Problem 57 - College Algebra and Calculus: An Applied Approach

Question

Cable TV Subscribers The table shows the average numbers \(S\) (in millions) of basic cable subscribers for the years 1995 to 2005. (Source: Kagan Research, LLC) $$ \begin{aligned} &\begin{array}{|l|l|l|l|l|} \hline \text { Year } & 1995 & 1996 & 1997 & 1998 \\ \hline \text { Subscribers, } S & 60.6 & 62.3 & 63.6 & 64.7 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|l|} \hline \text { Year } & 1999 & 2000 & 2001 & 2002 \\ \hline \text { Subscribers, } S & 65.5 & 66.3 & 66.7 & 66.5 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|} \hline \text { Year } & 2003 & 2004 & 2005 \\ \hline \text { Subscribers, } S & 66.1 & 65.7 & 65.3 \\ \hline \end{array} \end{aligned} $$ (a) Use a graphing utility to create a scatter plot of the data. Let \(t\) represent the year, with \(t=5\) corresponding to \(1995 .\) (b) Use the regression feature of a graphing utility to find a quadratic model for the data. (c) Use a graphing utility to graph the model from part (b) in the same viewing window as the scatter plot. (d) Use the graph of the model from part (c) to estimate when the number of basic cable subscribers was the greatest. Does this result agree with the actual data?

Step-by-Step Solution

Answer

The short answer could not be formulated precisely as it depends upon the analysis done using a graphing utility. Generally, the step-by-step guide encourages creation of a scatter plot, fitting a quadratic model to it using a regression function, plotting the model on the same graph and then interpreting the results which will help estimate when the number of subscribers was the greatest.

Step 1: Creating Scatter Plot

Using the given data, create a scatter plot using a graphing utility. This will help understand the general trend of the data. Here, the x-axis will be the years (with 5 corresponding to 1995) and the y-axis represents the number of subscribers.

Step 2: Find a Quadratic Model

Utilizing a regression feature in the graphing utility, fit the plotted data to a quadratic regression model. This model can be useful to predict future data points.

Step 3: Plotting the Quadratic Model

In the same graph, plot the quadratic model obtained from the regression with the scatter plot to compare the fit.

Step 4: Interpret the Graph

From the graph, identify the peak of the plot which will indicate the year with the greatest number of subscribers. Check if it corresponds with the actual data to verify the effectiveness of the model.

Key Concepts

Scatter Plot

When visualizing data, creating a scatter plot is an excellent first step, especially when you're dealing with two variables. In this case, we're looking at the number of cable TV subscribers over a specific period. By plotting each data point on a graph where the x-axis represents time (with the baseline year marked as 5 for 1995) and the y-axis represents the number of subscribers in millions, we can begin to see patterns or trends.

A scatter plot is essentially a collection of all the points that represent the given data. Each point corresponds to a year's subscriber count. Once the plot is made, we can look for trends, such as an increase or decrease in subscribers, or perhaps a cyclical pattern. This visual aid makes it considerably easier to understand complex data at a glance.

Graphing Utility

To analyze data efficiently, it's essential to use a graphing utility. This user-friendly software tool allows us to input our data and then does the heavy lifting for us, creating the scatter plot automatically. More than just plotting points, a graphing utility often comes with a suite of features that aid in advanced analysis, such as regression, which can help us find a line or curve that best fits the data.

For this exercise, we would input the years and corresponding subscriber counts into the utility. The graphing utility not only provides us with the visual representation but also supports us in calculating a quadratic regression model which can predict future trends based on historical data.

Regression Analysis

Regression analysis is a powerful statistical method used to understand the relationship between variables and predict future points. After creating a scatter plot using a graphing utility, we'd proceed to regression analysis to fit a curve to our data.

Here, given the subtle turns in the data suggest a non-linear trend, a quadratic model—a type of polynomial regression—is determined to be appropriate. It's termed 'quadratic' because the highest power of the independent variable is a square (exemplified by an equation like \( ax^2 + bx + c \)). Utilizing the graphing utility, the quadratic equation coefficients (a, b, and c) are calculated so that the resulting curve closely follows the trend indicated by the scatter plot points.

Data Prediction

Data prediction is where the true value of regression analysis comes into play. Using our quadratic regression model, we can estimate values that are not directly measured in our data set. For instance, we can predict future subscriber counts based on previous years' trends or determine when the number of subscribers was at its peak.

The predicted peak in the graph corresponds to the highest point on the model's curve. By interacting with the graphing utility, we may estimate the year at which subscribers were most numerous. It's important to verify this prediction with actual data when available, to assess the accuracy and reliability of our regression model for making predictions.