Let's address each part of the problem step-by-step:
### Part (a)
We need to determine the exponential regression equation [tex]\(P = a \times b^n\)[/tex] that models the given set of data.
From the calculations, we have:
- [tex]\(a = 4115.03\)[/tex] (rounded to two decimal places)
- [tex]\(b = 1.069\)[/tex] (rounded to three decimal places)
Thus, the exponential regression equation is:
[tex]\[ P = 4115.03 \times 1.069^n \][/tex]
### Part (b)
To find the percent increase per year based on the regression model, we consider the base [tex]\(b\)[/tex] of the exponential equation:
- [tex]\(b = 1.069\)[/tex]
The percent increase is calculated as [tex]\((b - 1) \times 100 \%\)[/tex]:
[tex]\[ \text{Percent Increase} = (1.069 - 1) \times 100 \% = 0.069 \times 100 \% = 6.88 \% \][/tex]
### Part (c)
Next, we use the regression model to find the population [tex]\(P\)[/tex] when [tex]\(n = 15\)[/tex]:
[tex]\[ P = 4115.03 \times 1.069^{15} \][/tex]
From the calculations, when [tex]\(n = 15\)[/tex], we have [tex]\(P \approx 11157.37\)[/tex] thousand people (rounded to two decimal places).
### Part (d)
Finally, we interpret the result. We need to fill in the sentence correctly.
The sentence to complete is:
"The population of the town after 15 years is 11157.37 thousand people."
So, after [tex]\(n = 15\)[/tex] years, the population [tex]\(P\)[/tex] will be [tex]\(11157.37\)[/tex] thousand people.
### Summary
1. The exponential regression equation is:
[tex]\[ P = 4115.03 \times 1.069^n \][/tex]
2. The percent increase per year is:
[tex]\[ 6.88 \% \][/tex]
3. When [tex]\(n = 15\)[/tex], the population [tex]\(P\)[/tex] is:
[tex]\[ P = 11157.37 \][/tex] thousand people
4. Interpretation:
"The population of the town after 15 years is 11157.37 thousand people."
This completes the detailed step-by-step solution.
