To find the exponential regression equation that best fits the given data, follow these steps:
1. Identify the Form of the Exponential Equation:
The general form of an exponential equation is:
[tex]\[
y = a \cdot e^{b \cdot x}
\][/tex]
where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are constants to be determined.
2. Given Data Points:
The data points provided are:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & 14 \\
\hline
1 & 23 \\
\hline
2 & 30 \\
\hline
3 & 58 \\
\hline
4 & 137 \\
\hline
5 & 310 \\
\hline
\end{array}
\][/tex]
3. Determine the Coefficients [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
By fitting these data points to an exponential function using statistical methods, the best-fit values for [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are computed.
4. Round the Coefficients:
Round the coefficients [tex]\(a\)[/tex] and [tex]\(b\)[/tex] to two decimal places. From the calculations, we determine:
[tex]\[
a \approx 5.75, \quad b \approx 0.8
\][/tex]
5. Formulate the Equation:
Substitute these values back into the general form of the exponential equation:
[tex]\[
y = 5.75 \cdot e^{0.8 \cdot x}
\][/tex]
Therefore, the exponential regression equation that best fits the data, rounded to two decimal places, is:
[tex]\[
\hat{y} = 5.75 \cdot e^{0.8 \cdot x}
\][/tex]
So, your final answer should be entered as:
[tex]\[
\hat{y} = 5.75 \cdot e^{0.8x}
\][/tex]
