A set of data points has a line of best fit of [tex]$y=-0.2 x+1.8$[/tex]. What is the residual for the point [tex]$(5,1)$[/tex]?

A. 0.2
B. -0.2
C. 1.6
D. 0.8



Answer :

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