Chapter 5.9, Problem 9AYU

(a)

To determine

To find: The scatter diagram a scatter diagram of the data using the number of years after 1970, t, as the independent variable and the number of subscribers as the dependent variable.

Answer to Problem 9AYU

The scatter plot is shown in Figure 4

Explanation of Solution

Given:

The given table for the number of subscribers is shown in Table 1

Table 1

Years	Subscribers
1975(t=5)	9,800
1980(t=10)	17,500
1985(t=15)	35,440
1990 (t= 20)	50,520
1992 (t= 22)	54,300
1994 (t= 24)	58,373
1996 (t= 26)	62,300
1998 (t= 28)	64,650
2000 (t=30)	66,250
2002 (t=32)	66,472
2004 (t=34)	65,727
2006(t=36)	65,319

Calculation:

From the given data TI-83 Calculator to plot the scatter diagram.

Consider $x=5$ corresponds to year 1975.

First turn On the stat plot 1, the diagram for the snip is shown in Figure 1

Figure 1

Then go to STAT+1 to enter the list.

Figure 2

Adjust the window so the calculator can show the scatter plot in the screen as shown in Figure 3

Figure 3

Graph the scatter plot as shown in Figure 4

Figure 4

(b)

To determine

To find: The logistic model of the given data.

Answer to Problem 9AYU

The logistic model is $y=\frac{67,860.50}{1+19.864e^{-0.20929x}}$ .

Explanation of Solution

In the graphing calculator press the STAT then click the CALC menu, the snip for the calculator is shown in Figure 5

Figure 5

To use, the logistic function to fit the data we pick the option B logistic then Enter button as shown in Figure 6

Figure 6

From above, the value model is,

  $y=\frac{67,860.50}{1+19.864e^{-0.20929x}}$

(c)

To determine

To find: The scatter plot for the logistic model.

Answer to Problem 9AYU

The scatter plot is shown in Figure 7

Explanation of Solution

Consider the expression for the logistic model is,

  $y=\frac{67,860.50}{1+19.864e^{-0.20929x}}$

Enter Y= button, the snip is shown in Figure 7

Figure 7

The scatter plot is shown in Figure 7

Figure 7

(d)

To determine

To find: The maximum number of cable TV subscribers in the United States.

Answer to Problem 9AYU

The maximum number of TV subscriber is $67,860,000$ .

Explanation of Solution

Consider the expression for the logistic model is,

  $y=\frac{67,860.50}{1+19.864e^{-0.20929x}}$

For $x$ tends to infinity, the denominator is equal to 1.

Then,

  $\begin{array}{c}y=\frac{67,860.50}{1}\\ =67,860\end{array}$

Thus, the maximum number of TV subscriber is $67,860,000$ .

(e)

To determine

To find: The model found in part (b) to predict the number of cable TV subscriber in the United States in 2015.

Answer to Problem 9AYU

The maximum number of TV subscriber is $67,711,000$ .

Explanation of Solution

Consider the expression for the logistic model is,

  $y=\frac{67,860.50}{1+19.864e^{-0.20929x}}$

To find the maximum number of subscriber in 2015 substitute 45 for $x$ in the equation of logistic model as,

Then,

  $\begin{array}{c}y=\frac{67,860.50}{1+19.864e^{-0.209259\left(45\right)}}\\ =67,711\end{array}$

Thus, the maximum number of TV subscriber is $67,711,000$ .