To find a power regression model for the given data, we aim to fit an equation of the form [tex]\( y = a \cdot x^b \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants to be determined. The provided dataset is:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & 1 & 2 & 4 & 5 & 6 & 8 \\
\hline
y & 112 & 41 & 28 & 31 & 18 & 8 \\
\hline
\end{array}
\][/tex]
Here are the steps to find the parameters [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
1. Define the power regression equation: [tex]\( y = a \cdot x^b \)[/tex].
2. Transform the data: One common approach to linearize the equation is to take the natural logarithm of both sides. Doing so, we get:
[tex]\[
\ln(y) = \ln(a) + b \ln(x)
\][/tex]
This transforms our equation into a linear relationship in terms of [tex]\( \ln(y) \)[/tex] and [tex]\( \ln(x) \)[/tex].
3. Fit the model: Using the transformed data, apply linear regression techniques or curve fitting to determine the best-fit parameters [tex]\( a \)[/tex] and [tex]\( b \)[/tex].
After applying these methods and calculations:
4. Solution: From the fitting process, the parameters are determined to be:
[tex]\[
a \approx 109.1 \quad \text{and} \quad b \approx -1.1
\][/tex]
Thus, the power regression model that best fits the provided data is:
[tex]\[
y \approx 109.1 \cdot x^{-1.1}
\][/tex]
Finally, rounding these parameters to the nearest tenth, we have:
[tex]\[
a = 109.1 \quad \text{and} \quad b = -1.1
\][/tex]
So the model is:
[tex]\[
y = 109.1 \cdot x^{-1.1}
\][/tex]
