To determine the residual for the point [tex]\((3,6)\)[/tex] given the line of best fit equation [tex]\( y = 5x - 2.5 \)[/tex], follow these steps:
1. Identify the given point and equation:
- Point [tex]\((x, y_{\text{actual}}) = (3, 6)\)[/tex]
- Line of best fit equation: [tex]\( y = 5x - 2.5 \)[/tex]
2. Calculate the predicted value [tex]\( y_{\text{predicted}} \)[/tex]:
Substitute [tex]\( x = 3 \)[/tex] into the equation [tex]\( y = 5x - 2.5 \)[/tex]:
[tex]\[
y_{\text{predicted}} = 5(3) - 2.5
\][/tex]
[tex]\[
y_{\text{predicted}} = 15 - 2.5
\][/tex]
[tex]\[
y_{\text{predicted}} = 12.5
\][/tex]
3. Determine the actual value [tex]\( y_{\text{actual}} \)[/tex]:
From the given point, [tex]\( y_{\text{actual}} = 6 \)[/tex].
4. Calculate the residual:
The residual is the difference between the actual [tex]\( y \)[/tex] value and the predicted [tex]\( y \)[/tex] value:
[tex]\[
\text{Residual} = y_{\text{actual}} - y_{\text{predicted}}
\][/tex]
Substitute the values:
[tex]\[
\text{Residual} = 6 - 12.5
\][/tex]
[tex]\[
\text{Residual} = -6.5
\][/tex]
Thus, the residual for the point [tex]\((3,6)\)[/tex] is [tex]\(-6.5\)[/tex]. Therefore, the correct answer is:
A. -6.5
