To calculate the residual value when [tex]\( x = 3 \)[/tex] given the line of best fit [tex]\( y = 2.69x - 7.95 \)[/tex]:
1. Identify the observed value from the data table:
- For [tex]\( x = 3 \)[/tex], the observed [tex]\( y \)[/tex] value is [tex]\( 1.0 \)[/tex].
2. Calculate the predicted [tex]\( y \)[/tex] value using the line of best fit:
- The line of best fit is given by the equation [tex]\( y = 2.69x - 7.95 \)[/tex].
- Substitute [tex]\( x = 3 \)[/tex] into this equation to get the predicted [tex]\( y \)[/tex] value:
[tex]\[
y_{\text{predicted}} = 2.69 \times 3 - 7.95
\][/tex]
- This simplifies to:
[tex]\[
y_{\text{predicted}} = 8.07 - 7.95
\][/tex]
[tex]\[
y_{\text{predicted}} = 0.12
\][/tex]
3. Calculate the residual:
- The residual is the difference between the observed [tex]\( y \)[/tex] value and the predicted [tex]\( y \)[/tex] value:
[tex]\[
\text{residual} = y_{\text{observed}} - y_{\text{predicted}}
\][/tex]
- Using the observed [tex]\( y \)[/tex] value [tex]\( 1.0 \)[/tex] and the predicted [tex]\( y \)[/tex] value [tex]\( 0.12 \)[/tex]:
[tex]\[
\text{residual} = 1.0 - 0.12
\][/tex]
[tex]\[
\text{residual} = 0.88
\][/tex]
Therefore, the residual value when [tex]\( x = 3 \)[/tex] is [tex]\( \boxed{0.88} \)[/tex].
