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:
| Enrollment | 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.
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.
Answer (2)
Calculate the sum of the residuals: 90 − 50 − 100 − 100 + 150 − 50 = − 60 .
Analyze the sum of the residuals: -60 is relatively close to zero.
Conclude that the equation is a good fit because the sum of the residuals is a small number.
The final answer is: Yes, the equation is a good fit because the sum of the residuals is a small number.
Explanation
Understanding the Problem We are given a table of actual and predicted enrollment values for a dance studio over six months, along with the residuals (the difference between the actual and predicted values). We need to determine if the equation used to generate the predicted values is a good line of best fit. A good line of best fit should have residuals that are randomly distributed around zero, and the sum of the residuals should be close to zero.
Calculating the Sum of Residuals To determine if the equation is a good fit, we need to calculate the sum of the residuals. The residuals are given as 90, -50, -100, -100, 150, and -50.
Sum of Residuals The sum of the residuals is calculated as follows: 90 + ( − 50 ) + ( − 100 ) + ( − 100 ) + 150 + ( − 50 ) = 90 − 50 − 100 − 100 + 150 − 50 = − 60 The sum of the residuals is -60.
Analyzing the Sum of Residuals Now, we analyze the sum of the residuals. A sum of -60 is relatively close to zero compared to the magnitude of the actual and predicted values, which range from 400 to 750. Therefore, the equation that produced the predicted values is a reasonable line of best fit.
Conclusion Based on our analysis, the equation is a good fit because the sum of the residuals is a small number.
Examples
In business, a line of best fit can be used to predict future sales based on past data. If the sum of the residuals is small, it indicates that the predictions are generally accurate. For example, a retail store might use a line of best fit to predict monthly revenue based on historical sales data. If the residuals are small, the store can have confidence in its revenue projections, which can inform decisions about inventory, staffing, and marketing.
The equation producing the predicted values is a good fit because the sum of the residuals is -60, which is relatively small compared to the actual enrollment values. This suggests the predictions are close to the actual values and thus represent a reasonable model. Therefore, the correct answer is option D.
;
