Question

Suppose the mean value E(y) of a response y is related to the quantitative independent variables x₁and x₂

$\mathit{E}\left(y\right)\mathbf{=}\mathbf{2}\mathbf{+}{\mathit{x}}_{\mathbf{1}}\mathbf{-}\mathbf{3}{\mathit{x}}_{\mathbf{2}}\mathbf{-}{\mathit{x}}_{\mathbf{1}}{\mathit{x}}_{\mathbf{2}}$

a) Identify and interpret the slope for${\mathit{x}}_{\mathbf{2}}$

b) Plot the linear relationship between E(y) and$x_2$for role="math" localid="1649796003444" ${\mathit{x}}_{\mathbf{1}}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{2}$, whererole="math" localid="1649796025582" $\mathbf{1}\mathbf{\le }{\mathit{x}}_{\mathbf{2}}\mathbf{\le }\mathbf{3}$

c) How would you interpret the estimated slopes?

d) Use the lines you plotted in part b to determine the changes in E(y) for eachrole="math" localid="1649796051071" ${\mathit{x}}_{\mathbf{1}}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{2}$.

e) Use your graph from part b to determine how much E(y) changes whenrole="math" localid="1649796075921" $\mathbf{3}\mathbf{\le }{\mathit{x}}_{\mathbf{1}}\mathbf{\le }\mathbf{5}$androle="math" localid="1649796084395" $\mathbf{1}\mathbf{\le }{\mathit{x}}_{\mathbf{2}}\mathbf{\le }\mathbf{3}$.

Short answer

a) The slope of _{$x_2$}from the equation can be seen is -3. A negative value indicates that $x_2$has an inverse relation with y and a higher value denotes that its of high magnitude.

b) Graph

c) For every change in the value of $x_1$out slope of the line changes and the line becomes steeper.

d) For the given value of $x_2$between $1\le x_2\le 3$, the changes in the value of $x_1$makes the slope of the line becomes steeper as the slope parameter increases from 3 to 4 to 5 for the values of $x_1$ as 0, 1, and 2.

e) E(y) changes by 1 to 17 to units when the value of $x_2$is $1\le x_2\le 3$and $x_1$ is$3\le x_1\le 5$

Step-by-step solution

Slope of x2

The slope of$x_2$from the equation that can be seen is -3. A negative value indicates that$x_2$ has an inverse relation with y and a higher value denotes that its of high magnitude.

Graph

Given

$$\begin{gathered} E\left(y\right)=2+x_1-3x_2-x_1{x}_{2}for{x}_1=0 \\ =2+0-3x_2-0\times x_2 \\ =2-3x_2 \end{gathered}$$

Now to plot this equation, make a table

Y	-1	-7
X2	1	3

Given

$$\begin{gathered} E\left(y\right)=2+x_1-3x_2-x_1{x}_{2}for{x}_1=1 \\ =2+1-3x_2-1\times x_2 \\ =3-4x_2 \end{gathered}$$

Now to plot this equation, make a table

Y	-1	-9
X2	1	3

Given

$$\begin{gathered} E\left(y\right)=2+x_1-3x_2-x_1{x}_{2}for{x}_1=2 \\ =2+2-3x_2-2\times x_2 \\ =4-5x_2 \end{gathered}$$

Now to plot this equation, make a table

Y	-1	-11
X2	1	3

Interpretation of graph

As it can be seen in the graph, for the value of$x_1=0,1,2$ , E(y) passes through (1, -1) when$x_2$ is. $1\le x_2\le 3$And for every change in the value of$x_1$out slope of the line changes and the line becomes steeper.

Explanation of the slope

For the given value of $x_2$between $1\le x_2\le 3$, the changes in the value of $x_1$makes the slope of the line becomes steeper as the slope parameter increases from 3 to 4 to 5 for the values of $x_1$as 0, 1, and 2.

Changes in E(y)

E(y) changes by 1 to 17 to units when the value of $x_2$is $1\le x_2\le 3$and $x_1$is $3\le x_1\le 5$

Given,

role="math" localid="1649798013525" $$\begin{gathered} E\left(y\right)=2+x_1-3x_2-x_1{x}_{2}for{x}_1=3and{x}_2=1 \\ y=2+3-3\times 1-3\times 1 \\ y=-1 \end{gathered}$$

Given,

$$\begin{gathered} E\left(y\right)=2+x_1-3x_2-x_1{x}_{2}for{x}_1=5and{x}_2=3 \\ y=2+5-3\times 3-5\times 3 \\ y=-17 \end{gathered}$$