To analyze if the line of best fit is appropriate for the data, we need to compute the residuals and then interpret the residual plot. A residual is the difference between an observed (given) value and a predicted value.
The formula for residuals [tex]\( R_i \)[/tex] is:
[tex]\[ R_i = \text{Given}_i - \text{Predicted}_i \][/tex]
Let's calculate the residuals for the given data:
1. For [tex]\( x = 1 \)[/tex]:
[tex]\[ R_1 = -2.7 - (-2.84) = 0.13999999999999968 \][/tex]
2. For [tex]\( x = 2 \)[/tex]:
[tex]\[ R_2 = -0.9 - (-0.81) = -0.08999999999999997 \][/tex]
3. For [tex]\( x = 3 \)[/tex]:
[tex]\[ R_3 = 1.1 - 1.22 = -0.11999999999999988 \][/tex]
4. For [tex]\( x = 4 \)[/tex]:
[tex]\[ R_4 = 3.2 - 3.25 = -0.04999999999999982 \][/tex]
5. For [tex]\( x = 5 \)[/tex]:
[tex]\[ R_5 = 5.4 - 5.28 = 0.1200000000000001 \][/tex]
Now we list the residuals in the table and analyze them:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
$x$ & Given & Predicted & Residual \\
\hline
1 & -2.7 & -2.84 & 0.14 \\
\hline
2 & -0.9 & -0.81 & -0.09 \\
\hline
3 & 1.1 & 1.22 & -0.12 \\
\hline
4 & 3.2 & 3.25 & -0.05 \\
\hline
5 & 5.4 & 5.28 & 0.12 \\
\hline
\end{tabular}
\][/tex]
To determine if the residual plot shows that the line of best fit is appropriate, we analyze the pattern of the residuals.
If the residuals are scattered randomly around the horizontal axis (with no distinct pattern), it suggests that the line of best fit is appropriate. Conversely, if there is a clear pattern in the residual plot (such as a curved pattern), this indicates that the line of best fit may not be the best model for the data.
Given the residuals:
[tex]\[ [0.14, -0.09, -0.12, -0.05, 0.12] \][/tex]
We observe that these residuals do not form a distinct pattern and are relatively evenly distributed about the [tex]\( x \)[/tex]-axis.
Therefore, the correct interpretation is:
- Yes, the points have no pattern.
This suggests that the line of best fit is appropriate for the given data.
