Chapter 14.1, Problem 11E

a.

To determine

Construct a graph for the given function.

Answer to Problem 11E

The graph for variable y is given below:

Explanation of Solution

Calculation:

The given model is $y=1.8+0.1x_1+0.8x_2+e$ and the values of $x_2$ vary, whereas the values of $x_1$ are fixed at three different values 10, 20, and 30.

For $x_1=10$:

$\begin{array}{c}y=1.8+0.1\left(10\right)+0.8x_2\\ =1.8+1+0.8x_2\\ =2.8+0.8x_2\end{array}$

For $x_1=20$:

$\begin{array}{c}y=1.8+0.1\left(20\right)+0.8x_2\\ =1.8+2+0.8x_2\\ =3.8+0.8x_2\end{array}$

For $x_1=30$:

$\begin{array}{c}y=1.8+0.1\left(30\right)+0.8x_2\\ =1.8+3+0.8x_2\\ =4.8+0.8x_2\end{array}$

Substitute different values of $x_2$ in the equations to obtain the corresponding values of y as follows:

$x_2$	$y=2.8+0.8x_2$	$y=3.8+0.8x_2$	$y=4.8+0.8x_2$
0	2.8	3.8	4.8
2	4.4	5.4	6.4
4	6	7	8
6	7.6	8.6	9.6
8	9.2	10.2	11.2
10	10.8	11.8	12.8

Procedure:

Step-by-step procedure to draw a graph for the function y:

• On the X–axis, mark the axis as the values of $x_2$.
• On the Y-axis, mark the axis at equidistant points, such as 2, 4, 6, 8, 10, 12, and 14.
• Plot the pairs of values $\left(x_2,y\right)$ obtained for $x_1=10$, $x_1=20$, and $x_1=30$, separately.
• Join the points for each value of $x_1$ to obtain the three lines separately.

Thus, the graph is obtained.

b.

To determine

Construct a graph for depicting the relationship between mean y and $x_1$.

Answer to Problem 11E

The graph for variable y is given below:

Explanation of Solution

Calculation:

The given model is $y=1.8+0.1x_1+0.8x_2+e$ and the values of $x_1$ vary, whereas the values of $x_2$ are 50, 55, and 60.

For $x_2=50$:

$\begin{array}{c}y=1.8+0.1x_1+0.8\left(50\right)\\ =1.8+0.1x_1+40\\ =41.8+0.1x_1\end{array}$

For $x_2=55$:

$\begin{array}{c}y=1.8+0.1x_1+0.8\left(55\right)\\ =1.8+0.1x_1+44\\ =45.8+0.1x_1\end{array}$

For $x_2=60$:

$\begin{array}{c}y=1.8+0.1x_1+0.8\left(60\right)\\ =1.8+0.1x_1+48\\ =49.8+0.1x_1\end{array}$

Substitute different value of $x_1$ in the equations to obtain the corresponding values of y as follows:

$x_1$	$y=41.8+0.1x_1$	$y=45.8+0.1x_1$	$y=49.8+0.1x_1$
0	41.8	45.8	49.8
5	42.3	46.3	50.3
10	42.8	46.8	50.8
15	43.3	47.3	51.3
20	43.8	47.8	51.8
25	44.3	48.3	52.3

Procedure:

Step-by-step procedure to draw a graph for the function y:

• On the X–axis, mark the axis as the values of $x_1$.
• On the Y-axis, mark the axis at equidistant points, such as 42, 44, 46, 48, 50, 52, and 54.
• Plot the pairs of values $\left(x_1,y\right)$ obtained for $x_2=50$, $x_2=55$, and $x_2=60$ separately.
• Join the points for each value of $x_2$ to obtain the three lines separately.

Thus, the graph is obtained.

c.

To determine

Explain the aspect of the graphs in Parts (a) and (b) that can be attributed to the lack of an interaction between $x_1$ and $x_2$.

Explanation of Solution

From Parts (a) and (b), the lines in each graph are parallel to each other. This implies that there is no interaction between $x_1$ and $x_2$,

d.

To determine

Construct the graph in Parts (a) and (b) with interaction effect.

