To find a model for the given data using power regression, we aim to fit a power-law function of the form:
[tex]\[ y = a \cdot x^b \][/tex]
Given the data points:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & 1 & 3 & 4 & 5 & 6 & 8 \\
\hline
y & 95 & 39 & 30 & 25 & 21 & 17 \\
\hline
\end{array}
\][/tex]
we need to determine the coefficients [tex]\( a \)[/tex] and [tex]\( b \)[/tex] that best fit the data using this power-law equation.
### Step-by-Step Process:
1. Data Plotting (Conceptual Step):
- First, plot the given data points on a graph with x-values on the horizontal axis and y-values on the vertical axis.
- By observing the plotted data, note that the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] suggests a decreasing trend which may fit a power-law model.
2. Power-Law Regression Model:
- Assume the model [tex]\( y = a \cdot x^b \)[/tex].
3. Parameter Estimation:
- Using a suitable regression method (like least squares), we estimate the parameters [tex]\( a \)[/tex] and [tex]\( b \)[/tex]. The goal is to minimize the sum of squared differences between the observed values and the values predicted by the model.
4. Fitted Model:
- After performing the regression analysis, the estimated parameters for [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are obtained. According to the result:
[tex]\[
a \approx 95.09887832336803
\][/tex]
[tex]\[
b \approx -0.8283949226773465
\][/tex]
Therefore, the power-law model that best describes the given data is:
[tex]\[ y = 95.09887832336803 \cdot x^{-0.8283949226773465} \][/tex]
### Final Regression Model:
Substituting the computed values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex], we get the final regression equation as:
[tex]\[ y = 95.09887832336803 \cdot x^{-0.8283949226773465} \][/tex]
This equation captures the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] for the provided data points using a power regression approach.
