Answer :
To determine the exponential regression function that represents the fish population and identify the percentage by which the population is declining each year, we can proceed by carefully analyzing the data and performing regression analysis to fit an exponential decay model.
### Step-by-Step Solution:
1. Collect the Data Points:
We have the following paired data points (Time (years), Fish Population):
[tex]\[ \begin{aligned} & (0, 2700) \\ & (1, 2300) \\ & (2, 1950) \\ & (3, 1660) \\ & (4, 1400) \\ & (5, 1200) \\ \end{aligned} \][/tex]
2. Transform the Exponential Equation:
The general form of the exponential decay function is:
[tex]\[ f(x) = a \times e^{bx} \][/tex]
where [tex]\(a\)[/tex] is the initial population and [tex]\(b\)[/tex] is the rate of decay.
3. Logarithmic Transformation:
To find the parameters [tex]\(a\)[/tex] and [tex]\(b\)[/tex], we take the natural logarithm of both sides of the equation:
[tex]\[ \ln(f(x)) = \ln(a) + bx \][/tex]
This transformation allows us to use linear regression techniques to find [tex]\(\ln(a)\)[/tex] (intercept) and [tex]\(b\)[/tex] (slope).
4. Linear Regression Analysis:
Applying linear regression to the transformed data, we seek to minimize the errors and find the best fit line for the log-transformed values of the fish population data.
5. Calculate the Coefficients:
From the regression analysis, we can determine the coefficients:
[tex]\[ \begin{aligned} \ln(a) &= 7.901479308633354 \\ b &= -0.16299924314944378 \\ \end{aligned} \][/tex]
Converting [tex]\(\ln(a)\)[/tex] back to [tex]\(a\)[/tex]:
[tex]\[ a \approx e^{7.901479308633354} \approx 2702.29 \][/tex]
6. Formulate the Exponential Regression Function:
Using the values obtained for [tex]\(a\)[/tex] and [tex]\(b\)[/tex], the exponential regression function is:
[tex]\[ f(x) = 2702.29 \times e^{-0.16299924314944378x} \][/tex]
7. Calculate the Decay Rate:
The decay rate [tex]\(r\)[/tex] is found by:
[tex]\[ r = 1 - e^{b} \][/tex]
Substituting the value of [tex]\(b\)[/tex]:
[tex]\[ r = 1 - e^{-0.16299924314944378} \approx 0.15040816857160289 \][/tex]
Converting the decay rate to a percentage:
[tex]\[ \text{Decay Rate} \approx 15.04\% \][/tex]
8. Verify with Given Data:
The exponential function given in the problem is [tex]\(f(x) = 2700 \times (0.85)^x\)[/tex], indicating a 15% decay since [tex]\(0.85 = 1 - 0.15\)[/tex].
Based on the above detailed steps and the results obtained:
- The exponential regression function is:
[tex]\[ f(x) = 2702.29 \times e^{-0.16299924314944378x} \][/tex]
- The population is declining approximately by [tex]\(15.04\%\)[/tex] each year. This aligns closely with the given decay rate of [tex]\(15\%\)[/tex].
Therefore, the biologist's observations about the decline in the fish population fit well with an exponential decay function showing a similar rate of decline.
### Step-by-Step Solution:
1. Collect the Data Points:
We have the following paired data points (Time (years), Fish Population):
[tex]\[ \begin{aligned} & (0, 2700) \\ & (1, 2300) \\ & (2, 1950) \\ & (3, 1660) \\ & (4, 1400) \\ & (5, 1200) \\ \end{aligned} \][/tex]
2. Transform the Exponential Equation:
The general form of the exponential decay function is:
[tex]\[ f(x) = a \times e^{bx} \][/tex]
where [tex]\(a\)[/tex] is the initial population and [tex]\(b\)[/tex] is the rate of decay.
3. Logarithmic Transformation:
To find the parameters [tex]\(a\)[/tex] and [tex]\(b\)[/tex], we take the natural logarithm of both sides of the equation:
[tex]\[ \ln(f(x)) = \ln(a) + bx \][/tex]
This transformation allows us to use linear regression techniques to find [tex]\(\ln(a)\)[/tex] (intercept) and [tex]\(b\)[/tex] (slope).
4. Linear Regression Analysis:
Applying linear regression to the transformed data, we seek to minimize the errors and find the best fit line for the log-transformed values of the fish population data.
5. Calculate the Coefficients:
From the regression analysis, we can determine the coefficients:
[tex]\[ \begin{aligned} \ln(a) &= 7.901479308633354 \\ b &= -0.16299924314944378 \\ \end{aligned} \][/tex]
Converting [tex]\(\ln(a)\)[/tex] back to [tex]\(a\)[/tex]:
[tex]\[ a \approx e^{7.901479308633354} \approx 2702.29 \][/tex]
6. Formulate the Exponential Regression Function:
Using the values obtained for [tex]\(a\)[/tex] and [tex]\(b\)[/tex], the exponential regression function is:
[tex]\[ f(x) = 2702.29 \times e^{-0.16299924314944378x} \][/tex]
7. Calculate the Decay Rate:
The decay rate [tex]\(r\)[/tex] is found by:
[tex]\[ r = 1 - e^{b} \][/tex]
Substituting the value of [tex]\(b\)[/tex]:
[tex]\[ r = 1 - e^{-0.16299924314944378} \approx 0.15040816857160289 \][/tex]
Converting the decay rate to a percentage:
[tex]\[ \text{Decay Rate} \approx 15.04\% \][/tex]
8. Verify with Given Data:
The exponential function given in the problem is [tex]\(f(x) = 2700 \times (0.85)^x\)[/tex], indicating a 15% decay since [tex]\(0.85 = 1 - 0.15\)[/tex].
Based on the above detailed steps and the results obtained:
- The exponential regression function is:
[tex]\[ f(x) = 2702.29 \times e^{-0.16299924314944378x} \][/tex]
- The population is declining approximately by [tex]\(15.04\%\)[/tex] each year. This aligns closely with the given decay rate of [tex]\(15\%\)[/tex].
Therefore, the biologist's observations about the decline in the fish population fit well with an exponential decay function showing a similar rate of decline.
