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Chapter 4: Problem 4

Magazine circulation: The following table shows the circulation, in thousands, of a magazine at the start of the given year. $$ \begin{array}{|c|c|} \hline \text { Year } & \begin{array}{c} \text { Circulation } \\ \text { (thousands) } \end{array} \\ \hline 2004 & 2.64 \\ \hline 2005 & 2.77 \\ \hline 2006 & 2.94 \\ \hline 2007 & 3.08 \\ \hline 2008 & 3.25 \\ \hline 2009 & 3.42 \\ \hline \end{array} $$ 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 of the logarithm.

Short Answer

Expert verified
The data can be approximated with an exponential function due to the linear fit of the log-transformed data.

Step by step solution

01

List the Data

Start by listing the given data in terms of the year and circulation. This is the first step before any transformation or computation: - 2004: 2.64 (thousands of subscribers) - 2005: 2.77 - 2006: 2.94 - 2007: 3.08 - 2008: 3.25 - 2009: 3.42
02

Calculate the Natural Logarithm

Take the natural logarithm of each circulation value to transform the data into a format suitable for linear regression.- 2004: \( \ln(2.64) \approx 0.971 \)- 2005: \( \ln(2.77) \approx 1.019 \)- 2006: \( \ln(2.94) \approx 1.078 \)- 2007: \( \ln(3.08) \approx 1.124 \)- 2008: \( \ln(3.25) \approx 1.178 \)- 2009: \( \ln(3.42) \approx 1.229 \)
03

Plot the Logarithmic Data

Using a graph, plot the calculated natural logarithms against the corresponding years. This will visually show whether the data points form a straight line, suggesting an exponential relationship in the original data.
04

Analyze the Plot

Observe the graph of the logarithmic data. If the points closely form a straight line, the data can be reasonably approximated by an exponential function. In this case, the points should approximately align linearly, suggesting an exponential trend in the original data.
05

Determine the Regression Line

Using linear regression techniques, calculate the best-fit line for the logarithmic data. The general form of the equation will be:\[ y = mx + c \]For this data, calculate 'm' (slope) and 'c' (y-intercept) such that they minimize the sum of squares of residuals between the observed \( y \) values (logarithms) and the values predicted by the line.
06

Draw the Regression Line

Plot the regression line on the same graph as the logarithmic data. This visual comparison will help confirm the accuracy of the fit.
07

Interpret Results in Context

Understand from the regression line's fit that since it closely follows the transformations of the original data, the assumption of an exponential model is reasonable. This interpretation aligns with the mathematical observation of the linearity in the log-transformed values.

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Key Concepts

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

Natural Logarithm
The natural logarithm is a mathematical function denoted as \(\ln(x)\), which is the logarithm to the base \(e\), where \(e\) is approximately 2.718. It is widely used in mathematics and science to transform exponential growth data into a linear format.
This is particularly useful in our exercise, as it helps simplify the analysis of the magazine's circulation over the years.

By converting the circulation numbers into their natural logarithms, each data point becomes the exponent to which \(e\) must be raised to obtain the original circulation value. For example, \(\ln(2.64) \approx 0.971\) means that \(e^{0.971} \approx 2.64\).

This transformation helps in linearizing exponential growth data, making it easier to apply linear regression techniques, which assume a linear relationship between variables.
Linear Regression
Linear regression is a statistical method used to model the relationship between two variables by fitting a linear equation to observed data.
In this context, after applying the natural logarithm to the circulation data, linear regression helps us find the best-fit line through the transformed data.

The general form of the linear equation is \(y = mx + c\), where:
  • \(y\) is the dependent variable (here, the natural logarithm of circulation)
  • \(x\) is the independent variable (the year)
  • \(m\) is the slope of the line
  • \(c\) is the y-intercept

By performing linear regression on the log-transformed circulation data, we can determine the slope \(m\) and y-intercept \(c\) that minimize the difference between the observed and predicted logarithmic values.
This linear relationship supports the hypothesis of exponential growth in the original data.
Logarithmic Transformation
A logarithmic transformation is a process of applying the logarithm to data points to reduce skewness and approximate an exponential relationship with linear trends.
It is a crucial step in analyzing data that grows exponentially, like the magazine circulation in the exercise.

