91Ó°ÊÓ

Tires The following data represent the cost of tires (in dollars) along with a variety of potential explanatory variables. Slalom time is the amount of time it took for a 3-series BMW to get through a slalom track, lap time is the amount of time it took the same car to complete a \(1 / 3\) -mile lap, and stopping distance is the distance it took the BMW to stop on wet pavement traveling 60 miles per hour. Find the best regression model using each of the three techniques presented in the section. What do you notice? $$ \begin{array}{lcccccc} \text { TIRE } & \begin{array}{c} \text { Cost } \\ \text { (dollars) } \end{array} & \text { MPG } & \begin{array}{c} \text { Slalom Time } \\ \text { (seconds) } \end{array} & \begin{array}{c} \text { Lap Time } \\ \text { (seconds) } \end{array} & \begin{array}{c} \text { Stopping } \\ \text { Distance } \\ \text { (feet) } \end{array} & \begin{array}{c} \text { Cornering } \\ \text { g-Force } \end{array} \\ \hline \text { BFGoodrich g-Force Sport COMP-2 } & 114 & 30.5 & 5.13 & 30.24 & 80.0 & 0.90 \\ \hline \text { Bridgestone Potenza RE760 Sport } & 126 & 30.2 & 5.08 & 30.14 & 79.4 & 0.91 \\ \hline \text { Firestone Firehawk Wide Oval Indy } 500 & 111 & 30.4 & 5.16 & 30.58 & 83.3 & 0.88 \\ \hline \text { Yokohama S.drive } & 119 & 31.0 & 5.20 & 30.61 & 82.2 & 0.90 \\\ \hline \text { Bridgestone Turanza Serenity Plus } & 154 & 32.2 & 5.10 & 31.13 & 90.4 & 0.84 \\ \hline \text { Continental PureContact } & 134 & 32.7 & 5.15 & 31.18 & 91.2 & 0.85 \\ \hline \text { Michelin Primacy MXV4 } & 135 & 32.3 & 5.15 & 31.18 & 90.2 & 0.85 \\ \hline \text { Yokohama AVID Ascend } & 134 & 32.3 & 5.17 & 31.11 & 91.4 & 0.86 \\ \hline \end{array} $$

Step by step solution

01

- Input the Data

Input the provided data into a statistical software or a programming language such as R or Python. Ensure that all variables (Cost, MPG, Slalom Time, Lap Time, Stopping Distance, Cornering g-Force) are correctly entered.

02

- Explore the Data

Generate summary statistics and visualize the data to understand the distributions and relationships between variables. This can include scatter plots, correlation matrices, and histograms.

03

- Simple Linear Regression

Perform a simple linear regression using each explanatory variable separately (MPG, Slalom Time, Lap Time, Stopping Distance, Cornering g-Force) to predict the Cost. Evaluate the model by checking the R-squared value and the p-value for each predictor.

04

- Multiple Linear Regression

Create a multiple linear regression model that includes all the explanatory variables (MPG, Slalom Time, Lap Time, Stopping Distance, Cornering g-Force) to predict the Cost. Evaluate the model by checking the Adjusted R-squared value and the p-values for each predictor.

05

- Stepwise Regression

Use stepwise regression to automatically select the most significant variables. This involves adding and removing predictors based on their statistical significance in the model. Software tools can automate this process using algorithms such as forward selection, backward elimination, or both.

06

- Compare the Models

Compare the results from the simple linear regression, multiple linear regression, and stepwise regression methods. Look at the R-squared, Adjusted R-squared, and p-values of the predictors. Determine which model best fits the data and provides the most reliable predictions.

07

- Interpret the Results

Interpret the coefficients of the best model. Consider the practical significance of each predictor in addition to their statistical significance. Discuss what the results imply about the relationship between tire cost and the explanatory variables.

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Key Concepts

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

simple linear regression

In simple linear regression, we examine the relationship between the dependent variable (tire cost) and one explanatory variable at a time. The goal is to find a straight line (regression line) that best predicts the cost of the tire based on the chosen variable (e.g., MPG, Slalom Time, Lap Time, Stopping Distance, or Cornering g-Force).

We use the formula: \( \text{Cost} = \beta_0 + \beta_1 \times \text{Explanatory Variable} \), where \( \beta_0 \) is the intercept, and \( \beta_1 \) is the slope of the line.

**Steps to Perform Simple Linear Regression:**

• Input the data into statistical software or a programming language.
• Generate scatter plots to visualize the relationship between the cost and each explanatory variable.
• Use the least squares method to fit a regression line for each explanatory variable.
• Evaluate the model using the R-squared value (which shows how well the data fits the model) and the p-value (which indicates the statistical significance of the predictors).

By performing simple linear regression, we can begin to understand how each individual explanatory variable impacts tire cost. This is a good first step before considering more complex models.

multiple linear regression

Multiple linear regression involves examining the relationship between the dependent variable (tire cost) and multiple explanatory variables simultaneously. In this case, we include all the provided variables (MPG, Slalom Time, Lap Time, Stopping Distance, and Cornering g-Force) in one model.

We use the formula: \( \text{Cost} = \beta_0 + \beta_1 \times \text{MPG} + \beta_2 \times \text{Slalom Time} + \beta_3 \times \text{Lap Time} + \beta_4 \times \text{Stopping Distance} + \beta_5 \times \text{Cornering g-Force} \), where each \( \beta \) represents the coefficient for each explanatory variable.

**Steps to Perform Multiple Linear Regression:**

• Input the data into software and ensure the variables are entered correctly.
• Create a multiple linear regression model using all the explanatory variables.
• Check the Adjusted R-squared value to see how well the model explains the variability in the data.
• Evaluate the p-values for each predictor to determine their statistical significance.

By using multiple linear regression, we can see how combinations of variables impact the cost of tires, providing a more comprehensive view of the factors that matter.

stepwise regression

Stepwise regression is an automated method that helps in selecting the most meaningful explanatory variables for the regression model. It combines both forward selection (adding the most significant variables) and backward elimination (removing the least significant variables), or uses either one of these methods exclusively.

**Steps to Perform Stepwise Regression:**

• Input the data into statistical software capable of stepwise regression (like R or Python).
• Choose the stepwise regression method - forward selection, backward elimination, or both.
• Let the software run the stepwise algorithm, which will add or remove variables based on their statistical significance.
• Analyze the final model, which includes only the predictors that have significant explanatory power.

Stepwise regression helps in simplifying the model and ensuring only the most impactful variables are included, thus making it easier to interpret and potentially improving the model’s predictive accuracy.