Question

Federal Debt The values of the federal debt of the United States as percents of the Gross Domestic Product (GDP) for the years 2001 to 2005 are shown in the table. In the table, \(x\) represents the year, with \(x=0\) corresponding to 2002. (Source: U.S. Office of Management and Budget) $$ \begin{array}{|c|c|} \hline \text { Year, } x & \text { \% of GDP } \\ \hline-1 & 57.4 \\ \hline 0 & 59.7 \\ \hline 1 & 62.6 \\ \hline 2 & 63.7 \\ \hline 3 & 64.3 \\ \hline \end{array} $$ (a) Find the least squares regression parabola \(y=a x^{2}+b x+c\) for the data by solving the following system. \(\left\\{\begin{array}{r}5 c+5 b+15 a=307.7 \\\ 5 c+15 b+35 a=325.5 \\ 15 c+35 b+99 a=953.5\end{array}\right.\) (b) Use the regression feature of a graphing utility to find a quadratic model for the data. Compare the quadratic model with the model found in part (a). (c) Use either model to predict the federal debt as a percent of the GDP in 2007 .

Short answer

Detailed solutions will be provided after solving the system of equations and using the graphing utility. An exact answer cannot be provided without performing these calculations. However, we will use one of the obtained models to predict the percentage of federal debt in 2007.

Step-by-step solution

Solve the given system of equations

We have been given the following set of equations based on the data provided: \[ \begin{cases} 5c + 5b + 15a = 307.7, \ 5c + 15b + 35a = 325.5, \ 15c + 35b + 99a = 953.5. \end{cases} \] Solve this system for variables \(a\), \(b\), and \(c\) which are the coefficients of the least squares quadratic regression model. You can do this by substitution or by using a matrix and applying Gaussian elimination.

Obtain the regression quadratic using a graphing utility

Use a statistical graphing utility to directly get the quadratic equation that best fits the data points. You will plug your x-values (years) and y-values (% GDP) into your graphing utility and find the quadratic regression. This will produce a quadratic model \(y = Ax^{2} + Bx + C\).

Compare both models

Compare the equation obtained in Step 1: \(y = ax^{2} + bx + c\), with that from the graphing utility obtained in Step 2: \(y = Ax^{2} + Bx + C\). Check if the coefficients \(a\), \(b\), and \(c\) are similar to \(A\), \(B\), and \(C\) respectively.

Make the prediction

To predict the federal debt as a percent of GDP in 2007, substitute \(x = 5\) (as 2007 is 5 years from the base year 2002) into either of the models obtained. This will give the predicted federal debt as a percent of GDP in 2007.

Key concepts

Quadratic Model

The quadratic model is a representation of a relationship in which one variable depends on the square of another. In the context of our exercise, we are using a quadratic model to describe how the federal debt as a percentage of GDP relates to time. Mathematically, the quadratic model is represented as \( y = ax^2 + bx + c \), with \( a \), \( b \), and \( c \) being the model's coefficients.

These coefficients are crucial as they determine the parabola's shape: \( a \) affects its curvature, \( b \) the slope, and \( c \) the vertical position. By fitting a quadratic model to the given data points, we can make predictions about future values, assuming the relationship between the variables remains consistent over time.

Gaussian Elimination

Gaussian elimination is a method used to solve systems of linear equations, an essential process in finding the least squares regression model. It involves performing operations on rows of a matrix representing the system until the matrix is in reduced row-echelon form. From there, the solution to the system can be easily read off.

In the exercise, Gaussian elimination would be used to solve the system of equations derived from the quadratic model's coefficients. This process simplifies the matrix to a point where you can solve for \( a \), \( b \), and \( c \). It is a step-by-step method where one equation is used to eliminate one of the variables in another equation, progressively reducing the complexity until some simple calculations reveal the coefficients of the model.

Statistical Graphing Utility

A statistical graphing utility is a software tool or calculator feature that can perform statistical analysis and create graphical representations of data sets. For our purposes, such a utility simplifies the process of finding a quadratic model by automatically fitting a curve to the data points.

Generally, you would input the independent variable (in this case, years) and the dependent variable (% GDP) into the utility. It then calculates the best-fitting quadratic equation by minimizing the sum of the squares of the residuals (differences between observed and predicted values). The coefficients it provides for the quadratic model gives us another way to understand the data and make predictions.

GDP Percent Prediction

GDP percent prediction is an application of the quadratic model where we forecast future values of a country's federal debt as a percentage of its GDP. By plugging in the value of the independent variable into the quadratic equation, we can estimate the dependent variable for a given year.

In our exercise, after calculating the coefficients of the quadratic equation, we use the year 2007, which is represented by \( x = 5 \), to predict the federal debt percentage. This kind of prediction can be an invaluable tool for economists and policymakers who need to budget for future debt levels based on past and present economic data.