Question

How is the cost of a plane flight related to the length of the trip? The table shows the average round-trip coach airfare paid by customers of American Airlines on each of 18 heavily traveled U.S. air routes. $$ \begin{array}{lrr} & \text { Distance } & \\ \text { Route } & \text { (miles) } & \text { Cost } \\ \hline \text { Dallas-Austin } & 178 & \$ 125 \\ \text { Houston-Dallas } & 232 & 123 \\ \text { Chicago-Detroit } & 238 & 148 \\ \text { Chicago-St. Louis } & 262 & 136 \\ \text { Chicago-Cleveland } & 301 & 129 \\ \text { Chicago-Atlanta } & 593 & 162 \\ \text { New York-Miami } & 1092 & 224 \\ \text { New York-San Juan } & 1608 & 264 \\ \text { New York-Chicago } & 714 & 287 \\ \text { Chicago-Denver } & 901 & 256 \\ \text { Dallas-Salt Lake } & 1005 & 365 \\ \text { New York-Dallas } & 1374 & 459 \\ \text { Chicago-Seattle } & 1736 & 424 \\ \text { Los Angeles-Chicago } & 1757 & 361 \\ \text { Los Angeles-Atlanta } & 1946 & 309 \\ \text { New York-Los Angeles } & 2463 & 444 \\ \text { Los Angeles-Honolulu } & 2556 & 323 \\ \text { New York-San Francisco } & 2574 & 513 \end{array} $$ a. If you want to estimate the cost of a flight based on the distance traveled, which variable is the response variable and which is the independent predictor variable? b. Assume that there is a linear relationship between cost and distance. Calculate the least-squares regression line describing cost as a linear function of distance. c. Plot the data points and the regression line. Does it appear that the line fits the data? d. Use the appropriate statistical tests and measures to explain the usefulness of the regression model for predicting cost.

Short answer

Question: Based on the provided step-by-step solution, briefly describe the purpose of calculating the correlation coefficient and the coefficient of determination in evaluating the usefulness of a regression model for predicting cost. Answer: The purpose of calculating the correlation coefficient and the coefficient of determination is to measure the strength and direction of the linear relationship between the variables, and how much variation in the response variable (cost) is explained by the predictor variable (distance). These values can indicate if the linear relationship between distance and cost of flight is strong and useful for predicting cost. The closer the correlation coefficient is to 1 or -1, and the closer the coefficient of determination is to 1, the more useful the regression model is for predicting cost.

Step-by-step solution

a. Identifying Variables

The response variable is the variable you want to estimate, which is the cost of a flight. The independent predictor variable is the distance traveled since we want to estimate the cost based on distance.

b. Calculate Regression Line

To find the least-squares regression line, we need to calculate the slope and intercept of the line. The slope is calculated using the formula: $$ b = \frac{n (\sum xy) - (\sum x)(\sum y)}{n (\sum x^2) - (\sum x)^2} $$ And the intercept is calculated using the formula: $$ a = \frac{\sum y - b \sum x}{n} $$ First, we calculate the sums needed for the formulas: $$ \sum x, \sum y, \sum xy, \text{ and } \sum x^2 $$ Then, we plug the values into the slope and intercept formulas to find the equation of the least-squares regression line.

c. Plot the Data Points and Regression Line

To create the plot, follow these steps: 1. Set up a scatter plot with distance on the x-axis and cost on the y-axis. 2. Plot the data points from the table. 3. Draw the regression line that you calculated in step b. 4. Examine the plot to check if the line fits the data points.

d. Evaluate Model Usefulness

To evaluate the usefulness of the regression model for predicting cost, perform the following steps: 1. Calculate the correlation coefficient, which measures the strength and direction of the linear relationship between the variables. The formula is: $$ r = \frac{n (\sum xy) - (\sum x)(\sum y)}{\sqrt{[n (\sum x^2) - (\sum x)^2][n (\sum y^2) - (\sum y)^2]}} $$ 2. Calculate the coefficient of determination, which measures how much variation in the response variable is explained by the predictor variable. The formula is: $$ R^2 = r^2 $$ 3. Test for statistical significance by computing a t-statistic and finding the corresponding p-value to see if the slope of the regression line is significantly different from zero. The t-statistic can be calculated using the formula: $$ t = \frac{b - 0}{SE_b} $$ where SE_b is the standard error of the slope, which can be found using the formula: $$ SE_b = \sqrt{\frac{\sum (y - \hat{y})^2}{(n - 2) \sum (x - \bar{x})^2}} $$ 4. If the correlation coefficient is close to 1 or -1, and the coefficient of determination is close to 1, this indicates that the linear relationship between distance and cost of flight is strong and useful for predicting cost. If the p-value obtained in the t-test is less than a common threshold like 0.05, we can conclude that the slope of the regression line is statistically significantly different from zero, meaning the model is useful for predicting cost.

Key concepts

Response Variable

In linear regression, the response variable is the one you're trying to predict or estimate. Think of it as the outcome you care about. In the context of our example with airfare, the response variable is the **cost of the flight**. When analyzing data, the response variable changes based on other factors, like in this case, the distance traveled.

By focusing on the response variable, you can understand how changes in other variables influence it. Thus, the response variable is vital for modeling and making predictions.

Independent Predictor Variable

The independent predictor variable is what you use to explain changes in the response variable. It's like the driving factor behind the response. In our example, the **distance traveled** is the independent predictor variable.

Understanding this variable helps you gather insights into how it affects the response variable. By analyzing the independent predictor variable, you can establish a relationship between it and the response variable, using statistical models like linear regression to make informed predictions.

Correlation Coefficient

The correlation coefficient, denoted as **r**, is a statistical measure that describes the strength and direction of a linear relationship between two variables. It ranges from -1 to 1:

• **1** indicates a perfect positive relationship.
• **-1** indicates a perfect negative relationship.
• **0** means no linear relationship.

For the airfare example, calculating the correlation coefficient between distance and cost helps you understand how closely the two variables are related. A higher absolute value of **r** suggests a stronger linear relationship.

This is critical in assessing whether a model will be useful for predicting one variable based on another.

Coefficient of Determination

The coefficient of determination, often represented as **R²**, quantifies how much of the variation in the response variable can be explained by the independent predictor variable. It is calculated by squaring the correlation coefficient ( **R² = r²**). This value ranges from 0 to 1:

• **0** indicates no explanatory power.
• **1** means perfect prediction of the response variable by the predictor variable.

In the context of airfare costs, a high **R²** value would mean that most of the change in flight cost is explained by the distance traveled.

This provides insight into the model's effectiveness and helps in deciding whether the regression model is a reliable tool for making predictions.