Let's find a model for the given data using power regression. Power regression models data using the following relationship:
[tex]\[ y = a \cdot x^b \][/tex]
Given data points:
[tex]\[ x: [1, 2, 4, 5, 6, 8] \][/tex]
[tex]\[ y: [112, 41, 28, 31, 18, 8] \][/tex]
### Step-by-Step Solution
1. Transform the Data:
- We take the logarithm of both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] to linearize the data. This means transforming [tex]\(x\)[/tex] and [tex]\(y\)[/tex] into [tex]\(\log(x)\)[/tex] and [tex]\(\log(y)\)[/tex].
2. Perform Linear Regression on Transformed Data:
- The linear regression on the transformed data will help us find the coefficients for the linear relationship. This is akin to finding [tex]\(m\)[/tex] and [tex]\(c\)[/tex] in the equation of a line [tex]\( \log(y) = m \cdot \log(x) + c \)[/tex].
3. Extract the Model Parameters:
- From the linear regression, [tex]\(m\)[/tex] will be our exponent [tex]\(b\)[/tex], and [tex]\(c\)[/tex] will be [tex]\(\log(a)\)[/tex].
- To get [tex]\(a\)[/tex], we take the exponential of [tex]\(c\)[/tex], i.e., [tex]\(a = e^c\)[/tex].
4. Build the Power Regression Model:
- Using the parameters [tex]\(a\)[/tex] and [tex]\(b\)[/tex], our power model will be in the form [tex]\( y = a \cdot x^b \)[/tex].
### Calculation
After performing these steps, we get the following results for the parameters:
[tex]\[ a \approx 110.0 \][/tex]
[tex]\[ b \approx -1.1 \][/tex]
Thus, the power regression model for the given data is:
[tex]\[ y = 110.0 \cdot x^{-1.1} \][/tex]
Therefore, rounding the answers to the nearest tenth, the model is:
[tex]\[
y = 110.0 \cdot x^{-1.1}
\][/tex]
