To fit a logistic equation [tex]\( N(t) = \frac{c}{1 + a e^{-b t}} \)[/tex] to the given data of time [tex]\( t \)[/tex] and the number of people [tex]\( N(t) \)[/tex] who have heard the rumor, we follow these steps:
1. Understand the Logistic Model:
The logistic model used to fit the data is:
[tex]\[ N(t) = \frac{c}{1 + a e^{-b t}} \][/tex]
where [tex]\( c \)[/tex], [tex]\( a \)[/tex], and [tex]\( b \)[/tex] are the parameters that we need to determine through regression.
2. Given Data:
[tex]\[
\begin{array}{l|llllllllllll}
\text{Time, } t \text{ (in days)} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline
\text{Number, } N \text{ Who Have Heard} & 1 & 2 & 3 & 6 & 14 & 18 & 22 & 24 & 28 & 28 & 29 & 30
\end{array}
\][/tex]
3. Perform Regression Analysis:
To find the parameters [tex]\( c \)[/tex], [tex]\( a \)[/tex], and [tex]\( b \)[/tex] that best fit the data, we apply logistic regression. The fitting process finds values of [tex]\( c \)[/tex], [tex]\( a \)[/tex], and [tex]\( b \)[/tex] such that the difference between the actual data points [tex]\( N(t) \)[/tex] and the logistic equation is minimized.
4. Solution Parameters:
After performing the regression analysis, we find the following rounded values for the parameters:
[tex]\[
c = 29.336, \quad a = 62.883, \quad b = 0.758
\][/tex]
5. Final Logistic Equation:
Using the calculated values of [tex]\( c \)[/tex], [tex]\( a \)[/tex], and [tex]\( b \)[/tex], the logistic equation fits the given data as:
[tex]\[
N(t) = \frac{29.336}{1 + 62.883 e^{-0.758 t}}
\][/tex]
Thus, the logistic equation that fits the data is:
[tex]\[
N(t) = \frac{29.336}{1 + 62.883 e^{-0.758 t}}
\][/tex]
When expressed in the format requested in the problem statement, our logistic equation becomes:
[tex]\[
N(t) = \frac{29.336}{1 + e^{-0.758 \cdot t}}
\][/tex]
