Question

An equation was created for the line of best fit from actual enrollment data. It was used to predict the dance studio enrollment values shown in the table:

Enrollment Month
January February March April May June
Actual 500 400 550 550 750 400
Predicted 410 450 650 650 600 450
Residual 90 −50 −100 −100 150 −50


Analyze the data. Determine whether the equation that produced the predicted values represents a good line of best fit.
No, the equation is not a good fit because the sum of the residuals is a large number.
No, the equation is not a good fit because the residuals are all far from zero.
Yes, the equation is a good fit because the residuals are not all far from zero.
Yes, the equation is a good fit because the sum of the residuals is a small number.

Answer 1

The quality of the line of best fit cannot be concluded definitively from the sum or range of the residuals alone; additional information about the variability of the residuals or a goodness-of-fit statistic is required.The equation for the line of best fit is good or not, we need to analyze the given residuals. Residuals represent the differences between actual values and predicted values. A good line of best fit would have residuals that are relatively small and randomly distributed, showing no pattern. While the sum of residuals can help in identifying systematic errors (a non-zero sum may indicate bias), the more critical assessment is often the size of the residuals relative to the data range and whether there's a pattern.

In this case, the residuals are quite varied, ranging from -100 to 150. There is no clear pattern such as a consistent overestimation or underestimation across all months, which suggests that there is no systematic error. However, some residuals are relatively large compared to the actual enrollment numbers, which could be a sign of a poor fit. Nevertheless, to truly assess the quality of the fit, we would need more information such as the variability of the residuals and possibly the calculation of a goodness-of-fit statistic like R-squared.

Therefore, based on the given information, we cannot conclude that the equation is not a good fit simply because the sum of the residuals is not zero or because some residuals are far from zero. It is more complex than that, and so the appropriate response is that we don't have enough information to make a definitive judgment about the quality of the regression equation's fit.

Answer 2

Explanation:

AI-generated answer

No, the equation is not a good fit because the residuals are all far from zero.

To determine whether the equation that produced the predicted values represents a good line of best fit, we need to analyze the residuals. The residuals are the differences between the actual enrollment values and the predicted enrollment values.

Looking at the table provided, we can see that the residuals for each month are 90, -50, -100, -100, 150, and -50, respectively. The residuals represent the error in the prediction made by the equation.

In a good line of best fit, we would expect the residuals to be small and balanced, with some positive and some negative values. This indicates that the predicted values are close to the actual values. However, in this case, all the residuals are far from zero, with large positive and negative values.

Therefore, we can conclude that the equation that produced the predicted values does not represent a good line of best fit because the residuals are all far from zero.