Shanti wrote the predicted values for a data set using the line of best fit [tex]y=2.55x-3.15[/tex]. She computed two of the residual values.

[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
$x$ & Given & Predicted & Residual \\
\hline
1 & -0.7 & -0.6 & -0.1 \\
\hline
2 & 2.3 & 1.95 & 0.35 \\
\hline
3 & 4.1 & 4.5 & $a$ \\
\hline
4 & 7.2 & 7.05 & $b$ \\
\hline
\end{tabular}
\][/tex]

What are the values of [tex]$a$[/tex] and [tex]$b$[/tex]?

A. [tex]$a=0.4$[/tex] and [tex]$b=-0.15$[/tex]

B. [tex]$a=-0.4$[/tex] and [tex]$b=0.15$[/tex]

C. [tex]$a=8.6$[/tex] and [tex]$b=14.25$[/tex]

D. [tex]$a=-8.6$[/tex] and [tex]$b=-14.25$[/tex]



Answer :

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]