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Chapter 4: Problem 9

Nearly linear or exponential data: One of the two tables below shows data that are better approximated with a linear function, and the other shows data that are better approximated with an exponential function. Make plots to identify which is which, and then use the appropriate regression to find models for both. $$ \begin{aligned} &\begin{array}{|c|c|} \hline t & f(t) \\ \hline 1 & 3.62 \\ \hline 2 & 23.01 \\ \hline 3 & 44.26 \\ \hline 4 & 62.17 \\ \hline 5 & 83.25 \\ \hline \end{array}\\\ &\begin{array}{|c|c|} \hline t & g(t) \\ \hline 1 & 3.62 \\ \hline 2 & 5.63 \\ \hline 3 & 8.83 \\ \hline 4 & 13.62 \\ \hline 5 & 21.22 \\ \hline \end{array} \end{aligned} $$

Short Answer

Expert verified
\(g(t)\) is linear; \(f(t)\) is exponential with models: \(g(t) \approx 4.41t - 0.86\) and \(f(t) \approx 2.4 \cdot 1.8^t\).

Step by step solution

01

Plot the Data

Start by plotting both sets of data. For the table containing \(f(t)\), plot the points (1, 3.62), (2, 23.01), (3, 44.26), (4, 62.17), and (5, 83.25). For \(g(t)\), plot the points (1, 3.62), (2, 5.63), (3, 8.83), (4, 13.62), and (5, 21.22). Look for patterns: a straight-line pattern suggests linear data, while a curved, upward-sloping pattern suggests exponential growth.
02

Identify Linear Data

After plotting, examine the graph. If the points of a dataset lie approximately in a straight line, that dataset is better approximated with a linear function. In this case, the data of \(g(t)\) appear linear.
03

Fit a Linear Model for Linear Data

Apply linear regression to the data set \(g(t)\). The linear regression model is of the form \( g(t) = at + b \). By calculating or using a calculator/computer, find the values of \(a\) and \(b\). For the dataset \(g(t)\), \(g(t) \approx 4.41t - 0.86\).
04

Identify Exponential Data

Examine the plot again and notice the pattern of \(f(t)\); it follows a curved upward pattern, indicating exponential growth.
05

Fit an Exponential Model for Exponential Data

An exponential function is of the form \( f(t) = a \cdot b^t \). By using exponential regression tools, calculate the parameters \(a\) and \(b\). For \(f(t)\), we find \(f(t) \approx 2.4 \cdot 1.8^t\).

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Key Concepts

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

Linear Function
A linear function is a mathematical expression in which the output grows proportionally to the input. It means that you can represent such a function using the equation \( y = mx + b \), where:
  • \( m \) is the slope (the change in \( y \) for a unit change in \( x \)).
  • \( b \) is the y-intercept (the point where the line crosses the y-axis).
In the context of data modeling, when data points lie approximately on a straight line, they are well-represented by a linear function. By plotting the points and fitting a linear function using linear regression, we can determine the best-fit line, as done with the dataset \( g(t) \). Here, the equation was found to be \( g(t) = 4.41t - 0.86 \), suggesting a direct and proportional growth over time with an established beginning point.
Exponential Function
Exponential functions, represented by the equation \( y = a \, b^x \), describe situations where the rate of change of the dependent variable accelerates over time or space. Key elements include:
  • \( a \): the initial amount when \( x = 0 \).
  • \( b \): the growth factor. If \( b > 1 \), the function indicates growth; if \( b < 1 \), it indicates decay.
In practical terms, exponential functions model phenomena where change occurs rapidly, such as population growth or compound interest.
In this exercise, plotting the dataset \( f(t) \) showed an upward curved pattern, hinting at its exponential nature. By applying exponential regression, the relationship was approximated as \( f(t) = 2.4 \, 1.8^t \). This demonstrates how each subsequent point grows by a factor of approximately 1.8, underscoring the power of exponential functions in capturing dramatic growth.
Data Modeling
Data modeling is the process of creating a mathematical representation of a set of data. It allows for predictions and insights by finding relationships and trends within the data. When modeling data:
  • Start by plotting the data points to visually assess their pattern.
  • Determine whether the data set follows a linear pattern or an exponential one.
  • Choose the appropriate model (either linear or exponential) to apply based on observed patterns.
  • Use regression techniques to quantify the relationship with the best fit parameters for the chosen model.
This exercise required distinguishing between linear and exponential data through plotting and identification, leading to the selection of the appropriate regression models. Regression analysis then provided the means to extract meaningful relationships from the data, demonstrating how effective data modeling can be in uncovering these patterns and trends.

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Most popular questions from this chapter

National health care spending: The following table shows national health care costs, measured in billions of dollars. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Date } & 1960 & 1970 & 1980 & 1990 & 2000 \\ \hline \begin{array}{l} \text { Costs } \\ \text { in billions } \end{array} & 27.6 & 75.1 & 254.9 & 717.3 & 1358.5 \\ \hline \end{array} $$ a. Plot the data. Does it appear that the data on health care spending can be appropriately modeled by an exponential function? b. Find an exponential function that approximates the data for health care costs. c. By what percent per year were national health care costs increasing during the period from 1960 through 2000?

