An equation was created for the line of best fit from the actual enrollment data. It was used to predict the dance studio enrollment values shown in the table below: Analyze the data. Determine whether the equation that produced the predicted values represents a good line of best fit. A. No, the equation is not a good fit because the sum of the residuals is a large number. B. No, the equation is not a good fit because the residuals are all far from zero. C. Yes, the equation is a good fit because the residuals are not all far from zero. D. Yes, the equation is a good fit because the sum of the residuals is a small number.
Mathematics · High School · Thu Feb 04 2021
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To determine whether the equation for the line of best fit represents a good fit for the actual enrollment data, we need to consider the residuals. Residuals are the differences between the actual values and the predicted values given by the line of best fit. A residual is calculated as:
Residual = Actual value - Predicted value
To assess the quality of the line of best fit, you can analyze the residuals in two key ways:
1. If the residuals are all relatively small and scattered randomly above and below the line (positive and negative), this suggests the line of best fit is appropriate for the data.
2. If the sum of the residuals is close to zero, it may suggest a good fit. However, the sum alone is not enough because positive and negative residuals could cancel each other out, giving a misleading sense of accuracy if you only consider the sum.
Options B and C consider whether the residuals are close to zero, which is an important aspect of determining a good fit. Option D incorrectly focuses on the sum of the residuals, which is not a reliable method on its own. Thus, the best answer is:
B. No, the equation is not a good fit because the residuals are all far from zero.
This option indicates that if all residuals are large (whether positive or negative), the predicted values are not close to the actual values, implying the line of best fit does not represent the data well. The sum of residuals (whether large or small) is not as relevant because it could be misleading due to the cancellation of positive and negative values.