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Time, [tex]\( t \)[/tex] (in days): 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

Number, [tex]\( N \)[/tex], Who Have Heard the Rumor: 1, 2, 3, 6, 14, 18, 22, 24, 28, 28, 29, 30

a) Use regression to fit a logistic equation [tex]\( N(t) = \frac{c}{1 + a e^{-bt}} \)[/tex] to the data.
[tex]\[ N(t) = \frac{\square}{1 + \square e^{-\square t}} \][/tex]
(Round to three decimal places as needed.)



Answer :

GinnyAnswer
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]

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