To find an exponential regression model for the given data, we can follow these steps:
1. Identify the form of the exponential function: The general form for an exponential decay function is [tex]\( y = a \cdot b^x \)[/tex], where [tex]\(a\)[/tex] is the initial amount (when [tex]\(x = 0\)[/tex]), and [tex]\(b\)[/tex] is the decay factor per unit of [tex]\(x\)[/tex].
2. Given data:
- Time (hours): [tex]\([1, 4, 7, 10, 13]\)[/tex]
- Meds (mg): [tex]\([500, 255, 129, 66, 33]\)[/tex]
3. Initial guess for parameters:
- We start with an initial guess for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]. In this problem, those guesses are implied but can be estimated by closely observing the data pattern.
4. Fit the model to the data:
- To find the best values for [tex]\( a \)[/tex] and [tex]\( b \)[/tex], we use an optimization method that minimizes the difference between the observed data and the estimated data from the model.
5. Result of the regression:
- After fitting the model, the values obtained for the parameters are:
- [tex]\( a \approx 626.56 \)[/tex]
- [tex]\( b \approx 0.798 \)[/tex]
6. Final model:
- Substituting the parameters back into the exponential function, we get the equation:
[tex]\[
y = 626.56 \cdot (0.798)^x
\][/tex]
Given the choices:
- [tex]\( y = .79(512.87)^x \)[/tex]
- [tex]\( y = .79(646.65)^2 \)[/tex]
- [tex]\( y = 646.65(.79)^x \)[/tex]
- [tex]\( y = 512.87(.79)^x \)[/tex]
The model that best fits the obtained parameters is:
[tex]\[
y = 646.65(.79)^x
\][/tex]
However, based on the exact parameters determined from our fit which are closer to 626.56 for [tex]\(a\)[/tex] and 0.798 for [tex]\(b\)[/tex], the correct and closest model among the choices provided should be:
[tex]\[
y = 646.65(.79)^x
\][/tex]
