To analyze whether the equation that produced the predicted values represents a good line of best fit, we need to closely examine the residuals. Residuals are the differences between the actual enrollment values and the predicted enrollment values. Mathematically, a residual is calculated as:
[tex]\[ \text{Residual} = \text{Actual Value} - \text{Predicted Value} \][/tex]
Given the data in the table:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{Month} & \text{January} & \text{February} & \text{March} & \text{April} & \text{May} & \text{June} \\
\hline
\text{Actual} & 12 & 15 & 15 & 12 & 17 & 15 \\
\hline
\text{Predicted} & 8 & 15 & -1 & 1 & -1 & -1 \\
\hline
\text{Residual} & 4 & 0 & 16 & 11 & 18 & 16 \\
\hline
\end{array}
\][/tex]
From the table, the residuals are: 4, 0, 16, 11, 18, and 16.
To determine if the equation is a good fit, we analyze the residuals:
1. Check the residual values: If the model was a good fit, we would expect the residuals to be close to zero because the predicted values would closely match the actual values. However, in this case, most of the residuals (4, 16, 11, 18, 16) are far from zero, indicating large discrepancies between the predicted and actual values.
2. Sum of the residuals: While the sum of the residuals can provide insight, it is not a sufficient standalone measure of fit quality. The sum of the residuals is 65, which is a substantial number given the data provided.
Therefore, based on the large residual values, it is evident that the predicted values do not closely match the actual values. This analysis suggests that:
- No, the equation is not a good fit because the residuals are all far from zero.
The sum of the residuals being a small number is irrelevant here because the residuals themselves (4, 0, 16, 11, 18, and 16) are quite large, indicating poor predictive accuracy. Thus, the equation that produced these predicted values is not a good line of best fit.