After computing the natural logarithm of the circulation numbers, we can plot these against the years.
This allows for a visual assessment, as a roughly straight line will suggest an exponential pattern in the original data.

The transformation is not only a mathematical necessity for applying linear regression, but it also simplifies the complex exponential growth pattern, making it easier to interpret and predict future values based on past trends.
This clarity is essential for confidently making decisions based on the modeled data.

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Most popular questions from this chapter

Headway on four-lane highways: When traffic is flowing on a highway, the headway is the average time between vehicles. On four-lane highways, the probability \(P\) that the headway is at least \(t\) seconds is given to a good degree of accuracy \({ }^{6}\) by $$ P=e^{-q t}, $$ a. On a four-lane highway carrying an average of 500 vehicles per hour in one direction, what is the probability that the headway is at least 15 seconds? (Note: 500 vehicles per hour is \(\frac{500}{3600}=0.14\) vehicle per second.) b. On a four-lane highway carrying an average of 500 vehicles per hour, what is the decay factor for the probability that headways are at least \(t\) seconds? Reminder: An important law of exponents tells us that \(a^{b c}=\left(a^{b}\right)^{c}\). where \(q\) is the average number of vehicles per second traveling one way on the highway.

Economic growth of the United States: This exercise and the next refer to the gross domestic product (GDP), which is the market value of the goods and services produced in a country in a given year. The data in these exercises are adapted from a report published in 2006 that predicted rates of growth for the world economy over a 15-year period.43 The following table shows the GDP of the United States, in trillions of dollars. The data are based on figures and projections in the report. $$ \begin{array}{|c|c|} \hline \text { Year } & \text { GDP (trillions of dollars) } \\ \hline 2005 & 12.46 \\ \hline 2008 & 13.62 \\ \hline 2011 & 14.85 \\ \hline 2014 & 16.13 \\ \hline 2017 & 17.52 \\ \hline \end{array} $$ a. Plot the natural logarithm of the data, and find the equation of the regression line for the natural logarithm of the data. Round the regression line parameters to four decimal places. b. Use the logarithm as a link to find an exponential model of the GDP in the United States. Round the initial value and growth factor to three decimal places.

Research project: For this project, you should collect and analyze data for a population of M\&M's TM. Start with four candies, toss them on a plate, and add one for each candy that has the M side up; record the data. Repeat this seven times and see how close the data are to being exponential. For a detailed description, go to http://college.hmco.com/PIC/crauder4e.

Population growth: A population of animals is growing exponentially, and an ecologist has made the following table of the population size, in thousands, at the start of the given year. $$ \begin{array}{|c|c|} \hline \text { Year } & \begin{array}{c} \text { Population } \\ \text { in thousands } \end{array} \\ \hline 2003 & 5.25 \\ \hline 2004 & 5.51 \\ \hline 2005 & 5.79 \\ \hline 2006 & 6.04 \\ \hline 2007 & 6.38 \\ \hline 2008 & 6.70 \\ \hline \end{array} $$ Looking over the table, the ecologist realizes that one of the entries for population size is in error. Which entry is it, and what is the correct population? (Round the ratios to two decimal places.)

Aphid growth on broccoli: This problem and the following one are based on the results of a study by R. Root and A. Olson of population growth for aphids on various host plants.44 The value of r = 0.243 per day is valid for aphids reared on broccoli plants outdoors. The table on the next page describes the growth of an aphid population reared on broccoli plants indoors (in a controlled environment). Time t is measured in days. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Time } t & 0 & 1 & 2 & 3 & 4 \\ \hline \text { Population } & 25 & 30 & 37 & 44 & 54 \\ \hline \end{array} $$ Calculate r for this table. Did the aphids fare better indoors or outdoors?
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