Answer :
To determine whether the equation that produced the predicted values is a good fit for the actual dance studio enrollment data, we need to analyze the residuals. Residuals are the differences between the actual and predicted values. For each month, we have the residuals as follows:
- January: 500 (Actual) - 410 (Predicted) = 90
- February: 400 (Actual) - 450 (Predicted) = -50
- March: 550 (Actual) - 650 (Predicted) = -100
- April: 550 (Actual) - 650 (Predicted) = -100
- May: 750 (Actual) - 600 (Predicted) = 150
- June: 400 (Actual) - 450 (Predicted) = -50
These residuals are: 90, -50, -100, -100, 150, -50.
First, let's find the sum of these residuals:
[tex]\[ 90 + (-50) + (-100) + (-100) + 150 + (-50) = 90 - 50 - 100 - 100 + 150 - 50 = -60 \][/tex]
The sum of the residuals is -60.
Now we need to determine if this indicates a good fit.
1. Sum of Residuals Close to Zero:
- A good line of best fit usually has a sum of residuals that is close to zero, indicating that the positive and negative errors balance out.
- In this case, the sum is -60, which suggests it is not close enough to zero for a perfect balance.
2. Magnitude of Residuals:
- We should also look at whether the individual residuals are small or large. If they are all close to zero, it would indicate that the predicted values are close to the actual values even if the sum isn’t exactly zero.
- Here, we see residuals of: 90, -50, -100, -100, 150, and -50. Although some are close to zero, others (like 100, -100, and 150) are quite large.
Considering these points, the residuals are not all small, and the sum of the residuals is not very close to zero. Hence, the equation does not seem to represent a good line of best fit.
Thus, the correct answer is:
No, the equation is not a good fit because the residuals are all far from zero.
- January: 500 (Actual) - 410 (Predicted) = 90
- February: 400 (Actual) - 450 (Predicted) = -50
- March: 550 (Actual) - 650 (Predicted) = -100
- April: 550 (Actual) - 650 (Predicted) = -100
- May: 750 (Actual) - 600 (Predicted) = 150
- June: 400 (Actual) - 450 (Predicted) = -50
These residuals are: 90, -50, -100, -100, 150, -50.
First, let's find the sum of these residuals:
[tex]\[ 90 + (-50) + (-100) + (-100) + 150 + (-50) = 90 - 50 - 100 - 100 + 150 - 50 = -60 \][/tex]
The sum of the residuals is -60.
Now we need to determine if this indicates a good fit.
1. Sum of Residuals Close to Zero:
- A good line of best fit usually has a sum of residuals that is close to zero, indicating that the positive and negative errors balance out.
- In this case, the sum is -60, which suggests it is not close enough to zero for a perfect balance.
2. Magnitude of Residuals:
- We should also look at whether the individual residuals are small or large. If they are all close to zero, it would indicate that the predicted values are close to the actual values even if the sum isn’t exactly zero.
- Here, we see residuals of: 90, -50, -100, -100, 150, and -50. Although some are close to zero, others (like 100, -100, and 150) are quite large.
Considering these points, the residuals are not all small, and the sum of the residuals is not very close to zero. Hence, the equation does not seem to represent a good line of best fit.
Thus, the correct answer is:
No, the equation is not a good fit because the residuals are all far from zero.
