91Ó°ÊÓ

$$ \begin{array}{|c|c|} \hline \text { Date } & \begin{array}{c} \text { Cellular service revenue } \\ \text { (in billions) } \end{array} \\ \hline 2002 & 76.5 \\ \hline 2003 & 87.6 \\ \hline 2004 & 102.1 \\ \hline 2005 & 113.5 \\ \hline \end{array} $$ Cell phones: The table on the following page gives the amount spent on cellular service. a. Plot the data points. b. Find the equation of the regression line and add its graph to the plotted data. c. In \(2006, \$ 125.5\) billion was spent on cellular service. If you had been a financial strategist in 2005 with only the data in the table above available, what would have been your prediction for the amount spent on cellular service in 2006 ?

Step by step solution

01

Plotting Data Points

Take the given data points (years 2002, 2003, 2004, 2005) and their corresponding cellular service revenues (76.5, 87.6, 102.1, 113.5) and plot them on a graph. Set the x-axis as the year and the y-axis as the revenue in billions. Each year corresponds to a data point at the coordinates (year, revenue): (2002, 76.5), (2003, 87.6), (2004, 102.1), (2005, 113.5).

02

Calculating the Linear Regression Line

Use the method of least squares to find the linear regression line. First, calculate the means of the x-values (years) and y-values (revenues). Then, determine the slope (m) using the formula: \[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \]where \( n \) is the number of data points. Next, calculate the y-intercept (b) using:\[ b = \frac{(\sum y) - m (\sum x)}{n} \].Finally, write the equation of the line in the form \( y = mx + b \).

03

Drawing the Regression Line

Using the linear equation from the previous step, draw the regression line on the same graph as the data points. This line represents the best fit for the data.

04

Predicting 2006 Revenue

To predict the 2006 cellular service revenue, use the regression equation. Substitute \( x = 2006 \) into the equation \( y = mx + b \) and solve for \( y \). This will provide the predicted revenue for 2006 based on the regression line.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Regression

Linear regression is a statistical method used to model the relationship between two variables by fitting a linear equation to observed data. The key idea is to find a straight line that best captures the general trend of the data points.

In a typical linear regression setup, you have an independent variable (often denoted as \( x \)) and a dependent variable (denoted as \( y \)). The goal is to find the equation of the line:

• Slope (\( m \)): Indicates how much \( y \) changes for a unit change in \( x \).
• Y-intercept (\( b \)): The value of \( y \) when \( x \) is 0.

This equation is written as \( y = mx + b \). Linear regression helps in identifying trends and making predictions based on past data.

Data Plotting

Data plotting is a visual representation of data points on a graph, which makes it easier to identify trends, patterns, and anomalies. In our exercise, plotting involves placing data points in a coordinate system where the x-axis represents years and the y-axis represents revenue in billions.

By plotting the provided data:

• (2002, 76.5)
• (2003, 87.6)
• (2004, 102.1)
• (2005, 113.5)

we can see an upward trend, indicating a steady increase in revenue over the years. Such visualizations are crucial for understanding data at a glance and making informed decisions.

Prediction Modeling

Prediction modeling involves using statistical techniques to forecast future data points based on existing data.The linear regression model, once calculated, can be used for prediction by simply plugging in future values of \( x \), such as predicting future revenue based on past trends.

In the exercise, with the regression line equation calculated, predicting the revenue for the year 2006 is straightforward:

• Substitute \( x = 2006 \) into the equation \( y = mx + b \).
• Calculate \( y \), the predicted revenue.

By focusing on these calculations, predictions can be effectively made to direct strategic decisions for future outcomes.

Least Squares Method

The least squares method is a mathematical approach used to determine the best-fitting line through a set of points by minimizing the sum of the squares of the vertical distances of the points from the line. It's essential for finding the slope and intercept in linear regression.

The least squares method formula for slope (\( m \)) calculation is: \[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}\]
where \( n \) is the number of data points.

After calculating \( m \), the intercept (\( b \)) is found using:\[ b = \frac{(\sum y) - m (\sum x)}{n}\]
These calculations result in the regression line equation \( y = mx + b \), the foundation of prediction modeling.