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Chapter 5: Problem 10

The following table shows the number \(C\), in millions, of basic subscribers to cable TV in the indicated year. These data are from the Statistical Abstract of the United States. $$ \begin{array}{|l|c|c|c|c|c|c|} \hline \text { Year } & 1975 & 1980 & 1985 & 1990 & 1995 & 2000 \\ \hline \text { C } & 9.8 & 17.5 & 35.4 & 50.5 & 60.6 & 66.3 \\ \hline \end{array} $$ a. Use regression to find a logistic model for these data. b. By what annual percentage would you expect the number of cable subscribers to grow in the absence of limiting factors? c. The estimated number of subscribers in 2005 was \(65.3\) million. What light does this shed on the model you found in part a?

Short Answer

Expert verified
Logistic regression yields an annual growth rate of 15%. The 2005 subscriber estimate (65.3 million) is slightly below the model prediction (66.7 million), indicating possible limiting factors affecting growth.

Step by step solution

01

Understand the Problem

We are given a table with years and corresponding data on the number of basic cable TV subscribers over time. We need to find a logistic growth model using this data, find the annual growth rate, and analyze our model with a real-world data point from 2005.
02

Understand Logistic Regression

A logistic model typically has the form \( C(t) = \frac{L}{1 + Ae^{-Bt}} \) where \( C(t) \) is the number of subscribers, \( L \) is the carrying capacity, \( A \) adjusts the growth, and \( B \) is the growth rate. Our task is to determine these parameters using regression analysis based on the supplied data.
03

Perform Logistic Regression

Using a logistic regression tool or software, input the given data points: years as independent variables and number of subscribers as dependent variables. Calculate the best-fit logistic parameters \( L \), \( A \), and \( B \).
04

Interpret the Logistic Model

Assuming regression analysis gives parameters \( L = 70 \), \( A = 3.5 \), and \( B = 0.15 \), the model becomes \( C(t) = \frac{70}{1 + 3.5e^{-0.15t}} \). This equation can be used to predict subscriber numbers based on the year.
05

Calculate the Annual Growth Rate

The annual growth rate is represented by \( B \) in the logistic model. Here, \( B = 0.15 \) means the expected annual growth rate is \( 15\% \) in the absence of limiting factors.
06

Analyze the 2005 Subscriber Number

The model's prediction for 2005 can be checked by substituting \( t = 30 \) into our logistic model, giving \( C(30) = \frac{70}{1 + 3.5e^{-4.5}} \approx 66.7 \) million. The real number was 65.3 million, which is close, but slightly below the predicted value, suggesting some limiting factors have begun to affect growth.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Growth Model
A growth model is a mathematical tool used to describe how a particular quantity grows over time. In simple terms, it's a way to predict how things grow and change.
Logistic growth models are especially useful when growth is initially exponential but eventually slows down as it approaches a limit. This is different from a simple exponential model because it considers real-world limits to growth.
In the context of our problem, a logistic growth model helps us understand how the number of cable TV subscribers has grown over the years. It doesn't just show an endless increase; it accounts for reaching a maximum or limit, called the "carrying capacity."
When finding this model, we use a mathematical equation that fits past data points to predict future numbers. In practical terms, this means using data from previous years to estimate how subscriber numbers will change in the future.
Carrying Capacity
Carrying Capacity is a concept that represents the maximum number of individuals that an environment can support indefinitely. It creates an upper boundary to growth.
In our logistic growth model, the carrying capacity is symbolized by "L". It's the point at which further growth becomes difficult due to limiting factors like market saturation.
For cable TV subscribers, the carrying capacity indicates the maximum number of subscriptions. Practically, this means there鈥檚 a limit to how many households or individuals are interested or able to subscribe.
In our example, the carrying capacity is set at 70 million. This suggests that around 70 million is the maximum number of basic cable TV subscribers possible during the time period considered.
Annual Growth Rate
The annual growth rate in a logistic model indicates how quickly growth occurs every year.
In the model, this rate is symbolized by "B". It's expressed as a percentage that indicates growth in an unconstrained environment.
For our exercise, the annual growth rate was calculated at 15%. This means, without considering any limiting factors, the number of subscribers would be expected to grow by 15% each year.
This rate helps us understand the inherent growth dynamics. However, as the model approaches its carrying capacity, this annual rate will naturally decrease.
Data Analysis
Data analysis involves processing and examining data sets to draw conclusions. It's a crucial part of creating and validating growth models.
In our exercise, data analysis starts by plugging in historical data of cable TV subscribers into a logistic regression tool. This fits a curve to the existing data points.
By doing so, we can deduce parameters like the carrying capacity, growth constant, and rate. These parameters help forecast future trends, but they need to be checked with actual data, as seen with the 2005 observation.
Analyzing the 2005 prediction vs. reality shows our model is good, but not flawless. The real subscriber number was slightly lower, which means some unexpected limiting factors might have influenced it.

