The table shows the numbers of international travelers y (in...

The table shows the numbers of international travelers $y$ (in thousands) to the United States from South America from 2010 through 2012 . (a) The data can be approximated by a parabola. Create a system of linear equations for the data. Let $t$ represent the year, with $t=0$ corresponding to $2010 .$ (b) Use the matrix capabilities of a graphing utility to find an inverse matrix to solve the system in part (a) and find the least squares regression parabola $y=a t^{2}+b t+c$ (c) Use the graphing utility to graph the parabola with the data points. (d) Use the result of part (b) to estimate the numbers of international travelers to the United States from South America in $2013,2014,$ and 2015 (e) Are your estimates from part (d) reasonable? Explain.

The table shows the numbers of international travelers $y$ (in thousands) to the United States from South America from 2010 through 2012 .
(a) The data can be approximated by a parabola. Create a system of linear equations for the data. Let $t$ represent the year, with $t=0$ corresponding to $2010 .$
(b) Use the matrix capabilities of a graphing utility to find an inverse matrix to solve the system in part
(a) and find the least squares regression parabola $y=a t^{2}+b t+c$
(c) Use the graphing utility to graph the parabola with the data points.
(d) Use the result of part (b) to estimate the numbers of international travelers to the United States from South America in $2013,2014,$ and 2015
(e) Are your estimates from part (d) reasonable? Explain.

Key Concepts

Matrix Inversion and Linear Algebra Techniques

In solving the system of equations for the regression model, one can organize the equations into matrix form and use matrix inversion to compute the coefficients. This linear algebra approach simplifies the process of solving multiple simultaneous equations and is particularly efficient when using technological tools such as graphing utilities.

Least Squares Regression

Least squares regression is a statistical method used to minimize the sum of the squared differences between the observed values and the values predicted by the model. This method is especially useful when more data points are available than are needed to exactly determine the model, resulting in an overdetermined system. The goal is to find the coefficients that best fit the data overall.

Extrapolation and Model Validity

Once a model is fitted to the data, it can be used to make predictions beyond the range of the observed data, a process known as extrapolation. However, caution must be exercised because the reliability of such estimates depends on the appropriateness of the model beyond the range of the sample data. Assessing the reasonableness of these predictions is a crucial part of any modeling process.

Quadratic (Parabolic) Modeling

This concept involves modeling a set of data points using a quadratic function of the form y = at² + bt + c. It is particularly useful when the relationship between the variables appears to follow a curved pattern. In this context, fitting a parabola provides a simple yet effective way to approximate the trend in the data over time.

System of Linear Equations

To determine the coefficients a, b, and c of the quadratic function, one sets up a system of linear equations using the given data points. Each data point substituted into the quadratic equation yields an equation, and with enough data points, a system is formed that can be solved simultaneously. This forms the basis for finding the best fitting curve.

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