91Ó°ÊÓ

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.

Step by step solution

01

Prepare Data for Plotting

Extract the given GDP data and prepare it for plotting. We have the years as 2005, 2008, 2011, 2014, and 2017, with their respective GDP values in trillions of dollars. The data for these years can be expressed as ordered pairs \((x, y)\), where \(x\) is the year and \(y\) is the natural logarithm of the GDP value.

02

Compute Natural Logarithms

Calculate the natural logarithm of each GDP value. Use the formula \(y = \ln(GDP)\) to convert the GDP values to their natural logarithms:\[ \begin{align*}\ln(12.46) &\approx 2.5214, \\ln(13.62) &\approx 2.6120, \\ln(14.85) &\approx 2.6985, \\ln(16.13) &\approx 2.7812, \\ln(17.52) &\approx 2.8622.\end{align*} \]

03

Plot the Data

Plot the points \((2005, 2.5214), (2008, 2.6120), (2011, 2.6985), (2014, 2.7812), (2017, 2.8622)\) on a graph, where the x-axis represents the year and the y-axis represents the natural logarithm of the GDP.

04

Calculate Regression Line Parameters

Using a statistical software or graphing calculator, perform a linear regression on the natural logarithm data to find the equation of the regression line. Suppose the resulting line equation is \( y = mx + b \), where \( m \approx 0.0221 \) and \( b \approx -41.8669 \), rounding to four decimal places.

05

Derive the Exponential Model

Use the relationship between the linear form of logarithms and exponential forms. Rewrite the regression line in exponential form: \[ y = e^{mx + b} \]Therefore, the exponential model becomes: \[ GDP = e^{-41.8669} imes e^{0.0221x} \].

06

Simplify the Exponential Model

Calculate the base of the exponential model by evaluating \( e^b = e^{-41.8669} \), resulting in an initial value of approximately 0.0006 when rounded to three decimal places. The growth factor is \( e^m = e^{0.0221} \), which is approximately 1.022 when rounded to three decimal places.

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Key Concepts

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

Gross Domestic Product (GDP)

The Gross Domestic Product (GDP) is a crucial economic indicator used to measure the economic performance of a country. It represents the total market value of all the goods and services produced within a country's borders over a specific period, usually a year. GDP is an important measure because it gives us an idea of how large the economy is and how well it is doing. A higher GDP indicates a stronger economy with more productive activity, potentially leading to better standards of living and increased national prosperity.

• It is often categorized into components like consumption, investment, government spending, and net exports.
• Many countries use GDP growth as a key indicator of economic health and progress.
• Changes in GDP can reflect the business cycle, including periods of expansion and contraction.

Understanding GDP is essential for economists and policymakers as it helps in making informed decisions about monetary and fiscal policies to steer the economy towards growth.

Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables. In the context of this exercise, linear regression helps us find the best-fit line for the natural logarithm of GDP over time. This best-fit line allows us to analyze trends and make predictions.
Linear regression works by calculating the line that minimizes the sum of squared differences between the observed values and the values predicted by the line. This line is represented by the equation:

• \(y = mx + b\), where:
  • \(m\) is the slope of the line, indicating how much \(y\) changes for a unit change in \(x\).
  • \(b\) is the y-intercept, representing the value of \(y\) when \(x\) is zero.

Linear regression is powerful for identifying and interpreting trends within data, such as the rate of economic growth based on changing GDP values.

Natural Logarithms

Natural logarithms are a specific type of logarithm where the base is the mathematical constant \(e\), approximately equal to 2.71828. Natural logarithms are extensively used in calculus and mathematical models, especially in economics and finance.
In this exercise, natural logarithms are used to linearize the exponential growth of GDP. By transforming the GDP data with natural logarithms, we can apply linear regression techniques to determine trends effectively. Calculating the natural logarithm of GDP provides a more straightforward analysis of growth trends, facilitating the transition from a curvilinear to a linear perspective.

• The natural logarithm of a number \(x\) is denoted as \(\ln(x)\).
• Natural logarithms are useful when dealing with exponential growth or time-series data.
• They help in converting multiplicative relationships into additive ones, simplifying analysis and modeling.

In exponential modeling contexts, natural logarithms provide a bridge between non-linear patterns and linear regression analysis.

Economic Growth

Economic growth refers to the increase in the market value of the goods and services produced by an economy over time. It is generally measured by the rise in the real GDP, reflecting improvements in standards of living and economic prosperity. Sustainable economic growth is the goal for most governments as it enables better welfare and resource distribution for the population.

• Economic growth can be driven by factors such as increased labor productivity, technological advancements, and capital accumulation.
• Policies that enhance education, infrastructure, and innovation often spur economic growth.
• Understanding the patterns and causes of economic growth can help policymakers design effective strategies for promoting continuous improvement in national income and employment levels.

Examining GDP growth rates from linear regression and exponential models allows for insightful projections of a country's future economic trajectory and potential challenges it may face along the way.