Chapter 28, Problem 28.41E

(a)

To determine

To use the Minitab output to estimate each parameter in this multiple regression model for predicting calories burned with the Cybex machine and estimate $\sigma$ .

Answer to Problem 28.41E

  $\text{Calories}=64.75+145.841\text{MPH}-50.01\text{ Ind\_slow}-145.06\text{ NoIncline}-72.83\text{ 2\%Incline}$ .

Explanation of Solution

In the question, it is given that,

  $\begin{array}{l}Ind\_slow=1\text{ for MPH}\le \text{3}\\ Ind\_slow=0\text{ for MPH}>\text{3}\\ NonIncline=1\text{ for 0\%incline}\\ NonIncline=0\text{ for other inclines}\\ \text{2\%}Incline=1\text{ for 0\%incline}\\ \text{2\%}Incline=0\text{ for other inclines}\end{array}$

And a Minitab output is shown fitting a multiple regression model to predict calories form MPH, Ind_slow, NoIncline and $\text{2\%}$ Incline for a man.

Thus, the parameter in this multiple regression model for predicting calories burned with the Cybex machine can be modeled as, from the output Minitab:

  $\text{Calories}=64.75+145.841\text{MPH}-50.01\text{ Ind\_slow}-145.06\text{ NoIncline}-72.83\text{ 2\%Incline}$

And the estimated $\sigma$ is from the Minitab as:

  $s=34.2865$ .

(b)

To determine

To find out how many separate lines are fitted with this model and explain do the lines all have the same slope and identify each fitted line.

Answer to Problem 28.41E

There are eight separate lines that are fitted with this model and no, all the lines do not have the same slope.

Explanation of Solution

In the question, it is given that,

  $\begin{array}{l}Ind\_slow=1\text{ for MPH}\le \text{3}\\ Ind\_slow=0\text{ for MPH}>\text{3}\\ NonIncline=1\text{ for 0\%incline}\\ NonIncline=0\text{ for other inclines}\\ \text{2\%}Incline=1\text{ for 0\%incline}\\ \text{2\%}Incline=0\text{ for other inclines}\end{array}$

And a Minitab output is shown fitting a multiple regression model to predict calories form MPH, Ind_slow, NoIncline and $\text{2\%}$ Incline for a man.Thus, the parameter in this multiple regression model for predicting calories burned with the Cybex machine can be modeled as, from the output Minitab:

  $\text{Calories}=64.75+145.841\text{MPH}-50.01\text{ Ind\_slow}-145.06\text{ NoIncline}-72.83\text{ 2\%Incline}$

Now, there are eightseparate lines that are fitted with this model and no,all the lines do not have the same slope because as we see in the Minitab output for this model and all the slopes are different. The lines that are fitted with this model are as follows:

  $\begin{array}{l}\text{Calories}=64.75+145.841\text{MPH}-50.01\text{ Ind\_slow}-145.06\text{ NoIncline}-72.83\text{ 2\%Incline}\\ =64.75+145.841\text{MPH}-50.01(0)-145.06(1)-72.83(1)\\ =-153.14+145.841\text{MPH}\end{array}$

  $\begin{array}{l}\text{Calories}=64.75+145.841\text{MPH}-50.01\text{ Ind\_slow}-145.06\text{ NoIncline}-72.83\text{ 2\%Incline}\\ =64.75+145.841\text{MPH}-50.01(0)-145.06(0)-72.83(1)\\ =-8.08+145.841\text{MPH}\end{array}$

  $\begin{array}{l}\text{Calories}=64.75+145.841\text{MPH}-50.01\text{ Ind\_slow}-145.06\text{ NoIncline}-72.83\text{ 2\%Incline}\\ =64.75+145.841\text{MPH}-50.01(0)-145.06(0)-72.83(0)\\ =64.75+145.841\text{MPH}\end{array}$

  $\begin{array}{l}\text{Calories}=64.75+145.841\text{MPH}-50.01\text{ Ind\_slow}-145.06\text{ NoIncline}-72.83\text{ 2\%Incline}\\ =64.75+145.841\text{MPH}-50.01(0)-145.06(1)-72.83(0)\\ =-80.31+145.841\text{MPH}\end{array}$

