Which regression equation best fits these data?
| x | y |
| --- | --- |
| -4 | 8 |
| -3 | 12 |
| -2 | 14 |
| -1 | 16 |
| 1 | 15 |
| 2 | 12 |
| 3 | 9 |
| 4 | 5 |
A. [tex]y=-0.43 x+11.34[/tex]
B. [tex]y=-0.58 x^2-0.43 x+15.75[/tex]
C. [tex]y=10.72 \cdot 0.95^x[/tex]
Answer (1)
Calculate the Sum of Squared Errors (SSE) for each regression equation.
For equation A ( y = − 0.43 x + 11.34 ), the SSE is 88.6188.
For equation B ( y = − 0.58 x 2 − 0.43 x + 15.75 ), the SSE is 0.8252.
For equation C ( y = 10.72 × 0.9 5 x ), the SSE is 96.1949. Therefore, equation B best fits the data: y = − 0.58 x 2 − 0.43 x + 15.75
Explanation
Problem Analysis We are given a set of data points and three potential regression equations. Our goal is to determine which equation best fits the data. To do this, we will calculate the sum of squared errors (SSE) for each equation. The equation with the smallest SSE will be considered the best fit.
Equation A First, let's consider equation A: y = − 0.43 x + 11.34 . We calculate the predicted y-values for each x-value and then compute the SSE.
Equation B Next, we analyze equation B: y = − 0.58 x 2 − 0.43 x + 15.75 . We calculate the predicted y-values for each x-value and then compute the SSE.
Equation C Finally, we analyze equation C: y = 10.72 ⋅ 0.9 5 x . We calculate the predicted y-values for each x-value and then compute the SSE.
SSE Calculation Results After performing the calculations (using a tool for efficiency), we find the following SSE values:
SSE for Equation A: 88.6188 SSE for Equation B: 0.8252 SSE for Equation C: 96.1949
Conclusion Comparing the SSE values, we see that Equation B has the smallest SSE (0.8252). Therefore, Equation B provides the best fit to the data.
Examples
Regression equations are used in various real-world applications, such as predicting sales based on advertising expenditure, forecasting weather patterns based on historical data, or modeling the relationship between study time and exam scores. In this case, we found the quadratic equation that best fits the given data. This could be used, for example, to model the relationship between the price of a product and the demand for it, where the demand initially increases with price but then decreases as the price becomes too high.
