To solve this problem, we first need to calculate the residuals for each data point. The residual is the difference between the given (observed) value and the predicted value for each [tex]\( x \)[/tex].
The formula for the residuals is:
[tex]\[ \text{Residual} = \text{Given value} - \text{Predicted value} \][/tex]
Let's calculate the residuals step-by-step for each value of [tex]\( x \)[/tex]:
### Step-by-Step Calculation:
For [tex]\( x = 1 \)[/tex]:
[tex]\[ \text{Residual} = -2.7 - (-2.84) = -2.7 + 2.84 = 0.14 \][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[ \text{Residual} = -0.9 - (-0.81) = -0.9 + 0.81 = -0.09 \][/tex]
For [tex]\( x = 3 \)[/tex]:
[tex]\[ \text{Residual} = 1.1 - 1.22 = -0.12 \][/tex]
For [tex]\( x = 4 \)[/tex]:
[tex]\[ \text{Residual} = 3.2 - 3.25 = -0.05 \][/tex]
For [tex]\( x = 5 \)[/tex]:
[tex]\[ \text{Residual} = 5.4 - 5.28 = 0.12 \][/tex]
### Summary of Residuals:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline x & \text{Given} & \text{Predicted} & \text{Residual} \\
\hline 1 & -2.7 & -2.84 & 0.14 \\
\hline 2 & -0.9 & -0.81 & -0.09 \\
\hline 3 & 1.1 & 1.22 & -0.12 \\
\hline 4 & 3.2 & 3.25 & -0.05 \\
\hline 5 & 5.4 & 5.28 & 0.12 \\
\hline
\end{array}
\][/tex]
### Analysis of Residual Plot for Line of Best Fit:
To determine if the line of best fit is appropriate, we need to look at the pattern of the residuals. In a suitable line of best fit, residuals should not exhibit any specific pattern and should be randomly distributed about the x-axis.
From our residuals:
[tex]\[
[0.14, -0.09, -0.12, -0.05, 0.12]
\][/tex]
These residuals do not follow any specific increasing or decreasing trend, nor do they form any specific pattern (e.g., a curved pattern). Thus, they are randomly distributed.
### Conclusion on Line of Best Fit:
Based on the residuals we found, the residual plot shows that the points have no pattern.
Therefore, the answer to the question, "Does the residual plot show that the line of best fit is appropriate for the data?" is:
Yes, the points have no pattern.