Reaction time: For certain decisions, the time it takes to respond is a logarithmic function of the number of choices faced. \({ }^{32}\) One model is $$ R=0.17+0.44 \log N, $$ where \(R\) is the reaction time in seconds and \(N\) is the number of choices. a. Draw a graph of \(R\) versus \(N\). Include values of \(N\) from 1 to 10 choices. b. Express using functional notation the reaction time if there are seven choices, and then calculate that time. c. If the reaction time is to be at most \(0.5\) second, how many choices can there be? d. If the number of choices increases by a factor of 10 , what happens to the reaction time? e. Explain in practical terms what the concavity of the graph means.

Cell phones: The following table shows the number, in millions, of cell phone subscribers in the United States at the end of the given year. $$ \begin{array}{|c|c|} \hline \text { Year } & \text { Subscribers (millions) } \\ \hline 2001 & 128.4 \\ \hline 2002 & 140.8 \\ \hline 2003 & 158.7 \\ \hline 2004 & 182.1 \\ \hline 2005 & 207.9 \\ \hline \end{array} $$ a. Plot the natural logarithm of the data points. Does this plot make it look reasonable to approximate the original data with an exponential function? b. Find the regression line for the natural logarithm of the data and add its graph to the plot in part a. c. Construct an exponential model for the original subscribership data using the logarithm as a link.

Stochastic population growth: Many populations are appropriately modeled by an exponential function, at least for a limited period of time. But there are many factors contributing to the growth of any population, and many of them depend on chance. There are a number of ways to produce stochastic models. We consider one that is an illustration of a simple Monte Carlo method. We want to model a population that is initially 500 and grows at an average rate of 2% per year. To do this we make a table of values for population according to the following procedure. The first entry in the table is for time t = 0, and it records the initial value 500. To get the entry corresponding to t = 1, we roll a die and change the population according to the face that appears, using the following rule. $$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { Face } \\ \text { appearing } \end{array} & \begin{array}{c} \text { Population } \\ \text { change } \end{array} \\ \hline 1 & \text { Down 2\% } \\ \hline 2 & \text { Down } 1 \% \\ \hline 3 & \text { No change } \\ \hline 4 & \text { Up 2\% } \\ \hline 5 & \text { Up 4\% } \\ \hline 6 & \text { Up 9\% } \\ \hline \end{array} $$ To get the entry corresponding to \(t=2\), we roll a die and change the population from \(t=1\), again using the above rule. This procedure is then followed for \(t=3\), and so on. a. Using this procedure, make a table recording the population values for years 0 through 10 . b. Plot the data points from your table and the exponential model on the same screen. Comment on the level of agreement.

Relative abundance of species: A collection of animals may contain a number of species. Some rare species may be represented by only 1 individual in the collection. Others may be represented by more. In 1943 Fisher, Corbert, and Williams \({ }^{37}\) proposed a model for finding the number of species in certain types of collections represented by a fixed number of individuals. They associate with an animal collection two constants, \(\alpha\) and \(x\), with the following property: The number of species in the sample represented by a single individual is \(\alpha x\); the number in the sample represented by 2 individuals is \(\alpha\left(x^{2} / 2\right)\); and, in general, the number of species represented by \(n\) individuals is \(\alpha\left(x^{n} / n\right)\). If we add all these numbers up, we get the total number \(S\) of species in the collection. It is an important fact from advanced mathematics that this sum also yields a natural logarithm: $$ S=\alpha x+\alpha \frac{x^{2}}{2}+\alpha \frac{x^{3}}{3}+\cdots=-\alpha \ln (1-x) $$ If \(N\) is the total number of individuals in the sample, then it turns out that we can find \(x\) by solving the equation $$ \frac{x-1}{x} \ln (1-x)=\frac{S}{N} . $$ We can then find the value of \(\alpha\) using $$ \alpha=\frac{N(1-x)}{x} \text {. } $$ In 1935 at the Rothamsted Experimental Station in England, 6814 moths representing 197 species were collected and catalogued. a. What is the value of \(S / N\) for this collection? (Keep four digits beyond the decimal point.) b. Draw a graph of the function $$ \frac{x-1}{x} \ln (1-x) $$ using a horizontal span of 0 to 1 . c. Use your answer to part a and your graph in part b to determine the value of \(x\) for this collection. (Keep four digits beyond the decimal point.) d. What is the value of \(\alpha\) for this collection? (Keep two digits beyond the decimal point.) e. How many species of moths in the collection were represented by 5 individuals? f. For this collection, plot the graph of the number of species represented by \(n\) individuals against \(n\). Include values of \(n\) up to 20 .
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