  $\begin{array}{l}\text{Calories}=64.75+145.841\text{MPH}-50.01\text{ Ind\_slow}-145.06\text{ NoIncline}-72.83\text{ 2\%Incline}\\ =64.75+145.841\text{MPH}-50.01(1)-145.06(1)-72.83(1)\\ =-203.15+145.841\text{MPH}\end{array}$

  $\begin{array}{l}\text{Calories}=64.75+145.841\text{MPH}-50.01\text{ Ind\_slow}-145.06\text{ NoIncline}-72.83\text{ 2\%Incline}\\ =64.75+145.841\text{MPH}-50.01(1)-145.06(0)-72.83(1)\\ =-58.09+145.841\text{MPH}\end{array}$

  $\begin{array}{l}\text{Calories}=64.75+145.841\text{MPH}-50.01\text{ Ind\_slow}-145.06\text{ NoIncline}-72.83\text{ 2\%Incline}\\ =64.75+145.841\text{MPH}-50.01(1)-145.06(0)-72.83(0)\\ =14.74+145.841\text{MPH}\end{array}$

  $\begin{array}{l}\text{Calories}=64.75+145.841\text{MPH}-50.01\text{ Ind\_slow}-145.06\text{ NoIncline}-72.83\text{ 2\%Incline}\\ =64.75+145.841\text{MPH}-50.01(1)-145.06(1)-72.83(0)\\ =-130.32+145.841\text{MPH}\end{array}$

(c)

To determine

To explain do you think that this model provides a good fit for these data.

Answer to Problem 28.41E

Yes, this model provides a good fit for these data.

Explanation of Solution

In the question, it is given that,

  $\begin{array}{l}Ind\_slow=1\text{ for MPH}\le \text{3}\\ Ind\_slow=0\text{ for MPH}>\text{3}\\ NonIncline=1\text{ for 0\%incline}\\ NonIncline=0\text{ for other inclines}\\ \text{2\%}Incline=1\text{ for 0\%incline}\\ \text{2\%}Incline=0\text{ for other inclines}\end{array}$

And a Minitab output is shown fitting a multiple regression model to predict calories form MPH, Ind_slow, NoIncline and $\text{2\%}$ Incline for a man. Thus, the parameter in this multiple regression model for predicting calories burned with the Cybex machine can be modeled as, from the output Minitab:

  $\text{Calories}=64.75+145.841\text{MPH}-50.01\text{ Ind\_slow}-145.06\text{ NoIncline}-72.83\text{ 2\%Incline}$

Since the $R^2=99.3\%$ that means $99.3\%$ of the variation is explained by the MPH in the model and that is why this model provides a good fit for these data.

(d)

To determine

To explain is there significant that more calories are burned for higher speeds and state the hypotheses and identify the test statistics and P-value and provide a conclusion.

Answer to Problem 28.41E

Yes, there is significant that more calories are burned for higher speeds and P-value is zero and $F=1819.18$ .

Explanation of Solution

In the question, it is given that,

  $\begin{array}{l}Ind\_slow=1\text{ for MPH}\le \text{3}\\ Ind\_slow=0\text{ for MPH}>\text{3}\\ NonIncline=1\text{ for 0\%incline}\\ NonIncline=0\text{ for other inclines}\\ \text{2\%}Incline=1\text{ for 0\%incline}\\ \text{2\%}Incline=0\text{ for other inclines}\end{array}$

And a Minitab output is shown fitting a multiple regression model to predict calories form MPH, Ind_slow, NoIncline and $\text{2\%}$ Incline for a man. Thus, the parameter in this multiple regression model for predicting calories burned with the Cybex machine can be modeled as, from the output Minitab:

  $\text{Calories}=64.75+145.841\text{MPH}-50.01\text{ Ind\_slow}-145.06\text{ NoIncline}-72.83\text{ 2\%Incline}$

The hypothesis will be defined as:

Null hypothesis: There is no significant difference between them.

Alternative hypothesis: There is significant evidence that more calories are burned for higher speeds.

Since we can see that the P-value is zero i.e. $P<0.05\Rightarrow \text{Reject }{H}_0$ . Thus, we can conclude that we have sufficient evidence to show that there is significant that more calories are burned for higher speeds. The test statistics value is $F=1819.18$ .