Chapter 5.9, Problem 10AYU

(a)

To determine

To find: The logistic model from the given data.

Answer to Problem 10AYU

The logistic model of the given data is $y=\frac{345.11}{1+18783e^{-0.27x}}$ .

Explanation of Solution

Given:

The given data is shown in Table 1

Table 1

Years x	Subscribers1000, $y$
1	0.34
3	1.23
5	3.51
8	11.03
11	33.76
14	69.21
18	140.77
20	182.14
22	233.00

Calculation:

Consider the users as the independent variable and then a scatter plot of the functions as shown in Figure 1

Figure 1

Consider the logistic model with the help of graphing utility as,

  $y=\frac{345.11}{1+18783e^{-0.27x}}$

(b)

To determine

To find: The scatter plot for the logistic model.

Answer to Problem 10AYU

The scatter plot is shown in Figure 2

Explanation of Solution

Consider the logistic model is,

  $y=\frac{345.11}{1+18783e^{-0.27x}}$

Consider the scatter plot from the given data is shown in Figure 2

Figure 2

(c)

To determine

To find: The predicted capacity of US cell phone subscribers.

Answer to Problem 10AYU

The capacity is $345.11$ millions.

Explanation of Solution

Consider the logistic model is,

  $y=\frac{345.11}{1+18783e^{-0.27x}}$

The predicted capacity is obtained by using $\infty$ for $x$ in the logistic model as,

  $\begin{array}{c}y=\frac{345.11}{1+18783e^{-0.27\left(\infty \right)}}\\ =345.11\end{array}$

That is $345.11$ millions.

(d)

To determine

To find: The predicted number of cell phone users at the end of 2009 from the model found in part (b).

Answer to Problem 10AYU

The predicted number of cell phone users is $305\text{million}$ .

Explanation of Solution

Consider the logistic model is,

  $y=\frac{345.11}{1+18783e^{-0.27x}}$

Consider the number of years passed is 25. Then,

  $\begin{array}{c}y=\frac{345.11}{1+18783e^{-0.27\left(25\right)}}\\ =305\text{million}\end{array}$

(e)

To determine

To find: The comparison of the answer in part (d) with the answers of example 1 part (e) and explain the different predictions.

Answer to Problem 10AYU

The model is better prediction of the future in terms of the cell phone users as it is far more than the population.

Explanation of Solution

Consider the number of users in example 1 is 1247 million that is far more than the population of the country so the users is at the par of the population is the prediction for the future users.

Hence, the model is better prediction of the future in terms of the cell phone users.