Question

Sound Recordings The percents of sound recordings purchased over the Internet (not including digital downloads) in the years 1999 to 2005 are shown in the table. In the table, \(x\) represents the year, with \(x=0\) corresponding to 2000. (Source: The Recording Industry Association of America) $$ \begin{array}{|c|c|} \hline \text { Year, } x & \begin{array}{c} \text { Percent of sound } \\ \text { recordings, } y \end{array} \\ \hline-1 & 2.4 \\ \hline 0 & 3.2 \\ \hline 1 & 2.9 \\ \hline 2 & 3.4 \\ \hline 3 & 5.0 \\ \hline 4 & 5.9 \\ \hline 5 & 8.2 \\ \hline \end{array} $$ (a) Find the least squares regression parabola \(y=a x^{2}+b x+c\) for the data by solving the following system. \(\left\\{\begin{array}{r}7 c+14 b+56 a=31.0 \\\ 14 c+56 b+224 a=86.9 \\ 56 c+224 b+980 a=363.3\end{array}\right.\) (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 percent of Internet sales in 2008 . Does your result seem reasonable? Explain.

Short answer

The solution to the system is the coefficients of the regression parabola. By comparing this to the equation generated using regression analysis in a graphing utility, you should find that they are very similar. Substituting \(x=8\) should give a reasonable prediction considering the data trends

Step-by-step solution

Solve the System of Equations

First, solve the provided system of linear equations. You can do this using any method of solving systems you are comfortable with. It should lead to the solutions \(a\), \(b\), and \(c\) which make up the quadratic regression parabola

Use a Graphing Utility for Regression Analysis

Use a graphing utility, such as a graphing calculator or a computer tool, to conduct quadratic regression on the same data set. Compare the equation it gives with the equation from Step 1. They should be very similar, with slight differences due to round-off errors.

Predict Future Sales

Substitute \(x=8\) (for the year 2008) into the regression parabola to predict the percent of Internet sales for that year. The number you get should make sense considering the trends shown in the given data

Key concepts

Least Squares Method

The Least Squares Method is a mathematical approach to finding the best-fitting curve for a set of data points. This method minimizes the sum of the squares of the differences ("residuals") between the observed values and those predicted by the model. The main goal is to identify the coefficients of the model, in this case, those of a quadratic equation, that best represent the data. When applying the Least Squares Method to a quadratic model, we typically express it in the form:\[ y = ax^2 + bx + c \]Here, \(a\), \(b\), and \(c\) are the coefficients that need to be determined. For our problem, we're provided with a system of linear equations derived from the method:1. \(7c + 14b + 56a = 31.0\)2. \(14c + 56b + 224a = 86.9\)3. \(56c + 224b + 980a = 363.3\)By solving this system of equations, you'll find the values of \(a\), \(b\), and \(c\). These coefficients allow the quadratic model to closely fit the observed data points from 1999 to 2005. Solving the system involves using techniques like substitution or elimination, ensuring that the quadratic model reflects the buying trends accurately.

Graphing Utility

A graphing utility is a vital tool for visualizing and analyzing data. It helps to graph equations and perform complex calculations that would be tedious by hand. In the context of quadratic regression, a graphing utility assists in efficiently finding a quadratic model that fits the data points. Using a graphing calculator or computer software designed for statistical analysis, you input your data points, and the utility applies the regression technique to determine the best-fit quadratic equation.

• Simply enter the data into the utility tool.
• Select the quadratic regression function.
• Observe as it outputs the equation coefficients.

The key advantage of using a graphing utility is that it reduces human error and saves time while ensuring precision in results. When you compare the output from the utility with the equation derived manually through the Least Squares Method, minor discrepancies may appear due to rounding. However, both should be closely aligned, confirming the validity of your manual calculations.

Predictive Modeling

Predictive modeling is using historical data to predict future outcomes. In this exercise, the quadratic regression model developed helps in forecasting the percentage of sound recordings purchased over the Internet for a year beyond the given data.By setting \(x = 8\), which represents the year 2008 in this context, and substituting it into our regression model:\[ y = ax^2 + bx + c \]You can project the expected percentage for that year. Here's why predictive modeling is useful:- It provides insight into potential future trends.- Helps in making informed decisions based upon projections.In this specific case, your prediction should align with the trend seen from 1999 to 2005. Increases in the percentages would not be surprising given the growing reliance on the Internet for sound recordings. If the prediction seems plausible, it validates the model’s effectiveness in capturing the underlying pattern of the data.