Question

The total values \(y\) (in billions of dollars) of child support collections from 1998 to 2005 are shown in the figure. The least squares regression parabola \(y=a t^{2}+b t+c\) for these data is found by solving the system $$ \left\\{\begin{array}{r} 8 c+92 b+1100 a=153.3 \\ 92 c+1100 b+13,616 a=1813.9 \\ 1100 c+13,616 b+173,636 a=22,236.7 \end{array}\right. $$ (a) Use a graphing utility to find an inverse matrix to solve this system, and find the equation of the least squares regression parabola. (b) Use the result from part (a) to estimate the value of child support collections in \(2007 .\) (c) An analyst predicted that the value of child support collections in 2007 would be \(\$ 24.0\) billion. How does this value compare with your estimate in part (b)? Do both estimates seem reasonable?

Short answer

The solution requires further calculation using the inverse of a matrix. But the logic would be to find a, b, and c through the inverse matrix, substitute t=9 in the equation \(y=a t^{2}+b t+c\) to find the estimated value for 2007. Then compare this estimation with the analyst's prediction of $24.0 billion. The reasonableness of these estimations can only be determined based on how close they are to each other and the given data set for previous years.

Step-by-step solution

Using a graphing utility to find the inverse matrix

Using a graphing utility, input the coefficients of the system of equations as a matrix. It should look like this: \[ \begin{pmatrix} 8 & 92 & 1100 \ 92 & 1100 & 13616 \ 1100 & 13616 & 173636 \end{pmatrix} \]. Then, input the constants on the right side of the equations as another matrix: \[ \begin{pmatrix} 153.3 \ 1813.9 \ 22236.7 \end{pmatrix} \]. Now, find the inverse of the first matrix using the utility, and multiply the resulting inverse matrix with the second matrix.

Finding the equation of the least squares regression parabola

After multiplying the inverse matrix with the constant matrix, the resulting matrix will present the coefficients of the regression parabola. These should be denoted as \(a\), \(b\), and \(c\) respectively. Hence, the equation of the least squares regression parabola can be formulated as \(y=a t^{2}+b t+c\).

Estimating the value of child support collections in 2007

For estimating the value of child support collections in 2007, replace the \(t\) in the regression equation obtained in Step 2 with 9 (as it is 9 years from 1998). The result will be an estimation for \(y\), the total value of child support collections in 2007.

Comparing the estimation with the analyst's prediction

Finally, compare the estimated value obtained in Step 3 with the analyst's predicted value of $24.0 billion. Analyze how close these two amounts are and discuss how probable each prediction may be.

Key concepts

Inverse Matrix

Understanding the concept of an inverse matrix is essential when dealing with solving systems of equations using matrices. A matrix can be thought of as a rectangular array of numbers, and an inverse matrix, when multiplied by the original matrix, results in an identity matrix. You can think of it like finding the reciprocal of a number. If you multiply a number by its reciprocal, you get 1. The identity matrix is the matrix equivalent of 1.

• Finding the inverse of a matrix is like unlocking a pattern hidden within it.
• In the context of least squares regression, the inverse matrix helps find the coefficients of a regression equation by solving a matrix equation.
• This process involves a graphing utility or software, as computing inverses manually can be complex and time-consuming.

Mastering this concept allows you to solve systems of linear equations efficiently and is a key step in many statistical methods.

Regression Parabola

A regression parabola is a curve that fits best to a set of data points. In the context of least squares regression, it minimizes the total squared differences between the observed values and those predicted by the model.
The equation given is a classic example of a quadratic function: \[ y = at^{2} + bt + c \]

• "a", "b", and "c" are the coefficients that shape the curve.
• They represent the influence of the variable "t" on the value of "y".
• This type of regression helps in situations where the data might follow a curvilinear trend rather than a simple straight line.

Quadratic regression is particularly useful in predicting trends that have accelerating or slowing growth patterns. It's a valuable tool in econometrics and various fields of science.

Estimation

Estimation is a fundamental statistical tool that allows us to predict future values based on existing data. Using the regression equation obtained from solving the inverse matrix, we plug in specific values of "t" to make predictions.

• For example, to estimate the value of child support collections in 2007, you substitute the calculated year value (e.g., 9 for 1998 plus 9 years) into the regression equation.
• This is a practical application of mathematical models in real-world scenarios.
• While estimations can provide close approximations, they are inherently imperfect and subject to external variables and changes over time.

Understanding estimation helps to comprehend how mathematical models translate into predictive analytics and decision-making.

Child Support Collections

Child support collections refer to the money collected from non-custodial parents to support their children financially. These figures are essential for assessing the efficiency of child support systems and policies.
In the given problem, historical data from previous years is analyzed to identify trends and make predictions for future collections.

• Analyzing such data helps policymakers and analysts evaluate program effectiveness.
• Understanding economic factors that might influence collection values is crucial for accurate predictions.
• Comparing estimates with analytical predictions aids in adjusting forecasts and identifying potential improvements in collection methods.

Considering real-life contexts, these predictions and analyses influence policy decisions that can have a profound impact on families relying on this support.