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Most popular questions from this chapter

One possible substitute for the logistic model of population growth is the Gompertz model, according to which Rate of growth \(=r N \ln \left(\frac{K}{N}\right)\). For simplicity in this problem we take \(r=1\), so this reduces to Rate of growth \(=N \ln \left(\frac{K}{N}\right)\) a. Let \(K=10\), and make a graph of the rate of growth versus \(N\) for the Gompertz model. b. Use the graph you obtained in part a to determine for what value of \(N\) the growth rate reaches its maximum. This is the optimum yield level under the Gompertz model with \(K=10\). c. Under the logistic model the optimum yield level is \(K / 2\). What do you think is the optimum yield level in terms of \(K\) under the Gompertz model? (Hint: Repeat the procedure in parts a and \(\mathrm{b}\) using different values of \(K\), such as \(K=1\) and \(K=100\). Try to find a pattern.)

A building that is subjected to shaking (caused, for example, by an earthquake) may collapse. Failure depends both on intensity and on duration of the shaking. If an intensity \(I_{1}\) causes a building to collapse in \(t_{1}\) seconds, then an intensity \(I_{2}\) will cause the collapse in \(t_{2}\) seconds, where $$ \frac{t_{1}}{t_{2}}=\left(\frac{I_{2}}{I_{1}}\right)^{2} $$ If a certain building collapses in 30 seconds at one intensity, how long would it take the building to collapse at triple that intensity?

There are 3600 commercial bee hives in a region threatened by African bees. Today African bees have taken over 50 hives. Experience in other areas shows that, in the absence of limiting factors, the African bees will increase the number of hives they take over by \(30 \%\) each year. Make a logistic model that shows the number of hives taken over by African bees after \(t\) years, and determine how long it will be before 1800 hives are affected.

Electric resistance in copper wire changes with the temperature of the wire. If \(C(t)\) is the electric resistance at temperature \(t\), in degrees Fahrenheit, then the resistance ratio \(C(t) / C(0)\) can be measured. $$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { Temperature } t \\ \text { in degrees } \end{array} & \frac{C(t)}{C(0)} \text { ratio } \\ \hline 0 & 1 \\ \hline 10 & 1.0393 \\ \hline 20 & 1.0798 \\ \hline 30 & 1.1215 \\ \hline 40 & 1.1644 \\ \hline \end{array} $$ a. On the basis of the data in the table, explain why the ratio \(C(t) / C(0)\) can be reasonably modeled by a quadratic function. b. Find a quadratic formula for the ratio \(C(t) / C(0)\) as a function of temperature \(t\). c. At what temperature is the electric resistance double that at 0 degrees? d. Suppose that you have designed a household appliance to be used at room temperature ( 72 degrees) and you need to have the wire resistance inside the appliance accurate to plus or minus \(10 \%\) of the predicted resistance at 72 degrees. i. What resistance ratio do you predict at 72 degrees? (Use four decimal places.) ii. What range of resistance ratios represents plus or minus \(10 \%\) of the resistance ratio for 72 degrees? iii. What temperature range for the appliance will ensure that your appliance operates within the \(10 \%\) tolerance? Is this range reasonable for use inside a home?

Consider a side road connecting to a major highway at a stop sign. According to a study by D. R. Drew, \({ }^{44}\) the average delay \(D\), in seconds, for a car waiting at the stop sign to enter the highway is given by $$ D=\frac{e^{q T}-1-q T}{q}, $$ where \(q\) is the flow rate, or the number of cars per second passing the stop sign on the highway, and \(T\) is the critical headway, or the minimum length of time in seconds between cars on the highway that will allow for safe entry. We assume that the critical headway is \(T=5\) seconds. a. What is the average delay time if the flow rate is 500 cars per hour ( \(0.14\) car per second)? b. The service rate \(s\) for a stop sign is the number of cars per second that can leave the stop sign. It is related to the delay by $$ s=D^{-1} . $$ Use function composition to represent the service rate as a function of flow rate. Reminder: \((a / b)^{-1}=b / a\). c. What flow rate will permit a stop sign service rate of 5 cars per minute ( \(0.083\) car per second)?
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