Answer to Problem 11E

The graph for variable y is given below:

Explanation of Solution

Calculation:

The given model with interaction effect is $y=1.8+0.1x_1+0.8x_2+0.03x_1{x}_2+e$. When the values of $x_2$ vary, the values of $x_1$ are fixed at 10, 20, and 30. When the values of $x_1$ vary, the values of $x_2$ are fixed at 50, 55, and 60.

For variable $x_2$:

For $x_1=10$:

$\begin{array}{c}y=1.8+0.1\left(10\right)+0.8x_2+0.03x_1{x}_2\\ =1.8+1+0.8x_2+\left(0.03\times 10\times x_2\right)\\ =2.8+0.8x_2+0.3x_2\\ =2.8+1.1x_2\end{array}$

For $x_1=20$:

$\begin{array}{c}y=1.8+0.1\left(20\right)+0.8x_2+0.03x_1{x}_2\\ =1.8+2+0.8x_2+\left(0.03\times 20\times x_2\right)\\ =3.8+0.8x_2+0.6x_2\\ =3.8+1.4x_2\end{array}$

For $x_1=30$:

$\begin{array}{c}y=1.8+0.1\left(30\right)+0.8x_2+0.03x_1{x}_2\\ =1.8+3+0.8x_2+\left(0.03\times 30\times x_2\right)\\ =4.8+0.8x_2+0.9x_2\\ =4.8+1.7x_2\end{array}$

Substitute different values of $x_2$ to obtain the values of y as follows:

$x_2$	$y=2.8+1.1x_2$	$y=3.8+1.4x_2$	$y=4.8+1.7x_2$
0	2.8	3.8	4.8
2	5	6.6	8.2
4	7.2	9.4	11.6
6	9.4	12.2	15
8	11.6	15	18.4
10	13.8	17.8	21.8

Procedure:

Step-by-step procedure to draw a graph for the function y:

• On the X–axis, mark the axis as the values of $x_1$.
• On the Y-axis, mark the axis at equidistant points, such as 0, 5, 10, 15, and 20.
• Plot the pairs of values $\left(x_2,y\right)$ obtained for $x_1=10$, $x_1=20$, and $x_1=30$ separately.
• Join the points for each value of $x_1$ to obtain the three lines separately.

Thus, the graph is obtained.

For variable $x_1$:

For $x_2=50$:

$\begin{array}{c}y=1.8+0.1x_1+0.8\left(50\right)+0.03x_1\left(50\right)\\ =1.8+0.1x_1+40+1.5x_1\\ =41.8+1.6x_1\end{array}$

For $x_2=55$:

$\begin{array}{c}y=1.8+0.1x_1+0.8\left(55\right)+0.03x_1\left(55\right)\\ =1.8+0.1x_1+44+1.65x_1\\ =45.8+1.75x_1\end{array}$

For $x_2=60$:

$\begin{array}{c}y=1.8+0.1x_1+0.8\left(60\right)+0.03x_1\left(60\right)\\ =1.8+0.1x_1+48+1.8x_1\\ =49.8+1.9x_1\end{array}$

Substitute different values of $x_1$ to obtain the values of y as follows:

$x_1$	$y=41.8+0.6x_1$	$y=45.8+1.75x_1$	$y=49.8+1.9x_1$
0	41.8	45.8	49.8
5	44.8	54.55	59.3
10	47.8	63.3	68.8
15	50.8	72.05	78.3
20	53.8	80.8	87.8
25	56.8	89.55	97.3

Procedure:

Step-by-step procedure to draw a graph for y function:

• On the X–axis, mark the axis as the values of $x_2$.
• On the Y-axis, mark the axis at some random places like 0, 5, 10, 15, 20, 25.
• Plot the pairs of values $\left(x_1,y\right)$ obtained for $x_2=50$, $x_2=55$, and $x_2=60$ separately.
• Join the points for each value of $x_2$ to obtain the three lines separately.

Thus, the graph is obtained.

Both the graphs, for the fixed values of $x_1$ and for the fixed values of $x_2$, show that the lines are not parallel. This indicates that the two variables $x_1$ and $x_2$ are not free from interaction.

Thus, the graphs drawn in this part reflect the interaction between $x_1$ and $x_2$, which is absent in the graphs drawn in Parts (a) and (b).