What is the exponential regression equation that fits these data?
| x | y |
|---|---|
| 1 | 3 |
| 2 | 8 |
| 3 | 27 |
| 4 | 85 |
| 5 | 240 |
| 6 | 570 |
A. [tex]$y=102.54 x-203.4$[/tex]
B. [tex]$y=2.93(1.03^x)$[/tex]
C. [tex]$y=1.03(2.93^x)$[/tex]
Answer (2)
The best-fitting exponential regression equation for the given data points is y = 1.03 ( 2.9 3 x ) . This equation was determined to have the smallest sum of squared differences compared to the other options. Thus, the correct answer is option C.
;
Calculate the sum of squared differences for each of the given equations using the data points.
Compare the sums of squared differences to determine the best fit.
Identify the equation with the smallest sum of squared differences.
The exponential regression equation that fits the data is y = 1.03 ( 2.9 3 x ) .
Explanation
Understanding the Problem We are given a set of data points (x, y) and asked to find the exponential regression equation that best fits the data. The given data points are (1, 3), (2, 8), (3, 27), (4, 85), (5, 240), and (6, 570). We are given three possible equations:
A. y = 102.54 x − 203.4 B. y = 2.93 ( 1.0 3 x ) C. y = 1.03 ( 2.9 3 x )
Our goal is to determine which of these equations best fits the given data.
Solution Strategy To determine the best fit, we will calculate the sum of squared differences between the calculated y values (using each equation) and the actual y values from the table. The equation with the smallest sum of squared differences will be considered the best fit.
Setting up the Calculations Let's calculate the sum of squared differences for each equation using the provided data points.
For equation A: y = 102.54 x − 203.4 For equation B: y = 2.93 ( 1.0 3 x ) For equation C: y = 1.03 ( 2.9 3 x )
Calculating Sum of Squared Differences After performing the calculations (using a Python script), we obtain the following sums of squared differences:
Sum of squared differences for equation A: 61432.3436 Sum of squared differences for equation B: 384170.547282225 Sum of squared differences for equation C: 7067.251118640862
Determining the Best Fit Comparing the sums of squared differences, we see that equation C has the smallest value (7067.251118640862). Therefore, equation C is the best fit for the given data.
Final Answer The exponential regression equation that best fits the given data is:
y = 1.03 ( 2.9 3 x )
Examples
Exponential regression is used in various fields, such as finance, biology, and physics, to model data that grows or decays exponentially. For example, it can be used to model the growth of a population, the decay of a radioactive substance, or the growth of an investment. Understanding exponential regression can help in making predictions and informed decisions based on data trends.
