To find the exponential regression of AIDS cases over time, we need to fit an exponential model to the given data points. Here are the detailed steps:
1. Collect the Data Points:
- Time (years) [tex]\[ \(0\), \(5\), \(10\), \(15\), \(20\), \(25\) \][/tex]
- AIDS Cases (thousands) [tex]\[ \(345\), \(264\), \(139\), \(98\), \(23\) \][/tex]
2. Transform the Data:
- Since we are fitting an exponential function of the form [tex]\(\(f(x) = a \cdot b^x\)[/tex]\), we need to transform our data to fit a linear model using logarithms.
- Let [tex]\(y = \log(\text{AIDS Cases})\)[/tex]. Then the exponential model [tex]\(f(x) = a \cdot b^x\)[/tex] can be linearized to:
[tex]\[
\log(\text{AIDS Cases}) = \log(a) + x \cdot \log(b)
\][/tex]
This represents a linear equation in the form [tex]\(y = mx + c\)[/tex], where:
- [tex]\(y\)[/tex] is [tex]\(\log(\text{AIDS Cases})\)[/tex]
- [tex]\(m\)[/tex] is [tex]\(\log(b)\)[/tex]
- [tex]\(c\)[/tex] is [tex]\(\log(a)\)[/tex]
3. Compute the Linear Regression on the Transformed Data:
- Compute the logarithm of each AIDS cases value:
[tex]\[
\log(345) \approx 5.843,
\log(264) \approx 5.576,
\log(139) \approx 4.934,
\log(98) \approx 4.584,
\log(23) \approx 3.135
\][/tex]
- Use these logarithmic values and the corresponding time values to perform a linear regression to find the best fit line [tex]\(y = mx + c\)[/tex].
4. Determine the Parameters [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
- From the regression, the coefficients (slope [tex]\(m\)[/tex] and intercept [tex]\(c\)[/tex]) are determined.
- Given the coefficient results from the regression:
[tex]\[
a = e^{\text{intercept}} \approx 843.0366540860003
\][/tex]
[tex]\[
b = e^{\text{slope}} \approx 0.8797287650679395
\][/tex]
5. Construct the Exponential Function:
- The resulting exponential regression model is:
[tex]\[
f(x) = 843.0366540860003 \cdot 0.8797287650679395^x
\][/tex]
Therefore, the correct exponential regression equation that models the AIDS cases over time is:
[tex]\[ f(x) = 843.0366540860003 \cdot 0.8797287650679395^x \][/tex]
