Question

Super Bowl Ad Cost The table shows the costs \(C\) (in millions of dollars) of a 30 -second TV ad during the Super Bowl for several years from 1987 to \(2006 .\) (Source: TNS Media Intelligence)$$ \begin{array}{|c|c|} \hline \text { Year } & \text { Cost } \\ \hline 1987 & 0.6 \\ \hline 1992 & 0.9 \\ \hline 1997 & 1.2 \\ \hline 2002 & 2.2 \\ \hline 2006 & 2.5 \\ \hline \end{array} $$(a) Use a graphing utility to create a scatter plot of the data. Let \(t\) represent the year, with \(t=7\) corresponding to \(1987 .\) (b) Use the regression feature of a graphing utility to find an exponential model for the data. Use the Inverse Property \(b=e^{\ln b}\) to rewrite the model as an exponential model in base \(e\). (c) Use a graphing utility to graph the exponential model in base \(e\). (d) Use the exponential model in base \(e\) to predict the costs of a 30 -second ad during the Super Bowl in 2009 and in 2010 .

Short answer

The cost of a 30-second Super Bowl ad in 2009 and 2010 can be predicted by substituting \(t = 29\) and \(t = 30\) respectively into the exponential model equation \(C = ae^{rt}\). The actual values would depend on the parameters of the model.

Step-by-step solution

Converting Years

First, let's begin by making the year conversion. Since \(t=7\) corresponds to \(1987\), we have to subtract \(1980\) from all years. This will give an array of years represented as \(t = [7, 12, 17, 22, 26]\).

Scatter plot

Next, plot the cost \(C\) as a function of \(t\) using a graphing utility. The x-axis should represent years (in the converted form), and the y-axis should represent the cost in millions.

Exponential Model

With the scatter plot, use the 'regression' feature of the graphing utility to find an exponential model of the form \(C = ab^t\). This formula represents an exponential growth model, where \(C\) is the cost, \(a\) is the initial amount, \(b\) is the growth factor, and \(t\) is the time in years.

Model in Base \(e\)

Rewrite the exponential model using the inverse property, converting the base to \(e\). This is done by changing the formula to \(C = ae^{rt}\), where \(r = \ln(b)\). This form is useful as it provides the rate of change r directly.

Create Exponential Graph

Create a new graph using the model in base \(e\), \(C = ae^{rt}\), and overlay it on top of the original scatter plot.

Make Predictions

For making predictions, substitute \(t = 29\) and \(t = 30\) into the model equation to find the expected cost of a 30-second Super Bowl ad in 2009 and 2010 respectively.

Key concepts

Scatter Plot

A scatter plot is a type of graph that shows the relationship between two variables. It helps visualize how a dataset behaves on a plane. For the given exercise, you start by plotting the years against the cost of a 30-second Super Bowl ad to form dots on a graph. Each dot represents a data point with its coordinate according to year and cost.

To create this plot, assign the x-axis to represent years, translated into the form of time steps since 1980 (e.g., year 1987 becomes 7). The y-axis displays the costs in millions. This type of visualization is beneficial because:

• It reveals trends and patterns in the data.
• It makes outliers easily identifiable.
• It sets the groundwork for further analyses like regression.

Scatter plots are typically the first step in examining how data points are distributed, which helps inform the choice of model to use for predictions.

Regression Analysis

Regression Analysis is a statistical method used to determine the relationships between variables. In the context of our exercise, it involves finding an exponential trend that fits the scatter plot data. This step fits an exponential model to the data points on the graph, using a formula of the form: \( C = ab^t \). Here, \( C \) is the cost, \( a \) is a constant, \( b \) is the growth factor, and \( t \) is the time variable.

The process of regression helps in finding a smooth curve that minimally deviates from all data points. This smooth curve "predicts" the patterns of the dataset more accurately.

• Helps to explain variation that might not be immediately visible in the raw data.
• Can be used to make future predictions based on past trends.
• Minimizes the distance to all data points, achieving the best fit line.

Additionally, our exercise features rewriting the model using the inverse property, so it takes the form \( C = ae^{rt} \), with \( r = \ln(b) \), transforming it into base \( e \) for more direct readability of the growth rate.

Graphing Utilities

Graphing utilities are software tools or calculator functions that help visualize mathematical functions and data plots. In the realm of exponential models and regression, graphing utilities play a central role for several reasons:

• They allow the creation of scatter plots to visualize data at a glance.
• They provide tools for applying regression analysis to find the best-fitting model to the dataset.
• They help visualize the exponential growth curves by drawing them directly on the graph.

Graphing utilities extend beyond simple plotting. They often include features like overlaying multiple graphs for comparison, letting you visually compare your model against actual data, which confirms how well your model fits.

In this exercise, using a graphing utility allows overlaying the model developed via regression upon the data scatter plot, thus validating the trend seen from actual data points.