Let's go through the calculation of each residual step by step for the given values in the table. The residual is computed by subtracting the predicted value from the given value.
1. For [tex]\( z = 1 \)[/tex]:
- Given value: [tex]\( 6.1 \)[/tex]
- Predicted value: [tex]\( 52 \)[/tex]
- Residual [tex]\( a \)[/tex]:
[tex]\[
a = 6.1 - 52 = -45.9
\][/tex]
2. For [tex]\( z = 2 \)[/tex]:
- Given value: [tex]\( 7.4 \)[/tex]
- Predicted value: [tex]\( 8.4 \)[/tex]
- Residual [tex]\( b \)[/tex]:
[tex]\[
b = 7.4 - 8.4 = -1.0
\][/tex]
3. For [tex]\( z = 3 \)[/tex]:
- Given value: [tex]\( 11 \)[/tex]
- Predicted value: [tex]\( 11.6 \)[/tex]
- Residual [tex]\( c \)[/tex]:
[tex]\[
c = 11 - 11.6 = -0.6
\][/tex]
4. For [tex]\( z = 4 \)[/tex]:
- Given value: [tex]\( 15 \)[/tex]
- Predicted value: [tex]\( 14.8 \)[/tex]
- Residual [tex]\( d \)[/tex]:
[tex]\[
d = 15 - 14.8 = 0.2
\][/tex]
So, the individual residual values needed for the table are:
[tex]\[
\begin{array}{l}
a = -45.9 \\
b = -1.0 \\
c = -0.6 \\
d = 0.2
\end{array}
\][/tex]
