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Find a model for this data using power regression.

[tex]\[
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$x$ & 2 & 3 & 5 & 6 & 8 \\
\hline
$y$ & 96 & 27 & 6 & 3 & 1 \\
\hline
\end{tabular}
\][/tex]

[tex]\[
y = [?] \cdot x^{[?]}
\][/tex]

Round answers to the nearest tenth.


Sagot :

Let's find a model for the given data using power regression, step by step.

Power regression is a method to find a relationship of the form [tex]\( y = a \cdot x^b \)[/tex], where constants [tex]\( a \)[/tex] and [tex]\( b \)[/tex].

The provided data points are:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & 2 & 3 & 5 & 6 & 8 \\ \hline y & 96 & 27 & 6 & 3 & 1 \\ \hline \end{array} \][/tex]

To use power regression, we undertake the following steps:

1. Log-transform the data: Both [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values are transformed using the natural logarithm:
[tex]\[ \ln(x) \text{ and } \ln(y) \][/tex]

2. Fit a linear model to the transformed data: This involves finding the best fitting line for the transformed data points [tex]\((\ln(x_i), \ln(y_i))\)[/tex].

The natural logs of the [tex]\( x \)[/tex] values [tex]\((2, 3, 5, 6, 8)\)[/tex] are:
[tex]\[ \begin{align*} \ln(2) &= 0.6931 \\ \ln(3) &= 1.0986 \\ \ln(5) &= 1.6094 \\ \ln(6) &= 1.7918 \\ \ln(8) &= 2.0794 \\ \end{align*} \][/tex]

The natural logs of the [tex]\( y \)[/tex] values [tex]\((96, 27, 6, 3, 1)\)[/tex] are:
[tex]\[ \begin{align*} \ln(96) &= 4.5643 \\ \ln(27) &= 3.2958 \\ \ln(6) &= 1.7918 \\ \ln(3) &= 1.0986 \\ \ln(1) &= 0 \\ \end{align*} \][/tex]

3. Find the linear regression line [tex]\( \ln(y) = \ln(a) + b \ln(x) \)[/tex], resulting in the coefficients for [tex]\( \ln(a) \)[/tex] and [tex]\( b \)[/tex].

After fitting a linear model, we find:

[tex]\[ \ln(y) = -3.2 \ln(x) + \ln(955.3) \][/tex]

4. Exponentiate to find the original scale:
- [tex]\( \ln(a) \)[/tex] corresponds to [tex]\( 6.8590473 \)[/tex] (which is calculated as [tex]\( 955.3 \)[/tex] when exponentiated),
- [tex]\( b \)[/tex] is simply the slope value, i.e., [tex]\( b \)[/tex].

So the final model is given by:
[tex]\[ y = 955.3 \cdot x^{-3.2} \][/tex]

Therefore, the model for the given data using power regression is:
[tex]\[ y = 955.3 \cdot x^{-3.2} \][/tex]