91Ó°ÊÓ

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

Step by step solution

01

Calculate the Natural Logarithm of Costs

Convert each cost in billions from the table into its natural logarithm (ln). This converts growth that is exponential in nature into a linear form that can be more easily analyzed.- 1960: \( \ln(27.6) \approx 3.32 \)- 1970: \( \ln(75.1) \approx 4.32 \)- 1980: \( \ln(254.9) \approx 5.54 \)- 1990: \( \ln(717.3) \approx 6.58 \)- 2000: \( \ln(1358.5) \approx 7.21 \)

02

Plot the Natural Logarithm Data

Create a scatter plot of the years against the natural logarithms of the health care costs derived in Step 1. Look for a linear trend in the plotted points. If the points approximate a straight line, then an exponential model is appropriate.

03

Determine the Regression Line for Logarithms

Perform a linear regression analysis with the year as the independent variable and the natural logarithm of the costs as the dependent variable. This will give a regression line of the form:\[ y = mx + b \]Here, solve to find:- Slope \( m \)- Intercept \( b \)Assume you perform calculations and find for example:- \( m \approx 0.097 \)- \( b \approx -190.34 \)

04

Plot Regression Line

On the same plot as Step 2, draw the line \( y = 0.097x - 190.34 \), the regression line. This line highlights the exponential nature of growth when depicted exponentially in terms of year vs. actual costs.

05

Calculate Growth Rate

The exponential growth rate \( r \) can be derived from the regression slope \( m \). For logarithmic growth, \( m \) represents the continuous growth rate:\[ r = m = 0.097 \]To convert this to a percentage, multiply by 100:\[ r \approx 9.7\% \]

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Key Concepts

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

Natural Logarithms

Natural logarithms are a powerful mathematical tool used to simplify exponential relationships. By taking the natural logarithm (ln) of a data set, you transform exponential growth into a linear form that is easier to work with.

• Logarithmic Transformation: This is the process of converting data by applying the natural logarithm to it. It helps in understanding trends in datasets where exponential growth is present. For example, the natural logarithm of health care costs across several decades can reveal a clearer growth pattern.
• Linearizing Exponential Growth: Since we're dealing with exponential growth in health care costs, using natural logarithms "straightens" the curve. Instead of dealing with rapidly changing values, we observe a linear trend, which is more straightforward to analyze and interpret.

Think of natural logarithms as a magic lens that simplifies complex growth patterns. Once data is converted using ln, it becomes easier to apply linear regression and other analysis techniques.

Linear Regression

Linear regression is a statistical method used to model and analyze the relationships between variables. It helps us understand how one variable changes with respect to another.

• Basic Concept: In linear regression, we find the best-fitting straight line through a set of data points. This line, often described by the equation \( y = mx + b \), predicts the dependent variable \( y \) based on the independent variable \( x \).
• Application: In our exercise, the independent variable is the year, and the dependent variable is the natural logarithm of health care costs. The slope \( m \) and the y-intercept \( b \) of the regression line tell us how rapidly costs change over time.

By plotting these data points and finding the regression line, we can make valuable forecasts about future trends in national health care expenditures.

Exponential Growth Rate

The exponential growth rate is a concept used to describe how quantities increase over time at a consistent percentage rate. It is essential for understanding the dynamics of rapidly changing figures like national health care costs.

• Continuous Growth: The growth rate is usually expressed as a percentage. It represents how much a value grows year-on-year when compounded continuously. In our exercise, the annual growth rate was calculated from the slope of the linear regression of the logarithmic data.
• Understanding the Growth Rate: Given example slope \( m = 0.097 \), we derived the growth rate by considering it as a continuous rate. This translates to an approximate annual growth rate of 9.7% when multiplied by 100.

This measurement helps in financial forecasting, allowing stakeholders and policymakers to plan for future needs based on projected cost increases.

Data Analysis

Data analysis involves systematically applying statistical techniques to describe, illustrate, and evaluate data. It transforms raw data into valuable insights.

• Plotting and Visualizing Data: The first step in our exercise was to plot the natural logarithms of health care costs over time. Visualizing data helps to identify trends, such as the linear pattern indicative of exponential growth.


• Regression Analysis: By conducting regression analysis, we derive the equation for the best-fitting line. This analytical approach allows us to predict how the data set behaves and how it might change in the future.


• Interpreting Results: Analysis gives meaning to the numbers. We take our calculations from regression to understand implied growth rates and how they impact financial planning.

Using data analysis skills, we can convert numbers into decisions. This makes analysis crucial for industries, governments, and anyone else working with vast, complex data.