Chapter 15, Problem 26E

(a)

To determine

To graph: A scatterplot for the provided data set A.

Explanation of Solution

Graph: A scatterplot is the graphical representation of the relationship between two quantitative variables for the same individuals. The explanatory variable is represented on the horizontal axis and the response variable is represented on the vertical axis.

In the provided problem, x is the explanatory variable and y is the response variable. To obtain the scatterplot for the provided data, Minitab is used. The steps followed to obtain the scatterplot are as follows:

Step 1: Open the Minitab file that contains the provided data.

Step 2: Go to Graph and then select Scatterplot.

Step 3: Select Simple and click Ok.

Step 4: Enter “y (A)” in the Y variables column and “x (A)” in the X variables column and click Ok.

The scatterplot appears as obtained in the Minitab file.

In the scatterplot, the horizontal axis shows the x values and the vertical axis shows the y values.

Graph: The scatterplot shows that there is a moderate linear association between the variables.

To graph: The regression line for the provided regression equation for data set A.

Explanation:

Calculation:

To draw the regression line, use the provided equation to predict the y for $x=5$.

$\begin{array}{c}y=3+0.5x\\ =3+\left(0.5\times 5\right)\\ =3+2.5\\ =5.5\end{array}$

The predicted y for $x=10$ is calculated as follows:

$\begin{array}{c}y=3+0.5x\\ =3+\left(0.5\times 10\right)\\ =3+5\\ =8\end{array}$

Therefore, the points are $\left(5,5.5\right)\text{ and }\left(10,8\right)$. Use the following steps to draw the least-squares equation on the scatterplot.

Graph:

Step 1: Open the scatterplot obtained in previous part.

Step 2: Mark the two obtained points $\left(5,5.5\right)\text{ and }\left(10,8\right)$ on the scatterplot of data set A.

Step 3: Connect the two points through a line.

The regression line appears on the scatterplot as follows:

The line that passes through the two points $\left(5,5.5\right)\text{ and }\left(10,8\right)$ is the required regression line for data set A.

To graph: A scatterplot for the provided data set B.

Explanation:

Graph: A scatterplot is the graphical representation of the relationship between two quantitative variables for the same individuals. The explanatory variable is represented on the horizontal axis and the response variable is represented on the vertical axis.

In the provided problem, x is the explanatory variable and y is the response variable. To obtain the scatterplot for the provided data, Minitab is used. The steps followed to obtain the scatterplot are as follows:

Step 1: Open the Minitab file that contains the provided data.

Step 2: Go to Graph and then select Scatterplot.

Step 3: Select Simple and click Ok.

Step 4: Enter “y (B)” in the Y variables column and “x (B)” in the X variables column and click Ok.

The scatterplot appears as obtained in the Minitab file.

In the scatterplot, the horizontal axis shows the x values and the vertical axis shows the y values.

Interpretation: The scatterplot shows that there is a nonlinear relationship or a curved relationship between the variables.

To graph: The regression line for the provided regression equation for data set B.

Explanation:

Calculation:

To draw the regression line, use the provided equation to predict the y for $x=5$.

$\begin{array}{c}y=3+0.5x\\ =3+\left(0.5\times 5\right)\\ =3+2.5\\ =5.5\end{array}$

The predicted y for $x=10$ is calculated as follows:

$\begin{array}{c}y=3+0.5x\\ =3+\left(0.5\times 10\right)\\ =3+5\\ =8\end{array}$

Therefore, the points are $\left(5,5.5\right)\text{ and }\left(10,8\right)$. Use the following steps to draw the least-squares equation on the scatterplot.

Graph:

Step 1: Open the scatterplot obtained in previous part.

Step 2: Mark the two obtained points $\left(5,5.5\right)\text{ and }\left(10,8\right)$ on the scatterplot of data set B.

Step 3: Connect the two points through a line.

The regression line appears on the scatterplot as follows:

The line that passes through the two points $\left(5,5.5\right)\text{ and }\left(10,8\right)$ is the required regression line for data set B.

To graph: A scatterplot for the provided data set C.

Explanation:

Graph: A scatterplot is the graphical representation of the relationship between two quantitative variables for the same individuals. The explanatory variable is represented on the horizontal axis and the response variable is represented on the vertical axis.

In the provided problem, x is the explanatory variable and y is the response variable. To obtain the scatterplot for the provided data, Minitab is used. The steps followed to obtain the scatterplot are as follows:

Step 1: Open the Minitab file that contains the provided data.

