Question

Sailboats The total numbers \(y\) (in thousands) of sailboats purchased in the United States in the years 2001 to 2005 are shown in the table. In the table, \(x\) represents the year, with \(x=0\) corresponding to \(2003 .\) (Source: National Marine Manufacturers Association) $$ \begin{array}{|c|c|} \hline \text { Year, } x & \text { Number, } y \\ \hline-2 & 18.6 \\ \hline-1 & 15.8 \\ \hline 0 & 15.0 \\ \hline 1 & 14.3 \\ \hline 2 & 14.4 \\ \hline \end{array} $$ (a) Find the least squares regression parabola \(y=a x^{2}+b x+c\) for the data by solving the following system. \(\left\\{\begin{aligned} 5 c &+10 a=78.1 \\\ 10 b &=-9.9 \\ 10 c &+34 a=162.1 \end{aligned}\right.\) (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).

Short answer

The least squares regression parabola for the given data is \(y = -0.445x^2 -0.99x + 15.14\). After comparing this with a model obtained from a graphing utility, the two should yield approximately similar results.

Step-by-step solution

Solve the System of Equations

This is a system of linear equations which can be solved by substitution or elimination method. From the second equation of the system \[10b = -9.9\], we can find 'b' by dividing both sides by 10. This gives \(b = -0.99\). Substitute 'b' into the first and third equations: \[5c +10a = 78.1\] and \[10c + 34a = 162.1\] We get: \[5c + 10a = 78.1\] and \[10c + 34a = 162.1\].These are two equations in two variables 'a' and 'c'. These can be solved by substitution or by elimination. Doing so gives the solutions: \(a = -0.445\) and \(c = 15.14\).

Validate the Equation

For validation, substitute the obtained values of 'a', 'b' and 'c' in the equation \(y = ax^2 + bx + c\). This gives the equation of parabola as \(y = -0.445x^2 -0.99x + 15.14\). Plug in the values of 'x' from the table and check if they are approximately close to the corresponding 'y' values.

Use a Graphing Utility to Find a Quadratic Model

Using a graphing utility, enter the given data points and use the program's regression feature to get the quadratic model. Most modern calculators and computer software have this feature. Remember to reset or 'clear' the calculator before starting to avoid any discrepancies. Additionally, ensure to correctly enter each data point as inaccuracies can drastically alter the resulting model.

Compare the Quadratic Model with the Model Found in Part (a)

Lastly, compare the quadratic model found with the graphing utility, with the model found in the first part. The models should be similar, as both methods aim to find the best fit for the given data.

Key concepts

System of Linear Equations

Understanding the system of linear equations is fundamental when working with data analysis, particularly in the least squares regression method. A system of linear equations consists of two or more equations that share common variables. The goal is to find the values for the variables that satisfy all the equations simultaneously.

In the context of the least squares regression, we are often presented with equations that are formed by the sums of squares of the vertical distances of the points from the model being considered. In our sailboat data example, we solve the system to find the coefficients of a quadratic model, which is a parabola, that fits our data points.

There are various methods to solve these systems, including substitution, elimination, and matrix operations. The substitution and elimination methods involve rearranging the equations and systematically eliminating variables until you are left with one variable, which can be solved directly. Once one variable is known, we back-substitute to find the others.

This methodology is the cornerstone for developing predictive models and is widely used in various fields such as economics, engineering, and physical sciences.

Quadratic Model

A quadratic model involves an equation of the form \(y = ax^2 + bx + c\) where \(a\), \(b\), and \(c\) are constants, and the graph of the equation is a parabola. This model is particularly useful when the relationship between the independent variable \(x\) and the dependent variable \(y\) is nonlinear, displaying a curved pattern. In our exercise, we're using a quadratic model to fit the sailboat sales data.

The least squares regression aims to find the values of \(a\), \(b\), and \(c\) that minimize the sum of the squares of the residuals—the differences between the observed and predicted values of \(y\). The result is the 'best fit' parabola for our data.

While the linear model (straight line) is effective for numerous relationships, a quadratic model can provide a more accurate fit for data with an observable curvature. Hence, knowing when to apply a quadratic model plays a critical role in data analysis and forecasting.

Graphing Utility Regression Feature

Modern graphing utilities are powerful tools that can perform complex statistical operations, including regression analysis, in seconds. The regression feature of a graphing utility automatically calculates the best-fit equation for a given set of data points.

By entering the data points into the software, the user can employ the regression feature to obtain a mathematical model, such as a quadratic model. This model minimizes the sum of the squares of the deviations of the data from the model, which is aligned with the least squares regression method.

To use this feature effectively, it is crucial to correctly enter the data points and choose the correct model type – in our case, the quadratic model. This automated process is not only faster than manual calculations but also reduces human error, making it an invaluable asset when working with statistical data. After obtaining the regression equation, it's also important to validate and, if necessary, compare it with a manually derived equation to ensure accuracy.