To find the residual values, we need to subtract the predicted values from the given values for each corresponding [tex]\(x\)[/tex]:
[tex]\[
\text{Residual} = \text{Given} - \text{Predicted}
\][/tex]
Let's compute these residuals step-by-step for each [tex]\(x\)[/tex]:
1. For [tex]\(x = 1\)[/tex]:
[tex]\[
\text{Residual} = 3.5 - 4.06 = -0.56
\][/tex]
2. For [tex]\(x = 2\)[/tex]:
[tex]\[
\text{Residual} = 2.3 - 2.09 = 0.21
\][/tex]
3. For [tex]\(x = 3\)[/tex]:
[tex]\[
\text{Residual} = 1.1 - 0.12 = 0.98
\][/tex]
4. For [tex]\(x = 4\)[/tex]:
[tex]\[
\text{Residual} = -2.2 - (-1.85) = -0.35
\][/tex]
5. For [tex]\(x = 5\)[/tex]:
[tex]\[
\text{Residual} = -4.1 - (-3.82) = -0.28
\][/tex]
So, the residual values are:
[tex]\[
[-0.56, 0.21, 0.98, -0.35, -0.28]
\][/tex]
Now that we have the residuals, let's complete the table:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
x & Given & Predicted & Residual \\
\hline
1 & 3.5 & 4.06 & -0.56 \\
\hline
2 & 2.3 & 2.09 & 0.21 \\
\hline
3 & 1.1 & 0.12 & 0.98 \\
\hline
4 & -2.2 & -1.85 & -0.35 \\
\hline
5 & -4.1 & -3.82 & -0.28 \\
\hline
\end{tabular}
\][/tex]
To determine if the line of best fit is appropriate, we would typically construct a residual plot, which is a graph of the residuals on the y-axis and the corresponding [tex]\(x\)[/tex] values on the x-axis.
The residual plot is objectively evaluated based on the distribution of residuals:
- If the residuals are randomly scattered around the horizontal axis (zero residual line), it indicates that the line of best fit is appropriate.
- If there is a discernable pattern in the residual plot (like a clear curve or linear trend), it suggests that the model may not be the best fit.
Given the residuals:
[tex]\[
[-0.56, 0.21, 0.98, -0.35, -0.28]
\][/tex]
When these residuals are plotted against the [tex]\(x\)[/tex] values, the points are somewhat scattered around the horizontal axis without any apparent pattern.
Thus, the correct interpretation based on the residual plot would be:
[tex]\[
\textbf{Yes, because the points have no clear pattern.}
\][/tex]
