To find the residuals [tex]\( a \)[/tex] and [tex]\( b \)[/tex] in the table, we need to calculate the difference between the given values and the predicted values for the corresponding [tex]\( x \)[/tex] values.
The residual is defined as:
[tex]\[ \text{Residual} = \text{Given value} - \text{Predicted value} \][/tex]
Let's calculate the residuals step-by-step.
1. For [tex]\( x = 3 \)[/tex]:
- Given value: [tex]\( 4.1 \)[/tex]
- Predicted value: [tex]\( 4.5 \)[/tex]
- Residual: [tex]\( a \)[/tex]
Calculate [tex]\( a \)[/tex]:
[tex]\[ a = 4.1 - 4.5 \][/tex]
[tex]\[ a = -0.4 \][/tex]
2. For [tex]\( x = 4 \)[/tex]:
- Given value: [tex]\( 7.2 \)[/tex]
- Predicted value: [tex]\( 7.05 \)[/tex]
- Residual: [tex]\( b \)[/tex]
Calculate [tex]\( b \)[/tex]:
[tex]\[ b = 7.2 - 7.05 \][/tex]
[tex]\[ b = 0.15 \][/tex]
Therefore, the values of the residuals are:
[tex]\[ a = -0.4 \][/tex]
[tex]\[ b = 0.15 \][/tex]
The correct option is:
[tex]\[ a = -0.4 \text{ and } b = 0.15 \][/tex]
