A final step in regression analysis is an examination of the residuals in a residual plot. This allows you to test the assumptions of the regression model itself by looking for patterns in the residuals. The residual plot for the problem is as follows: FIGURE. RESIDUAL PLOT FOR U.S. DEPARTMENT OF TRANSPORATION PROBLEM Percent Under 21 Residual Plot 2 1.5 1 0.5 半 to 16 10 12 14 18 20 -0.5 -1 -1.5 Percent under 21 Which statement offers the best interpretation of the residual plot? O It appears that the residual plot exhibits a pattern whereby a linear model may not be adequate or the best fit, indicating that an assumption of the linear model may have been violated. O It appears that the residual plot exhibits a good pattern of constant variance, indicating that the equal variance assumption of the model is supported. aEcumption of the model is not supported. Residuals -00

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As an auto insurance risk analyst, it is your job to research risk profiles for various types of drivers. One common area of concern for auto insurance companies is the risk involved when offering policies to younger, less experienced drivers. The U.S. Department of Transportation recently conducted a study in which it analyzed the relationship between 1) the number of fatal accidents per 1000 licenses, and 2) the percentage of licensed drivers under the age of 21 in a sample of 42 cities.

Your first step in the analysis is to construct a scatterplot of the data.

**Figure**: Scatterplot for U.S. Department of Transportation Problem

*Scatterplot Description*
The scatterplot titled "U.S. Department of Transportation: The Relationship Between Fatal Accident Frequency and Driver Age" shows data points representing various cities. The x-axis represents the percentage of drivers under age 21, ranging from 6 to 20. The y-axis represents fatal accidents per 1000 licenses, ranging from 0 to 4.5. The plot indicates a positive trend, where an increase in the percentage of drivers under 21 corresponds to an increase in fatal accidents per 1000 licenses.

Upon visual inspection, you determine that the variables do have a linear relationship. After a linear pattern has been established visually, you now proceed with performing linear regression analysis. The results are as follows:

**Table**: Linear Regression Output for U.S. Department of Transportation Problem

|                      | Coefficients | Standard Error | t Statistic | p-value |
|----------------------|--------------|----------------|-------------|---------|
| Intercept            | -1.5974      | 0.3717         | -4.2979     | 0.0001  |
| Percent Under 21     | 0.2871       | 0.0294         | 9.7671      | 0.0000  |

The table indicates a significant relationship between the percentage of drivers under 21 and the number of fatal accidents, with a positive coefficient for the percentage under 21 variable, suggesting that as the percentage of younger drivers increases, so do fatal accidents.
Transcribed Image Text:As an auto insurance risk analyst, it is your job to research risk profiles for various types of drivers. One common area of concern for auto insurance companies is the risk involved when offering policies to younger, less experienced drivers. The U.S. Department of Transportation recently conducted a study in which it analyzed the relationship between 1) the number of fatal accidents per 1000 licenses, and 2) the percentage of licensed drivers under the age of 21 in a sample of 42 cities. Your first step in the analysis is to construct a scatterplot of the data. **Figure**: Scatterplot for U.S. Department of Transportation Problem *Scatterplot Description* The scatterplot titled "U.S. Department of Transportation: The Relationship Between Fatal Accident Frequency and Driver Age" shows data points representing various cities. The x-axis represents the percentage of drivers under age 21, ranging from 6 to 20. The y-axis represents fatal accidents per 1000 licenses, ranging from 0 to 4.5. The plot indicates a positive trend, where an increase in the percentage of drivers under 21 corresponds to an increase in fatal accidents per 1000 licenses. Upon visual inspection, you determine that the variables do have a linear relationship. After a linear pattern has been established visually, you now proceed with performing linear regression analysis. The results are as follows: **Table**: Linear Regression Output for U.S. Department of Transportation Problem | | Coefficients | Standard Error | t Statistic | p-value | |----------------------|--------------|----------------|-------------|---------| | Intercept | -1.5974 | 0.3717 | -4.2979 | 0.0001 | | Percent Under 21 | 0.2871 | 0.0294 | 9.7671 | 0.0000 | The table indicates a significant relationship between the percentage of drivers under 21 and the number of fatal accidents, with a positive coefficient for the percentage under 21 variable, suggesting that as the percentage of younger drivers increases, so do fatal accidents.
### Analyzing Residual Plots in Regression Analysis

**Purpose of Residual Plots**

In regression analysis, a crucial step is examining the residuals using a residual plot. This examination helps test the assumptions of the regression model by identifying patterns in the residuals.

**Figure Overview**

**Title:** Residual Plot for U.S. Department of Transportation Problem

- **Graph Components:**
  - **X-axis:** Percent Under 21
  - **Y-axis:** Residuals
  - **Data Points:** Plotted residuals shown as "X" marks distributed across the range of the percent under 21 values.

**Interpretation**

The plot shows residuals scattered around a horizontal line at zero, with the residual spread appearing fairly consistent across different values of the "percent under 21." This consistency (or lack thereof) can provide insights into the suitability of the regression model.

**Questions for Evaluation:**

Which statement offers the best interpretation of the residual plot?

- ○ It appears that the residual plot exhibits a pattern whereby a linear model may not be adequate or the best fit, indicating that an assumption of the linear model may have been violated.
- ○ It appears that the residual plot exhibits a good pattern of constant variance, indicating that the equal variance assumption of the model is supported.
- ○ It appears that the residual plot exhibits a good pattern of constant variance, indicating that the equal variance assumption of the model is not supported.
- ○ It appears that the residual plot exhibits a pattern of non-constant variance, indicating that an assumption of the linear model may have been violated.

Understanding the pattern of residuals can indicate whether the assumptions about linearity and variance in the regression model are appropriate or if they have been violated.
Transcribed Image Text:### Analyzing Residual Plots in Regression Analysis **Purpose of Residual Plots** In regression analysis, a crucial step is examining the residuals using a residual plot. This examination helps test the assumptions of the regression model by identifying patterns in the residuals. **Figure Overview** **Title:** Residual Plot for U.S. Department of Transportation Problem - **Graph Components:** - **X-axis:** Percent Under 21 - **Y-axis:** Residuals - **Data Points:** Plotted residuals shown as "X" marks distributed across the range of the percent under 21 values. **Interpretation** The plot shows residuals scattered around a horizontal line at zero, with the residual spread appearing fairly consistent across different values of the "percent under 21." This consistency (or lack thereof) can provide insights into the suitability of the regression model. **Questions for Evaluation:** Which statement offers the best interpretation of the residual plot? - ○ It appears that the residual plot exhibits a pattern whereby a linear model may not be adequate or the best fit, indicating that an assumption of the linear model may have been violated. - ○ It appears that the residual plot exhibits a good pattern of constant variance, indicating that the equal variance assumption of the model is supported. - ○ It appears that the residual plot exhibits a good pattern of constant variance, indicating that the equal variance assumption of the model is not supported. - ○ It appears that the residual plot exhibits a pattern of non-constant variance, indicating that an assumption of the linear model may have been violated. Understanding the pattern of residuals can indicate whether the assumptions about linearity and variance in the regression model are appropriate or if they have been violated.
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