Modeling Data The table shows the numbers of cell phone...

Modeling Data The table shows the numbers of cell phone subscribers (in millions) in the United States for selected years. (Source: CTIA-The Wireless Association) $$\begin{array}{|c|c|c|c|c|}\hline \text { Year } & {2000} & {2002} & {2004} & {2006} \\ \hline \text { Number } & {109} & {141} & {182} & {233} \\ \hline \text { Year } & {2008} & {2010} & {2012} & {2014} \\ \hline \text { Number } & {270} & {296} & {326} & {355} \\ \hline\end{array}$$ (a) Use the regression capabilities of a graphing utility to find a mathematical model of the form $y=a t^{2}+b t+c$ for the data. In the model, $y$ represents the number of subscribers (in millions) and $t$ represents the year, with $t=0$ corresponding to $2000 .$ (b) Use a graphing utility to plot the data and graph the model. Compare the data with the model. (c) Use the model to predict the number of cell phone subscribers in the United States in the year 2024 .

Modeling Data The table shows the numbers of cell phone subscribers (in millions) in the United States for selected years. (Source: CTIA-The Wireless Association)

$$\begin{array}{|c|c|c|c|c|}\hline \text { Year } & {2000} & {2002} & {2004} & {2006} \\ \hline \text { Number } & {109} & {141} & {182} & {233} \\ \hline \text { Year } & {2008} & {2010} & {2012} & {2014} \\ \hline \text { Number } & {270} & {296} & {326} & {355} \\ \hline\end{array}$$

(a) Use the regression capabilities of a graphing utility to find a mathematical model of the form $y=a t^{2}+b t+c$ for the data. In the model, $y$ represents the number of subscribers (in millions) and $t$ represents the year, with $t=0$ corresponding to $2000 .$
(b) Use a graphing utility to plot the data and graph the model. Compare the data with the model.
(c) Use the model to predict the number of cell phone subscribers in the United States in the year 2024 .

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Key Concepts

Extrapolation

Extrapolation is the process of using an inferred model to predict values beyond the range of observed data. Once a model is established, it can be used to forecast future data points, such as estimating the number of cell phone subscribers in a future year.

Data Visualization

Data visualization entails graphically representing data to assess patterns, trends, and the fit of a model. Plotting both the observed data and the regression model on a graph helps in evaluating how accurately the model reflects the data.

Regression Analysis

Regression analysis is a statistical method used for estimating the relationships among variables. In this context, it involves using the least squares method to determine the coefficients in a quadratic equation that best fit the given data points.

Quadratic Modeling

Quadratic modeling involves representing data with a second-degree polynomial equation of the form y = at² + bt + c. This form is particularly useful for capturing curvature in the relationship between two variables, allowing for increased accuracy when trends are non-linear.

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