A data set is shown in the table. The line of best fit modeling the data is [tex]y=2.69 x-7.95[/tex].

\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
1 & -5.1 \\
\hline
2 & -3.2 \\
\hline
3 & 1.0 \\
\hline
4 & 2.3 \\
\hline
5 & 5.6 \\
\hline
\end{tabular}

What is the residual value when [tex]x=3[/tex]?

A. -0.88
B. -0.12
C. 0.12
D. 0.88



Answer :

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]