Sure! Let's go through the problem step by step to determine the correct values for [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
### Given Information
1. Shanti's line of best fit equation is [tex]\( y = 2.55x - 3.15 \)[/tex].
2. We are given the following table:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
x & \text{Given} & \text{Predicted} & \text{Residual} \\
\hline
1 & -0.7 & -0.6 & -0.1 \\
\hline
2 & 2.3 & 1.95 & 0.35 \\
\hline
3 & 4.1 & 4.5 & a \\
\hline
4 & 7.2 & 7.05 & b \\
\hline
\end{array}
\][/tex]
### Understanding Residuals
A residual is the difference between the given value (observed value) and the predicted value, which is calculated as:
[tex]\[ \text{Residual} = \text{Given} - \text{Predicted} \][/tex]
### Calculating Residual [tex]\( a \)[/tex]
For [tex]\( x = 3 \)[/tex]:
- Given value: 4.1
- Predicted value: From the line of best fit, [tex]\( y = 2.55(3) - 3.15 = 4.5 \)[/tex]
Thus, the residual [tex]\(a\)[/tex] is:
[tex]\[ a = 4.1 - 4.5 = -0.4 \][/tex]
### Calculating Residual [tex]\( b \)[/tex]
For [tex]\( x = 4 \)[/tex]:
- Given value: 7.2
- Predicted value: From the line of best fit, [tex]\( y = 2.55(4) - 3.15 = 7.05 \)[/tex]
Thus, the residual [tex]\(b\)[/tex] is:
[tex]\[ b = 7.2 - 7.05 = 0.15 \][/tex]
### Conclusion
The correct values for the residuals are:
[tex]\[ a = -0.4 \][/tex]
[tex]\[ b = 0.15 \][/tex]
Therefore, the correct option is:
[tex]\[ \boxed{a = -0.4 \text{ and } b = 0.15} \][/tex]
