To find the residual values, we will follow these steps:
1. Subtraction for Residuals: For each [tex]\( x \)[/tex]-value, subtract the Predicted Value from the Given Value to find the residual:
[tex]\[
\text{Residual} = \text{Given} - \text{Predicted}.
\][/tex]
Let's perform these calculations:
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
\text{Residual} = -3.5 - (-1.1) = -3.5 + 1.1 = -2.4.
\][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[
\text{Residual} = -2.9 - 2 = -4.9.
\][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[
\text{Residual} = -1.1 - 5.1 = -6.2.
\][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[
\text{Residual} = 2.2 - 8.2 = -6.0.
\][/tex]
- For [tex]\( x = 5 \)[/tex]:
[tex]\[
\text{Residual} = 3.4 - 11.3 = -7.9.
\][/tex]
2. Completing the Table:
Fill in the residual values in the table:
\begin{tabular}{|c|c|c|c|}
\hline [tex]$x$[/tex] & Given & Predicted & Residual \\
\hline 1 & -3.5 & -1.1 & -2.4 \\
\hline 2 & -2.9 & 2 & -4.9 \\
\hline 3 & -1.1 & 5.1 & -6.2 \\
\hline 4 & 2.2 & 8.2 & -6.0 \\
\hline 5 & 3.4 & 11.3 & -7.9 \\
\hline
\end{tabular}
3. Analyzing the Residual Plot Data:
To determine if the line of best fit is appropriate for the data, graph the residuals on a residual plot where the [tex]\( x \)[/tex]-axis represents the original [tex]\( x \)[/tex]-values, and the [tex]\( y \)[/tex]-axis represents the residuals.
Observing the residual values:
[tex]\[
(-2.4, -4.9, -6.2, -6.0, -7.9)
\][/tex]
indicates a pattern in the residuals. The residuals are not randomly scattered; instead, they show a trend or pattern.
4. Conclusion:
Therefore, the residual plot shows that the line of best fit is not appropriate for the data because the points are not randomly scattered around the x-axis but are in a linear pattern.
Thus, the correct answer is:
No, the points are in a linear pattern.
