Chapter 3.4, Problem 65E

Solution

(a)

To express : The given data in terms of their common logarithms and a scatter plot of $\left(\log w,\log r\right)$ .

The required data is:

Mammal	$\log w$	$\log r$
Rat	$-0.699$	$2.623$
Guinea Pig	$-0.523$	$2.477$
Rabbit	$0.301$	$2.312$
Small Dog	$0.699$	$2.079$
Large Dog	$1.477$	$1.929$
Sheep	$1.699$	$1.845$
Human	$1.845$	$1.857$

Given information :

The data:

Mammal	Body Weight (kg)	Pulse Rate (beats/min)
Rat	$0.2$	$420$
Guinea Pig	$0.3$	$300$
Rabbit	$2$	$205$
Small Dog	$5$	$120$
Large Dog	$30$	$85$
Sheep	$50$	$70$
Human	$70$	$72$

The scientists have found that the pulse rate $\left(r\right)$ of mammals are a power function of their body weight $\left(w\right)$ .

Calculation :

Consider the given table and express the given data in terms of their common logarithms.

Mammal	$\log w$	$\log r$
Rat	$-0.699$	$2.623$
Guinea Pig	$-0.523$	$2.477$
Rabbit	$0.301$	$2.312$
Small Dog	$0.699$	$2.079$
Large Dog	$1.477$	$1.929$
Sheep	$1.699$	$1.845$
Human	$1.845$	$1.857$

Plot $\log w$ on $x-$ axis and $\log r$ on $y-$ axis and make scatter diagram.

  

(b)

To calculate : The linear regression model $\left(\log r\right)=a\left(\log w\right)+b$ for $\left(\log w,\log r\right)$ .

The linear regression model is $\left(\log r\right)=-0.30\left(\log w\right)+2.364$ .

Given information :

The data:

Mammal	Body Weight (kg)	Pulse Rate (beats/min)
Rat	$0.2$	$420$
Guinea Pig	$0.3$	$300$
Rabbit	$2$	$205$
Small Dog	$5$	$120$
Large Dog	$30$	$85$
Sheep	$50$	$70$
Human	$70$	$72$

The scientists have found that the pulse rate $\left(r\right)$ of mammals are a power function of their body weight $\left(w\right)$ .

Calculation :

Consider the table expressed in terms of common logarithms of given data.

Mammal	$\log w$	$\log r$
Rat	$-0.699$	$2.623$
Guinea Pig	$-0.523$	$2.477$
Rabbit	$0.301$	$2.312$
Small Dog	$0.699$	$2.079$
Large Dog	$1.477$	$1.929$
Sheep	$1.699$	$1.845$
Human	$1.845$	$1.857$

The linear regression model is given by:

  $\left(\log r\right)=a\left(\log w\right)+b$

Use calculator to find the value of $a\text{ and }b$ for the given data.

  $\begin{array}{l}a\approx -0.30\\ b\approx 2.364\end{array}$

Therefore, the linear regression model is $\left(\log r\right)=-0.30\left(\log w\right)+2.364$ .

(c)

To graph : The regression curve on the scatter plot.

Given information :

The data:

Mammal	Body Weight (kg)	Pulse Rate (beats/min)
Rat	$0.2$	$420$
Guinea Pig	$0.3$	$300$
Rabbit	$2$	$205$
Small Dog	$5$	$120$
Large Dog	$30$	$85$
Sheep	$50$	$70$
Human	$70$	$72$

The scientists have found that the pulse rate $\left(r\right)$ of mammals are a power function of their body weight $\left(w\right)$ . The linear regression model is $\left(\log r\right)=-0.30\left(\log w\right)+2.364$ .

Graph :

Consider the linear regression model $\left(\log r\right)=-0.30\left(\log w\right)+2.364$ .

Make regression curve on the scatter plot using graphing calculator.

  

(d)

To calculate : The pulse rate for a $450\text{ kg}$ horse of using the regression equation and state if it’s closer to $38$ beats/min.

The pulse rate for a horse is $36.6$ beats/min.

Given information :

The data:

Mammal	Body Weight (kg)	Pulse Rate (beats/min)
Rat	$0.2$	$420$
Guinea Pig	$0.3$	$300$
Rabbit	$2$	$205$
Small Dog	$5$	$120$
Large Dog	$30$	$85$
Sheep	$50$	$70$
Human	$70$	$72$

The scientists have found that the pulse rate $\left(r\right)$ of mammals are a power function of their body weight $\left(w\right)$ . The regression equation is $\left(\log r\right)=-0.30\left(\log w\right)+2.364$ .

Calculation :

Consider the regression equation is $\left(\log r\right)=-0.30\left(\log w\right)+2.364$ .

Substitute $w=450$ .

  $\begin{array}{c}\left(\log r\right)=-0.30\left(\log 450\right)+2.364\\ =1.564\\ r={10}^{1.564}\\ \approx 36.6\end{array}$

Therefore, the pulse rate for a horse is $36.6$ beats/min and it is closer to $38$ beats/min.

(e)

To explain : The reason for using common or natural logarithms to re-express data that fits a power regression model.

The common or natural logarithms to re-express data that fits a power regression model because there is a linear relationship between $\log y\text{ and }\log x$ .

Given information :

The data:

Mammal	Body Weight (kg)	Pulse Rate (beats/min)
Rat	$0.2$	$420$
Guinea Pig	$0.3$	$300$
Rabbit	$2$	$205$
Small Dog	$5$	$120$
Large Dog	$30$	$85$
Sheep	$50$	$70$
Human	$70$	$72$

The scientists have found that the pulse rate $\left(r\right)$ of mammals are a power function of their body weight $\left(w\right)$ .

Explanation :

The power regression model is $y=a\cdot x^b$ .

Apply common logarithm to both sides.

  $\begin{array}{l}\log y=\log a+\log x^b\\ \log y=\log a+b\log x\end{array}$

It is a linear model between $\log y\text{ and }\log x$ .

Therefore, the common or natural logarithms to re-express data that fits a power regression model because there is a linear relationship between $\log y\text{ and }\log x$ .