 0.3z>0.384\end{array}$

Add 4.8 on both sides of the inequality.

$\begin{array}{c}4.8+0.3z>0.384+4.8\\ x>5.2\end{array}$

Hence, the x interval for $1.28<z$ is $x>5.2$.

(f)

To determine

Find the x interval for $-2.25<z<-1.00$.

Answer to Problem 11P

The x interval for $-2.25<z<-1.00$ is $4.12<x<4.50$.

Explanation of Solution

Calculation:

For the x interval consider,

$-2.25<z<-1.00$

Multiply 0.3 for each part of the inequality.

$\begin{array}{c}-2.25\left(0.3\right)<0.3z<-1.00\left(0.3\right)\\ -0.675<0.3z<-0.300\end{array}$

Add 4.8 for each part of the inequality.

$\begin{array}{c}-0.675+4.8<4.8+0.3z<-0.300+4.8\\ 4.12<x<4.50\end{array}$

Hence, the x interval for $-2.25<z<-1.00$ is $4.12<x<4.50$.

(g)

To determine

Identify whether the RBC count of 5.9 or higher can be considered unusually high or not using z values and Figure 6-15.

Answer to Problem 11P

The RBC count of 5.9 or higher is unusually high.

Explanation of Solution

Calculation:

The RBC count of 5.9 or higher, that is $x\ge 5.9$. Then the value of z is,

$x\ge 5.9$

Subtract 4.8 on both sides of the inequality.

$\begin{array}{c}x-4.8\ge 5.9-4.8\\ x-4.8\ge 1.1\end{array}$

Divide 0.3 on both sides of the inequality.

$\begin{array}{c}\frac{x-4.8}{0.3}\ge \frac{1.1}{0.3}\\ z\ge 3.67\end{array}$

The RBC count of 5.9 or higher is 3.67 standard deviations above the mean value. The z score value is greater than 3 indicating that the value is very unusual.

The figure 6-15 is the standard normal distribution curve. The z value is located on the curve as below.

The z value that is far from the mean (zero) is considered as unusual. If the value is closer to –3 is usually very small and value closer to 3 is usually very large.

If the value of z for RBC count of 5.9 or higher is greater to 3 then RBC count would be very large and unusual.