To find the residual value when [tex]\( x = 3 \)[/tex], we need to follow a series of steps:
1. Determine the predicted [tex]\( y \)[/tex] value using the line of best fit:
The equation of the line of best fit is [tex]\( y = 2.69 x - 7.95 \)[/tex].
For [tex]\( x = 3 \)[/tex]:
[tex]\[
y_{\text{pred}} = 2.69 \times 3 - 7.95
\][/tex]
2. Calculate [tex]\( y_{\text{pred}} \)[/tex]:
Substitute [tex]\( x = 3 \)[/tex] into the equation:
[tex]\[
y_{\text{pred}} = 2.69 \times 3 - 7.95 = 8.07 - 7.95 = 0.12
\][/tex]
3. Given the actual [tex]\( y \)[/tex] value from the table:
When [tex]\( x = 3 \)[/tex], the actual [tex]\( y \)[/tex] value ([tex]\( y_{\text{actual}} \)[/tex]) is 1.0.
4. Calculate the residual:
The residual is the difference between the actual [tex]\( y \)[/tex] value and the predicted [tex]\( y \)[/tex] value:
[tex]\[
\text{Residual} = y_{\text{actual}} - y_{\text{pred}}
\][/tex]
Substituting the values we have:
[tex]\[
\text{Residual} = 1.0 - 0.12 = 0.88
\][/tex]
Thus, the residual value when [tex]\( x = 3 \)[/tex] is 0.88. The correct answer is:
[tex]\[
\boxed{0.88}
\][/tex]
