The table shows the numbers of cellular phone subscribers...

The table shows the numbers of cellular phone subscribers (in millions) in the United States for selected years. (Source: $\mathrm{CTIA}$ -The Wireless) $$ \begin{array}{|c|c|c|c|c|c|c|}\hline \text { Year } & {1995} & {1998} & {2001} & {2004} & {2007} & {2010} \\ \hline \text { Number } & {34} & {69} & {128} & {182} & {255} & {303} \\ \hline\end{array} $$ (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 $t=5$ corresponding to $1995 .$ (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 cellular phone subscribers in the United States in the year 2020 .

The table shows the numbers of cellular phone subscribers (in millions) in the United States for selected years. (Source: $\mathrm{CTIA}$ -The Wireless)
$$
\begin{array}{|c|c|c|c|c|c|c|}\hline \text { Year } & {1995} & {1998} & {2001} & {2004} & {2007} & {2010} \\ \hline \text { Number } & {34} & {69} & {128} & {182} & {255} & {303} \\ \hline\end{array}
$$
(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 $t=5$ corresponding to $1995 .$
(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 cellular phone subscribers in the United States in the year 2020 .

Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two, generally expressed in the form y = ax² + bx + c. They are widely used in modeling because they can capture non-linear trends in data, such as acceleration or deceleration over time. In this context, using a quadratic function allows for a curved fit that can represent growth patterns more accurately than a linear model.

Regression Analysis

Regression analysis is a statistical technique used to determine the relationship between a dependent variable and one or more independent variables. It involves fitting a model to observed data so that predictions or insights about the underlying trends can be made. Here, regression is utilized to find the best-fitting quadratic model that describes the relationship between time and the number of cellular subscribers.

Data Visualization

Data visualization involves graphically representing data points along with models to aid in understanding patterns, trends, and deviations. Using a graphing utility to plot the original data and the corresponding quadratic model helps in comparing the model's predictions against the real values, thereby assessing the model's accuracy and appropriateness.

Extrapolation

Extrapolation is the process of extending a model beyond its original range to make predictions about future or unseen data. Once a regression model is established, it can be used to estimate values at points outside the range of the observed data, such as forecasting the number of subscribers in a later year. However, caution is advised as extrapolation assumptions may not always hold true in different contexts.

Time Variable Transformation

Time variable transformation involves redefining the independent variable, often to simplify the calculation and interpretation of the model. For instance, resetting the starting year to t = 0 or any other convenient value (like t = 5 for 1995) allows for a more balanced distribution of data points and can reduce computational complexity or numerical errors during regression analysis.

Transcript

00:02 We were given this table of data where we're given some column of year, and then this is the number of mobile subscribers in millions in the us, i believe.

00:14 And we are asked the following questions.

00:17 So we're asked first a to take this data and fit into a model of the form y is equal to some constant a, some primord of t squared, plus b, some of the constant b times t plus see where each of those there they tell us that y is the number of subscribers in millions and the variable t that's the years with t equals 5 starting at 1995 so in other words if we add another column here t then t would be 5 in 1999 5 then it would be 8 in 1998 11, 14, 17, and finally, 20.

01:23 And we want to fit this model to this equation.

01:26 So i'm going to go ahead and pull up an online quadratic regression, which you can just find by googling.

01:32 So this is the desmos quadratic equation.

01:34 It's just a free online quadratic regression you see on the left here.

01:38 And i went ahead and put in this table of data, as you can see here, where t is the time in years, and then y1 here would be our y.

01:49 And doing that, we get our plot on the right here, and we get our model data here where a, b, and c have been found using this online quadratic regression.

02:00 So that is what we get for our regression fit.

02:05 So in other words, this would give us the fit y is equal to 0 .24 t squared plus 12 .64t minus 39 .9.

02:19 So that is the regression model we get from the online fitter.

02:24 And you should get the same out of a calculated fitter, whatever you use.

02:29 And so in b, they ask us how, if the model fits the data well, if we go back to our plot.

02:38 So in red is the model fit, and then the black points are our data.

02:43 So we can zoom in on this, and we can see that we have to zoom, zoom in quite a bit in order for there to be any difference between the model and the data...

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