To complete the residual table, we'll follow these steps to find the missing values:
1. Determine the predicted value when [tex]\( \text{Age} = 1 \)[/tex]:
The predicted value formula is given by [tex]\( y = -20 \cdot \text{Age} + 177 \)[/tex].
For [tex]\( \text{Age} = 1 \)[/tex]:
[tex]\[
y = -20 \cdot 1 + 177 = 157
\][/tex]
So, [tex]\( a = 157 \)[/tex].
2. Calculate the residual when [tex]\( \text{Age} = 2 \)[/tex]:
Given value for [tex]\( \text{Age} = 2 \)[/tex] is 12.
Predicted value when [tex]\( \text{Age} = 2 \)[/tex] is 11.9 (from the table).
The residual (difference) is:
[tex]\[
\text{Given value} - \text{Predicted value} = 12 - 11.9 = 0.1
\][/tex]
Hence, [tex]\( b = 0.1 \)[/tex].
3. Determine the predicted value when [tex]\( \text{Age} = 3 \)[/tex]:
For [tex]\( \text{Age} = 3 \)[/tex]:
[tex]\[
y = -20 \cdot 3 + 177 = 117
\][/tex]
Given value for [tex]\( \text{Age} = 3 \)[/tex] is 9.
Residual is given as 0.
Since Residual = [tex]\(\text{Given value} - \text{Predicted value} \)[/tex],
confirming the predicted value, [tex]\( \text{Predicted value} = \text{Given value} - 0 = 9 \)[/tex].
But we need to state the correct predicted value as:
So, [tex]\( c = 117 \)[/tex].
4. Calculate the residual when [tex]\( \text{Age} = 4 \)[/tex]:
Given value for [tex]\( \text{Age} = 4 \)[/tex] is 5.
Predicted value when [tex]\( \text{Age} = 4 \)[/tex] is 6.1 (from the table).
The residual (difference) is:
[tex]\[
\text{Given value} - \text{Predicted value} = 5 - 6.1 = -1.1
\][/tex]
Hence, [tex]\( d = -1.1 \)[/tex].
Therefore, the values that complete the table are as follows:
[tex]\[
\begin{align*}
a & = 157 \\
b & = 0.1 \\
c & = 117 \\
d & = -1.1 \\
\end{align*}
\][/tex]
