To determine which regression equation best fits the given data, we need to evaluate the fit for each provided model and compare the errors. Let's analyze the errors associated with each regression choice:
1. Choice A: Linear Regression
The equation is given by:
[tex]\[
y = -0.43x + 11.34
\][/tex]
Calculating the error for this model provides a sum of squared differences (error) of:
[tex]\[
88.60833333333333
\][/tex]
2. Choice B: Exponential Regression
The equation is given by:
[tex]\[
y = 10.72 \cdot 0.95^x
\][/tex]
The error for this model is:
[tex]\[
96.39719170863424
\][/tex]
3. Choice C: Quadratic Regression
The equation is given by:
[tex]\[
y = -0.58x^2 - 0.43x + 15.75
\][/tex]
The error for this model might be:
[tex]\[
0.8166666666666671
\][/tex]
Now we can compare these errors:
- Error for Choice A: [tex]\( 88.60833333333333 \)[/tex]
- Error for Choice B: [tex]\( 96.39719170863424 \)[/tex]
- Error for Choice C: [tex]\( 0.8166666666666671 \)[/tex]
From these calculated errors, it is clear that the smallest error is for the quadratic regression, Choice C, with an error of [tex]\( 0.8166666666666671 \)[/tex].
Conclusion:
The best fitting regression equation for the given data is the quadratic regression:
[tex]\[
y = -0.58x^2 - 0.43x + 15.75
\][/tex]
Hence, the answer is:
C. [tex]\( y = -0.58 x^2 - 0.43 x + 15.75 \)[/tex]
