What is the exponential regression equation that fits these data?
| x | y |
| --- | ---- |
| -4 | 6.01 |
| -3 | 6.03 |
| -2 | 6.12 |
| -1 | 6.38 |
| 0 | 8 |
| 1 | 12 |
| 2 | 13 |
| 3 | 36 |
| 4 | 88 |
A. [tex]y=4.89 \cdot 1.47^x[/tex]
B. [tex]y=2.49 x^2+7.29 x+3.57[/tex]
C. [tex]y=1.36 \cdot 12.11^x[/tex]
Answer (2)
Calculate the Sum of Squared Errors (SSE) for each of the given equations.
Compare the SSE values to determine which equation has the smallest error.
Identify that Option B has the smallest SSE.
Conclude that the best fit equation is y = 2.49 x 2 + 7.29 x + 3.57 .
Explanation
Problem Analysis We are given a set of data points and three potential exponential 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 and choose the equation with the smallest SSE.
Given Data and Equations The x values are: -4, -3, -2, -1, 0, 1, 2, 3, 4. The y values are: 6.01, 6.03, 6.12, 6.38, 8, 12, 13, 36, 88. Option A: y = 4.89 × 1.4 7 x Option B: y = 2.49 x 2 + 7.29 x + 3.57 Option C: y = 1.36 × 12.1 1 x
Calculating SSE We calculate the SSE for each equation using the formula: SSE = \tSum ( y a c t u a l − y p re d i c t e d ) 2 . This involves plugging each x-value into each equation to get a predicted y-value, then finding the squared difference between the predicted and actual y-values, and summing these squared differences for all data points.
SSE Results After performing the calculations (using a python tool), we find the following SSE values: SSE for Option A: 4773.26 SSE for Option B: 809.17 SSE for Option C: 856077484.35
Conclusion The equation with the smallest SSE is Option B, with an SSE of 809.17. Therefore, the best fit for the given data is the quadratic equation y = 2.49 x 2 + 7.29 x + 3.57 .
Examples
Imagine you are tracking the growth of a plant over several weeks. You collect data on its height at different time points. Using regression analysis, you can find an equation that best describes the plant's growth pattern. This equation can then be used to predict the plant's height at future times, helping you understand its growth rate and plan accordingly. Regression analysis is a powerful tool in various fields, including biology, economics, and engineering, for modeling and predicting trends based on observed data.
To find the exponential regression equation that fits the data best, we calculated the Sum of Squared Errors (SSE) for each provided option. Option B, with the equation y = 2.49 x 2 + 7.29 x + 3.57 , had the smallest SSE, indicating it is the best fit for the data. Hence, the final answer is Option B.
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