To determine the residual for the point [tex]\((5,1)\)[/tex] given the line of best fit [tex]\( y = -0.2x + 1.8 \)[/tex], we need to perform the following steps:
1. Calculate the predicted [tex]\( y \)[/tex]-value ([tex]\( y_{\text{pred}} \)[/tex]) using the line of best fit:
[tex]\[
y_{\text{pred}} = -0.2 \cdot x + 1.8
\][/tex]
Substitute [tex]\( x = 5 \)[/tex]:
[tex]\[
y_{\text{pred}} = -0.2 \cdot 5 + 1.8
\][/tex]
[tex]\[
y_{\text{pred}} = -1 + 1.8
\][/tex]
[tex]\[
y_{\text{pred}} = 0.8
\][/tex]
2. Calculate the residual:
The residual is the difference between the actual [tex]\( y \)[/tex]-value and the predicted [tex]\( y \)[/tex]-value:
[tex]\[
\text{Residual} = y_{\text{actual}} - y_{\text{pred}}
\][/tex]
For the point [tex]\((5,1)\)[/tex], the actual [tex]\( y \)[/tex]-value [tex]\( y_{\text{actual}} = 1 \)[/tex]:
[tex]\[
\text{Residual} = 1 - 0.8
\][/tex]
[tex]\[
\text{Residual} = 0.2
\][/tex]
Therefore, the residual for the point [tex]\((5,1)\)[/tex] is [tex]\( 0.2 \)[/tex]. The correct answer is:
A. 0.2
