Exponential Regression
The table below shows the population \( P \) (in thousands) of a town after \( n \) years.
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
n & 0 & 3 & 7 & 12 & 14 & 19 \\
\hline
P & 3800 & 4069.32 & 4673.52 & 5580.43 & 5747.84 & 6330.16 \\
\hline
\end{array}
\][/tex]
(a) Use your calculator to determine the exponential regression equation \( P \) that models the set of data above. Round the value of \( a \) to two decimal places and round the value of \( b \) to three decimal places. Use the indicated variables.
The exponential regression model can be represented as:
[tex]\[
P = a \cdot e^{bn}
\][/tex]
After performing exponential regression on the given data, we find the parameters \( a \) and \( b \).
The value of \( a \) (the initial population) is approximately 3850.47, and the value of \( b \) (the rate parameter) is approximately 0.028.
Therefore, the exponential regression equation is:
[tex]\[
P \approx 3850.47 \cdot e^{0.028n}
\][/tex]
(b) Based on the regression model, what is the percent increase per year?
To find the percent increase per year, we use the value of \( b \) with the formula:
[tex]\[
\text{Percent Increase Per Year} = (e^b - 1) \times 100
\][/tex]
Substituting the value of \( b = 0.028 \):
[tex]\[
\text{Percent Increase Per Year} \approx (e^{0.028} - 1) \times 100 \approx 2.789 \%
\][/tex]
(c) Use your regression model to find \( P \) when \( n = 8 \). Round your answer to two decimal places.
Using the exponential regression equation:
[tex]\[
P \approx 3850.47 \cdot e^{0.028 \cdot 8}
\][/tex]
Calculating this value:
[tex]\[
P \approx 4798.27
\][/tex]
So, when \( n = 8 \):
[tex]\[
P \approx 4798.27 \text{ thousand people}
\][/tex]
(d) Interpret your answer by completing the following sentence.
The population of the town after [tex]\( 8 \)[/tex] years is [tex]\( 4798.27 \)[/tex] thousand people.
