To determine which of the given equations best fits the data in the table, we need to evaluate each model by calculating the sum of the squared errors (SSE) for each one. The model with the lowest SSE will be the best fit for the data. Here are the equations and the models:
1. [tex]\( y = -1.026x^2 + 1016.402x - 162075 \)[/tex]
2. [tex]\( y = -1.036x^2 + 1024.771x - 163710 \)[/tex]
3. [tex]\( y = 298.214x - 66317.667 \)[/tex]
4. [tex]\( y = 196.2x - 18710 \)[/tex]
Given data points:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Items produced } (x) & \text{Dollars of profit } (y) \\
\hline
100 & -70500 \\
200 & 50 \\
300 & 50100 \\
400 & 80300 \\
500 & 90400 \\
600 & 78000 \\
\hline
\end{array}
\][/tex]
Let's calculate the sum of squared errors (SSE) for each model.
### Model 1: [tex]\( y = -1.026x^2 + 1016.402x - 162075 \)[/tex]
Calculate the predicted [tex]\( y \)[/tex] values and SSE for Model 1:
[tex]\[
\text{SSE}_1 \approx 980515.6400000884
\][/tex]
### Model 2: [tex]\( y = -1.036x^2 + 1024.771x - 163710 \)[/tex]
Calculate the predicted [tex]\( y \)[/tex] values and SSE for Model 2:
[tex]\[
\text{SSE}_2 \approx 1981387.31
\][/tex]
### Model 3: [tex]\( y = 298.214x - 66317.667 \)[/tex]
Calculate the predicted [tex]\( y \)[/tex] values and SSE for Model 3:
[tex]\[
\text{SSE}_3 \approx 3930834054.897733
\][/tex]
### Model 4: [tex]\( y = 196.2x - 18710 \)[/tex]
Calculate the predicted [tex]\( y \)[/tex] values and SSE for Model 4:
[tex]\[
\text{SSE}_4 \approx 6601942100.0
\][/tex]
Now, comparing the SSE values for each model:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Model} & \text{SSE} \\
\hline
1 & 980515.6400000884 \\
2 & 1981387.31 \\
3 & 3930834054.897733 \\
4 & 6601942100.0 \\
\hline
\end{array}
\][/tex]
The model with the lowest SSE is Model 1 with an SSE of [tex]\( 980515.6400000884 \)[/tex].
### Conclusion
The equation that best represents the data is:
[tex]\[ y = -1.026x^2 + 1016.402x - 162075 \][/tex]
