Find a model for this data using power regression.

\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
[tex]$x$[/tex] & 1 & 3 & 4 & 5 & 6 & 8 \\
\hline
[tex]$y$[/tex] & 95 & 39 & 30 & 25 & 21 & 17 \\
\hline
\end{tabular}

[tex]\[ y = [?] \cdot x \][/tex]



Answer :

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.