Chapter 11.3, Problem 15E

(a)

To determine

To find: The probability that the person has responded after 5 days, 20 days and 90 days.

Answer to Problem 15E

The probability that he responded after 5 days is $20.9\%$ , for $t=20$ it is $60.9\%$ and for $t=90$ it is $98.5\%$

Explanation of Solution

Given:

The given model for probability is,

  $P=1-e^{-0.047t}$

Calculation:

Consider the model is $P=1-e^{-0.047t}$ .

Then,

  $\begin{array}{c}P=1-e^{-0.047\left(5\right)}\\ =20.9\%\end{array}$

For the t is equal to 20 then,

  $\begin{array}{c}P=1-e^{-0.047\left(20\right)}\\ =60.9\%\end{array}$

For the t is equal to 90 then,

  $\begin{array}{c}P=1-e^{-0.047\left(90\right)}\\ =98.5\%\end{array}$

Thus, the probability that he responded after 5 days is $20.9\%$ , for $t=20$ it is $60.9\%$ and for $t=90$ it is $98.5\%$

(b)

To determine

To find: The graph for the function to find the probability when the individual responds to the advertisement is 75%.

Answer to Problem 15E

The required graph is shown in Figure 4 and it is clear that the probability of a person responding after 29 days is 75%.

Explanation of Solution

Consider the probability is 75% then,

  $\begin{array}{l}75\%=1-e^{-0.047t}\\ 1-e^{-0.047t}=0.75\\ e^{-0.047t}=0.25\\ t=29\text{days}\end{array}$

By the use graphing utility press Y= and enter the model in the calculator as shown in Figure 1

  

Figure 1

Press the window button and input the parameters as shown in Figure 2

  

Figure 2

The graph for the given model is shown in Figure 3 and is obtained by pressing the graph button.

  

Figure 3

Press the graph button trace a point on the graph as shown in Figure 4

  

Figure 4

Thus, the required graph is shown in Figure 4 and it is clear that the probability of a person responding after 29 days is 75%.

(c)

To determine

To find: The way in which the model must be used to plan the introduction of the new advertisements.

Answer to Problem 15E

The ads can be introduced after 50 days.

Explanation of Solution

Consider that the probability that a person who is responding approaches 100% as the value of t approaches infinity. The new ads must be introduced after the height percentage of those that will respond. The graph then will appear to level after 50 days.

Thus, the ads can be introduced after 50 days.