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Find a model for this data using power regression.
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.
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