Sure, let's solve the problem step-by-step.
We have a table with given values, predicted values, and residuals, and we need to determine the missing residuals [tex]\( g \)[/tex] and [tex]\( h \)[/tex].
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
$x$ & Given & Predicted & Residual \\
\hline
1 & 6 & 7 & -1 \\
\hline
2 & 12 & 11 & 1 \\
\hline
3 & 13 & 15 & $g$ \\
\hline
4 & 20 & 19 & $h$ \\
\hline
\end{tabular}
\][/tex]
The residual value is calculated using the formula:
[tex]\[ \text{Residual} = \text{Given} - \text{Predicted} \][/tex]
We can apply this formula to find the missing residuals [tex]\( g \)[/tex] and [tex]\( h \)[/tex].
For [tex]\( x = 3 \)[/tex]:
[tex]\[ g = \text{Given value at } x=3 - \text{Predicted value at } x=3 \][/tex]
[tex]\[ g = 13 - 15 \][/tex]
[tex]\[ g = -2 \][/tex]
For [tex]\( x = 4 \)[/tex]:
[tex]\[ h = \text{Given value at } x=4 - \text{Predicted value at } x=4 \][/tex]
[tex]\[ h = 20 - 19 \][/tex]
[tex]\[ h = 1 \][/tex]
Thus, the missing residual values are:
[tex]\[ g = -2 \text{ and } h = 1 \][/tex]
Looking at the provided options:
- [tex]\( g = 2 \)[/tex] and [tex]\( h = -1 \)[/tex]
- [tex]\( g = 28 \)[/tex] and [tex]\( h = 39 \)[/tex]
- [tex]\( g = -2 \)[/tex] and [tex]\( h = 1 \)[/tex]
- [tex]\( g = -28 \)[/tex] and [tex]\( h = -39 \)[/tex]
The correct option is:
[tex]\[ g = -2 \text{ and } h = 1 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{-2 \text{ and } 1} \][/tex]
Which corresponds to the third option in the provided list:
[tex]\[ g = -2 \text{ and } h = 1 \][/tex]
