Let's determine the exponential regression equation that best fits the given data points. The data provided is:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 3 \\
\hline
2 & 8 \\
\hline
3 & 27 \\
\hline
4 & 85 \\
\hline
5 & 240 \\
\hline
6 & 570 \\
\hline
\end{array}
\][/tex]
To perform an exponential regression, we assume that the relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] follows the form:
[tex]\[ y = a \cdots b^x \][/tex]
### Step-by-Step Solution:
1. Transform the data:
- Take the natural logarithm of each [tex]\(y\)[/tex] value to linearize the exponential relationship. This becomes:
[tex]\[
\ln(y) = \ln(a) + x \cdots \ln(b)
\][/tex]
2. Perform a linear regression:
- We assume: [tex]\( y' = \ln(y) \)[/tex] and then fit the data points [tex]\((x, y')\)[/tex] to a linear equation [tex]\( y' = mx + c \)[/tex], where:
- [tex]\( m = \ln(b) \)[/tex] (slope)
- [tex]\( c = \ln(a) \)[/tex] (intercept)
3. Interpret the coefficients:
- Once the slope ([tex]\(m\)[/tex]) and intercept ([tex]\(c\)[/tex]) are found, [tex]\(b\)[/tex] and [tex]\(a\)[/tex] can be determined as:
[tex]\[
b = e^m \quad \text{and} \quad a = e^c
\][/tex]
### Matching the Exponential Regression Equation:
Given the possible answers, we need to match our form:
A. [tex]\( y = 1.03 \cdots (2.93^x) \)[/tex]\
B. [tex]\( y = 102.54 \cdots x - 203.4 \)[/tex] (not an exponential form)\
C. [tex]\( y = 2.93 \cdots (1.03^x) \)[/tex]\
D. [tex]\( y = 38.73 \cdots x^2 - 168.58 \cdots x + 158.1 \)[/tex] (not an exponential form)
Going through the correct transformations and interpretations, the equation that best represents the data in an exponential form is:
[tex]\[ y = 1.03 \cdots (2.93^x) \][/tex]
Thus, the correct answer is:
### Answer: A. [tex]\( y = 1.03 \cdots (2.93^x) \)[/tex]
