To find the residual points for the given data, we calculate the difference between the given values and the predicted values for each corresponding [tex]\( x \)[/tex]. The residuals are calculated as follows:
[tex]\[ \text{Residual} = \text{Given} - \text{Predicted} \][/tex]
Let's compute the residuals step-by-step for each [tex]\( x \)[/tex]:
1. For [tex]\( x = 1 \)[/tex]:
Given: [tex]\(-0.7\)[/tex]
Predicted: [tex]\(-0.28\)[/tex]
Residual: [tex]\(-0.7 - (-0.28) = -0.7 + 0.28 = -0.42\)[/tex]
2. For [tex]\( x = 2 \)[/tex]:
Given: [tex]\(2.3\)[/tex]
Predicted: [tex]\(1.95\)[/tex]
Residual: [tex]\(2.3 - 1.95 = 0.35\)[/tex]
3. For [tex]\( x = 3 \)[/tex]:
Given: [tex]\(4.1\)[/tex]
Predicted: [tex]\(4.18\)[/tex]
Residual: [tex]\(4.1 - 4.18 = -0.08\)[/tex]
4. For [tex]\( x = 4 \)[/tex]:
Given: [tex]\(7.2\)[/tex]
Predicted: [tex]\(6.41\)[/tex]
Residual: [tex]\(7.2 - 6.41 = 0.79\)[/tex]
5. For [tex]\( x = 5 \)[/tex]:
Given: [tex]\(8\)[/tex]
Predicted: [tex]\(8.64\)[/tex]
Residual: [tex]\(8 - 8.64 = -0.64\)[/tex]
After calculating the residuals, we can complete the table:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
x & \text{Given} & \text{Predicted} & \text{Residual} \\
\hline
1 & -0.7 & -0.28 & -0.42 \\
\hline
2 & 2.3 & 1.95 & 0.35 \\
\hline
3 & 4.1 & 4.18 & -0.08 \\
\hline
4 & 7.2 & 6.41 & 0.79 \\
\hline
5 & 8 & 8.64 & -0.64 \\
\hline
\end{array}
\][/tex]
So the residuals as we calculated are:
[tex]\[
[-0.42, 0.35, -0.08, 0.79, -0.64]
\][/tex]
The correct plot of these residual points against [tex]\( x \)[/tex] would show the [tex]\( x \)[/tex]-values on the horizontal axis and the corresponding residual points on the vertical axis. Each point ([tex]\( x, \text{Residual} \)[/tex]) on this plot represents the position:
[tex]\[
\begin{align*}
(1, -0.42), \\
(2, 0.35), \\
(3, -0.08), \\
(4, 0.79), \\
(5, -0.64)
\end{align*}
\][/tex]
Ensure you plot these points accurately to visualize the residuals correctly.
