Chapter 29, Problem 29.47E

a.

To determine

To estimate: The parameters for the given regression model and also estimate for $\sigma$.

Explanation of Solution

Given info:

The dataset shows the calories burnt after running in a treadmill for various speed.

The variables are listed below:

No incline – 1 for 0% incline and 0 for other incline positions.

2% incline – 1 for 2% incline and 0 for other incline positions.

Indslow – 1 for MPH less than or equal to 3 and 0 for greater than 3.

MPH – mile per hour.

Justification:

From the MINITAB output, it can be observed that the parameters in the regression model are as follows:

$\text{Constant}=64.75$

$\text{MPH}=145.841$

$\text{Indslow}=-50.01$

$\text{No incline}=-145.06$

$\text{2\% incline}=-72.83$

The estimate for $\sigma$ is the standard error which is observed to be 34.2865.

b.

To determine

To find: The number of lines fitted with the given model.

To check: Whether all the slope values are identical.

To identify: Each of the fitted line.

Answer to Problem 29.47E

There are 8 lines fitted with the given model.

The slope values are not identical.

The fitted values are given below:

When Indslow, 2% incline and  No incline as 0.

$\text{Calories}=\text{64}\text{.75 + 145}\text{.841 MPH}$

When Indslow as 1, 2% incline and No incline 0.

$\text{Calories = 14}\text{.74+ 145}\text{.841 MPH}$

When Indslow, 2% incline and No incline takes 1.

$\text{Calories =}-203.15\text{+ 145}\text{.841 MPH}$

When Indslow takes 0, 2% incline takes 1 and No incline takes 1.

$\text{Calories = }-81.91\text{+ 145}\text{.841 MPH}$

When Indslow and 2% incline takes 0, No incline takes 1.

$\text{Calories =}-8.08\text{+ 145}\text{.841 MPH}$

When Indslow and 2% incline takes 1, No incline takes 0.

$\text{Calories =}-130.32\text{+ 145}\text{.841 MPH}$

When Indslow takes 0, 2% incline and No incline takes 1.

$\text{Calories =}-153.14\text{+ 145}\text{.841 MPH}$

When Indslow takes 1, 2% incline takes 0 and No incline takes 1.

$\text{Calories =}-58.09\text{ + 145}\text{.841 MPH}$

Explanation of Solution

Calculation:

From the MINITAB output, it can be observed that the regression model predicting calories using the variables No incline, 2% incline, Indslow and MPH are given below:

$\text{Calories = }\left[\begin{array}{c}\text{64}\text{.75 + 145}\text{.841 MPH }-50.01\left(\text{Indslow}\right)-\\ 145.06\left(\text{No incline}\right)-72.83\left(\text{2\% incline}\right)\end{array}\right]$

There are 8 regression lines for predicting calories under different levels of Indslow, No incline, 2% incline:

When Indslow, No incline and 2% incline takes 0, then the regression equation for predicting calories is as follows,

$\begin{array}{c}\text{Calories = 64}\text{.75 + 145}\text{.841 MPH }-50.01\left(\text{0}\right)-145.06\left(\text{0}\right)-72.83\left(\text{0}\right)\\ =\text{64}\text{.75 + 145}\text{.841 MPH}\end{array}$

When Indslow takes 1, No incline and 2% incline takes 0, then the regression equation for predicting calories is as follows,

$\begin{array}{c}\text{Calories = 64}\text{.75 + 145}\text{.841 MPH }-50.01\left(\text{1}\right)-145.06\left(\text{0}\right)-72.83\left(\text{0}\right)\\ =\left(\text{64}\text{.75}-50.01\right)\text{+ 145}\text{.841 MPH}\\ \text{=14}\text{.74+ 145}\text{.841 MPH}\end{array}$

When Indslow, No incline and 2% incline takes 1, then the regression equation for predicting calories is as follows,

$\begin{array}{c}\text{Calories = 64}\text{.75 + 145}\text{.841 MPH }-50.01\left(\text{1}\right)-145.06\left(\text{1}\right)-72.83\left(\text{1}\right)\\ =\left(\text{64}\text{.75}-50.01-145.06-72.83\right)\text{+ 145}\text{.841 MPH}\\ \text{=}-203.15\text{+ 145}\text{.841 MPH}\end{array}$

When Indslow takes 0, No incline takes 1 and 2% incline takes 0, then the regression equation for predicting calories is as follows,

