To find the missing values in the residual table, we'll need to calculate both the predicted values and the given values. Residuals are given as the difference between the actual given value and the predicted value according to the line of best fit.
Starting with the equation for the line of best fit:
[tex]\[ y = -2.9x + 17.7 \][/tex]
We will calculate the predicted values for the ages provided (1, 2, 3, 4, and 5 years).
1. Age: 1 year
- Predicted Value:
[tex]\[ y = -2.9(1) + 17.7 = 14.8 \][/tex]
- Given Residual:
[tex]\[ \text{Residual} = 0.2 \][/tex]
- Given Value:
[tex]\[ \text{Given Value} = \text{Predicted Value} + \text{Residual} = 14.8 + 0.2 = 15 \][/tex]
Thus, [tex]\(a = 14.8\)[/tex].
2. Age: 2 years
- The Predicted Value provided is 11.9.
- Given Value:
[tex]\[ \text{Given Value} = 12 \][/tex]
- Residual:
[tex]\[ \text{Residual} = \text{Given Value} - \text{Predicted Value} = 12 - 11.9 = 0.1 \][/tex]
Thus, [tex]\(b = 0.1\)[/tex].
3. Age: 3 years
- Predicted Value:
[tex]\[ y = -2.9(3) + 17.7 = 9 \][/tex]
- Given Residual:
[tex]\[ \text{Residual} = 0 \][/tex]
- Given Value:
[tex]\[ \text{Given Value} = \text{Predicted Value} + \text{Residual} = 9 + 0 = 9 \][/tex]
Thus, [tex]\(c = 9\)[/tex].
4. Age: 4 years
- The Predicted Value provided is 6.1.
- Given Residual:
[tex]\[ \text{Residual} = 0.8 \][/tex]
- Given Value:
[tex]\[ \text{Given Value} = \text{Predicted Value} + \text{Residual} = 6.1 + 0.8 = 6.9 \][/tex]
Thus, [tex]\(d = 6.9\)[/tex].
5. Age: 5 years
- Predicted Value:
[tex]\[ y = -2.9(5) + 17.7 = 3.2 \][/tex]
- Given Residual:
[tex]\[ \text{Residual} = 0.8 \][/tex]
- Given Value:
[tex]\[ \text{Given Value} = \text{Predicted Value} + \text{Residual} = 3.2 + 0.8 = 4 \][/tex]
Now we have all the completed values for the residual table:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
\text{Age (years)} & \text{Given Value} & \text{Predicted Value} & \text{Residual} \\
\hline
1 & 15 & 14.8 & 0.2 \\
\hline
2 & 12 & 11.9 & 0.1 \\
\hline
3 & 9 & 9 & 0 \\
\hline
4 & 5 & 6.1 & 0.8 \\
\hline
5 & 4 & 3.2 & 0.8 \\
\hline
\end{array}
\][/tex]
The missing predicted values (a, c) and given values based on residuals (b, d) are:
[tex]\[
\begin{array}{l}
a = 14.8 \\
b = 0.1 \\
c = 9.0 \\
d = 6.9 \\
\end{array}
\][/tex]