Step 2: Go to Graph and then select Scatterplot.

Step 3: Select Simple and click Ok.

Step 4: Enter “y (C)” in the Y variables column and “x (C)” in the X variables column and click Ok.

The scatterplot appears as obtained in the Minitab file.

In the scatterplot, the horizontal axis shows the x values and the vertical axis shows the y values.

Interpretation: The scatterplot shows that there is an extreme outlier. The association can be linear if the outlier is omitted.

To graph: The regression line for the provided regression equation for data set C.

Explanation:

Calculation:

To draw the regression line, use the provided equation to predict the y for $x=5$.

$\begin{array}{c}y=3+0.5x\\ =3+0.5\times 5\\ =3+2.5\\ =5.5\end{array}$

The predicted y for $x=10$ is calculated as

$\begin{array}{c}y=3+0.5x\\ =3+0.5\times 10\\ =3+5\\ =8\end{array}$

Therefore, the points are $\left(5,5.5\right)\text{ and }\left(10,8\right)$. Use the following steps to draw the least-squares equation on the scatterplot.

Graph:

Step 1: Open the scatterplot obtained in previous part.

Step 2: Mark the two obtained points $\left(5,5.5\right)\text{ and }\left(10,8\right)$ on the scatterplot of data set C.

Step 3: Connect the two points through a line.

The regression line appears on the scatterplot as follows:

The line that passes through the two points $\left(5,5.5\right)\text{ and }\left(10,8\right)$ is the required regression line of the data set C.

To graph: A scatterplot for the provided data set D.

Explanation:

Graph: A scatterplot is the graphical representation of the relationship between two quantitative variables for the same individuals. The explanatory variable is represented on the horizontal axis and the response variable is represented on the vertical axis.

In the provided problem, x is the explanatory variable and y is the response variable. To obtain the scatterplot for the provided data, Minitab is used. The steps followed to obtain the scatterplot are as follows:

Step 1: Open the Minitab file that contains the provided data.

Step 2: Go to Graph and then select Scatterplot.

Step 3: Select Simple and click Ok.

Step 4: Enter “y (D)” in the Y variables column and “x (D)” in the X variables column and click Ok.

The scatterplot appears as obtained in the Minitab file.

In the scatterplot, the horizontal axis shows the x values and the vertical axis shows the y values.

Interpretation: The scatterplot shows that there is an extreme influential point. The other points alone do not predict any indication for slope of the line.

To graph: The regression line for the provided regression equation for data set D.

Explanation:

Calculation:

To draw the regression line, use the provided equation to predict the y for $x=5$.

$\begin{array}{c}y=3+0.5x\\ =3+0.5\times 5\\ =3+2.5\\ =5.5\end{array}$

The predicted y for $x=10$ is calculated as

$\begin{array}{c}y=3+0.5x\\ =3+0.5\times 10\\ =3+5\\ =8\end{array}$

Therefore the points are $\left(5,5.5\right)\text{ and }\left(10,8\right)$. Use the following steps to draw the least-squares equation on the scatterplot.

Graph:

Step 1: Open the scatterplot obtained in previous part.

Step 2: Mark the two obtained points $\left(5,5.5\right)\text{ and }\left(10,8\right)$ on the scatterplot of data set D.

Step 3: Connect the two points through a line.

The regression line appears on the scatterplot as follows:

The line that passes through the two points $\left(5,5.5\right)\text{ and }\left(10,8\right)$ is the required regression line of the data set D.

(b)

To determine

To find: The data set for which the provided regression line equation is best suited to predict y for provided $x=10$.

Answer to Problem 26E

Solution: The provided regression line equation is best suited to predict y for provided $x=10$ for data set A.

Explanation of Solution

Calculation:

For the data set A, the regression line seems to have a moderate linear association between the variables. Hence the use of the regression line equation is reasonable.

For the data set B, the scatterplot does not show any obvious linear relationship. It shows a curved relationship, and hence, the regression line equation cannot fit the data set B.

For the data set C, the point $\left(13,12.74\right)$ deviates from the linear pattern of the other points. The provided regression equation may suit the data set C if this extreme point is excluded.

For the data set D, the point $\left(19,12.50\right)$ is an influential point, which is much away from the pattern of other points. The other points do not indicate any slope of the line. Hence, the regression equation cannot fit the data set D.