To determine whether the equation that produced the predicted values represents a good line of best fit, we need to analyze several key components:
1. Sum of the Residuals:
- The residuals are calculated as the difference between the actual and predicted values for each month.
- Given residuals: [4, -1, -1, 1, -1, -1]
- Sum of the residuals: 4 + (-1) + (-1) + 1 + (-1) + (-1) = 1
2. Magnitude of Residuals:
- For a good fit, the residuals should not be excessively large in magnitude.
- Given residuals: [4, -1, -1, 1, -1, -1]
- Observing these residuals, we notice that 4 is significantly higher in magnitude compared to the others.
3. Good Fit Criteria:
- If the sum of the residuals is around zero, it indicates that the positive and negative errors balance each other out.
- Small residuals indicate a close prediction to the actual values, suggesting accuracy.
Conclusion:
- Sum of the Residuals: The sum of the residuals is 1, which is a small number. This suggests that overall the errors balance out well.
- Residual Magnitude: However, one of the residuals (4) is quite large in magnitude, deviating considerably from zero.
Given these observations:
- The small sum of the residuals indicates that the errors almost balance each other out.
- The statement "No, the equation is not a good fit because the sum of the residuals is a large number" is incorrect because the sum is actually small (1).
- The statement "No, the equation is not a good fit because the residuals are all far from zero" looks at the magnitude and the presence of a residual (4) far from zero could imply a poor fit.
- The statement "Yes, the equation is a good fit because the residuals are all far from zero" is incorrect because good residuals should be close to zero.
- The statement "Yes, the equation is a good fit because the sum of the residuals is a small number" correctly identifies the balance of residuals, despite one being large, the overall assessment is correct based on the given fit criteria.
Thus, the most appropriate conclusion is:
Yes, the equation is a good fit because the sum of the residuals is a small number.
