Chapter 6.3, Problem 33P

(a)

To determine

Find the standard deviation of the x distribution.

Answer to Problem 33P

The standard deviation of the x distribution is 12 beats per minute.

Explanation of Solution

Calculation:

Rule of Thumb:

The formula for standard deviation using range is,

$\sigma =\frac{\text{Range}}{4}$

In the formula, range is the obtained by subtracting the low value from the high value.

The variable x is a random variable that represents the resting heart rate for an adult horse.

The 95% of data range from 22 to 70 beats per minute.

The standard deviation is,

$\begin{array}{c}\sigma =\frac{70-22}{4}\\ =\frac{48}{4}\\ =12\end{array}$

Hence, the standard deviation of the x distribution is 12 beats per minute.

(b)

To determine

Find the probability that the heart rate is fewer than 25 beats per minute.

Answer to Problem 33P

The probability that the heart rate is fewer than 25 beats per minute is 0.0401.

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.

Substitute x as 25, $\mu$ as 46 and $\sigma$ as 12 in the formula of z score.

$\begin{array}{c}z=\frac{25-46}{12}\\ =\frac{-21}{12}\\ =-1.75\end{array}$

Use the Appendix II: Tables, Table 5: Areas of a Standard Normal Distribution: to obtain probability less than –1.75.

• Locate the value –1.7 in column z.
• Locate the value 0.05 in top row.
• The intersecting value of row and column is 0.0401.

The probability is,

$\begin{array}{c}P\left(x<25\right)=P\left(z<-1.75\right)\\ =0.0401\end{array}$

Hence, the probability that the heart rate is fewer than 25 beats per minute is 0.0401.

(c)

To determine

Find the probability that the heart rate is greater than 60 beats per minute.

Answer to Problem 33P

The probability that the heart rate is greater than 60 beats per minute is 0.1210.

Explanation of Solution

Calculation:

Substitute x as 60, $\mu$ as 46 and $\sigma$ as 12 in the formula of z score.

$\begin{array}{c}z=\frac{60-46}{12}\\ =\frac{14}{12}\\ =1.17\end{array}$

Use the Appendix II: Tables, Table 5: Areas of a Standard Normal Distribution: to obtain probability less than 1.17.

• Locate the value 1.1 in column z.
• Locate the value 0.07 in top row.
• The intersecting value of row and column is 0.8790.

The probability is,

$\begin{array}{c}P\left(x>60\right)=P\left(z>1.17\right)\\ =1-P\left(z<1.17\right)\\ =1-0.8790\\ =0.1210\end{array}$

Hence, the probability that the heart rate is greater than 60 beats per minute is 0.1210.

(d)

To determine

Find the probability that the heart rate is between 25 and 60 beats per minute.

Answer to Problem 33P

The probability that the heart rate is between 25 and 60 beats per minute is 0.8389.

Explanation of Solution

Calculation:

Substitute x as 25, $\mu$ as 46 and $\sigma$ as 12 in the formula of z score.

$\begin{array}{c}z=\frac{25-46}{12}\\ =\frac{-21}{12}\\ =-1.75\end{array}$

Substitute x as 60, $\mu$ as 46 and $\sigma$ as 12 in the formula of z score.

$\begin{array}{c}z=\frac{60-46}{12}\\ =\frac{14}{12}\\ =1.17\end{array}$

Use the Appendix II: Tables, Table 5: Areas of a Standard Normal Distribution: to obtain probability less than –1.75.

• Locate the value –1.7 in column z.
• Locate the value 0.05 in top row.
• The intersecting value of row and column is 0.0401.

Use the Appendix II: Tables, Table 5: Areas of a Standard Normal Distribution: to obtain probability less than 1.17.

• Locate the value 1.1 in column z.
• Locate the value 0.07 in top row.
• The intersecting value of row and column is 0.8790.

The probability is,

$\begin{array}{c}P\left(25<x<60\right)=P\left(-1.75<z<1.17\right)\\ =P\left(z<1.17\right)-P\left(z<-1.75\right)\\ =0.8790-0.0401\\ =0.8389\end{array}$

Hence, the probability that the heart rate is between 25 and 60 beats per minute is 0.8389.

(e)

To determine

Find heart rate corresponding to the upper 10% cutoff point of the probability distribution.

Answer to Problem 33P

The heart rate corresponding to the upper 10% cutoff point of the probability distribution is 61 beats per minute.

Explanation of Solution

Calculation:

Step by step procedure to obtain probability plot using MINITAB software is given below:

• Choose Graph > Probability Distribution Plot choose View Probability > OK.
• Enter the Mean as 46, and Standard deviation as 12.
• From Distribution, choose ‘Normal’ distribution.
• Click the Shaded Area tab.
• Choose Probability and Right Tail, for the region of the curve to shade.
• Enter the Probability as 0.10.
• Click OK.

Output using MINITAB software is given below:

From Minitab output, the heart rate corresponding to the upper 10% cutoff point of the probability distribution is 61.38.

Hence, the heart rate corresponding to the upper 10% cutoff point of the probability distribution is approximately 61 beats per minute.