$\begin{array}{c}\text{Calories = 64}\text{.75 + 145}\text{.841 MPH }-50.01\left(\text{0}\right)-145.06\left(\text{1}\right)-72.83\left(\text{0}\right)\\ =\left(\text{64}\text{.75}-145.06\right)\text{+ 145}\text{.841 MPH}\\ \text{=}-81.91\text{+ 145}\text{.841 MPH}\end{array}$

When Indslow and No incline takes 0 and 2% incline takes 1, then the regression equation for predicting calories is as follows,

$\begin{array}{c}\text{Calories = 64}\text{.75 + 145}\text{.841 MPH }-50.01\left(\text{0}\right)-145.06\left(\text{0}\right)-72.83\left(\text{1}\right)\\ =\left(\text{64}\text{.75}-72.83\right)\text{+ 145}\text{.841 MPH}\\ \text{=}-8.08\text{+ 145}\text{.841 MPH}\end{array}$

When Indslow and No incline takes 1 and 2% incline takes 0, then the regression equation for predicting calories is as follows,

$\begin{array}{c}\text{Calories = 64}\text{.75 + 145}\text{.841 MPH }-50.01\left(\text{1}\right)-145.06\left(1\right)-72.83\left(\text{0}\right)\\ =\left(\text{64}\text{.75}-50.01-145.06\right)\text{+ 145}\text{.841 MPH}\\ \text{=}-130.32\text{+ 145}\text{.841 MPH}\end{array}$

When Indslow takes 0, No incline and 2% incline takes 1, then the regression equation for predicting calories is as follows,

$\begin{array}{c}\text{Calories = 64}\text{.75 + 145}\text{.841 MPH }-50.01\left(\text{0}\right)-145.06\left(1\right)-72.83\left(\text{1}\right)\\ =\left(\text{64}\text{.75}-145.06-72.83\right)\text{+ 145}\text{.841 MPH}\\ \text{=}-153.14\text{+ 145}\text{.841 MPH}\end{array}$

When Indslow takes 1, No incline takes 0 and 2% incline takes 1, then the regression equation for predicting calories is as follows,

$\begin{array}{c}\text{Calories = 64}\text{.75 + 145}\text{.841 MPH }-50.01\left(\text{1}\right)-145.06\left(0\right)-72.83\left(\text{1}\right)\\ =\left(\text{64}\text{.75}-50.01-72.83\right)\text{+ 145}\text{.841 MPH}\\ \text{=}-58.09\text{ + 145}\text{.841 MPH}\end{array}$

Justification:

The slopes are not the same because of indicator variables Indslow, No incline, and 2% incline.

Indicator variable:

An indicator variable places the individual observation in one among the two categories; an indicator variable is coded by the values 0 and 1.

c.

To determine

To suggest: Whether the model provides a better fit for the given data.

Answer to Problem 29.47E

The model provides a better fit for the given data.

Explanation of Solution

Justification:

R-square:

The R-square is a multiple correlation coefficient and it is the square of correlation between the observed response variable and predicted response variable.

R-square tells about the fit of the model. If the R-square value is high then the model fits better.

From the MINITAB output, it can be seen that the R-square value is 99.3%. This tells that the explanatory variables could explain 99.3% of variation in predicting the calories burnt.

d.

To determine

To test: Whether there is any significance that more speed results in more calories burn.

To state: The hypotheses, test-statistic and the P-value.

Answer to Problem 29.47E

There is significance that more speed results in more calories burn.

The hypothesis used for testing the significant in the speed is given below:

$\begin{array}{l}{H}_0:{\beta }_1=0\\ H_a:{\beta }_1\ne 0\end{array}$

The test statistic value is 56.17 and P-value is 0.000.

Explanation of Solution

Justification:

The hypothesis used for testing the significant in the speed is given below:

$\begin{array}{l}{H}_0:{\beta }_1=0\\ H_a:{\beta }_1\ne 0\end{array}$

Where, ${\beta }_1$ represents the coefficient corresponding to MPH.

Test statistic:

$t=\frac{b}{S{E}_b}$

b represents the estimate for $\beta$,

$S{E}_b$ represents the standard error obtained while estimating $\beta$.

From the MINITAB output, it can be observed that the P-value corresponding to the variable MPH is 0.000 which is lesser than the level of significance $\left(\alpha =0.05\right)$. Thus, the null hypothesis is rejected. Also, the test statistic value is 56.17.

Hence, conclude that there is significance that more speed results in more calories burn.

Conclusion:

From this data and the output, it is clear that burning calories depend on the individual and it is not in the hands of other things like equipments because the variable MPH turns out to be significant and its coefficient is 145.841. Thus, if speed increases calories burn will also increase.