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The question asks about calculating the Durbin-Watson (DW) statistic, which measures the presence of autocorrelation (specifically, first-order autocorrelation) in the residuals from a regression analysis.

### **Steps to approach the problem:**

1. **Understanding the Residual Plot**: The residual plot provided shows the residuals versus the observation number, which helps in visualizing whether the residuals are randomly dispersed. If there is a pattern, it suggests autocorrelation.

2. **Durbin-Watson Statistic (DW)**: 
   - The DW statistic is used to detect the presence of autocorrelation in the residuals from a regression analysis.
   - The DW statistic ranges from 0 to 4, where:
     - DW = 2 indicates no autocorrelation.
     - DW < 2 suggests positive autocorrelation.
     - DW > 2 suggests negative autocorrelation.

3. **Interpreting the Graph**:
   - The residual plot shows a pattern of points, but without a clear upward or downward trend, implying the absence of significant autocorrelation.
   - Since the plot seems to show residuals randomly dispersed around zero without a clear pattern, a Durbin-Watson statistic close to 2 is expected.

4. **Proportion of Values**:
   - The question asks about the proportion of values (given in the options) that seem reasonable for the DW statistic. Since the values range from 1.50 to 3.75, we're looking for a value closest to 2.

5. **Analysis of Options**:
   - The provided options are:
     - A) 0.125 
     - B) 0.25 
     - C) 0.375 
     - D) 0.5 
     - E) 0.625

   Since the Durbin-Watson statistic is around 2 (indicating no autocorrelation), we need to choose the option with a proportion indicating this value.

### **Answer**: 

Given that the correct DW statistic should be close to 2, the **proportion that best represents this value** is **0.5** (option D), since it indicates the middle value between the lower (1.50) and upper (3.75) range.

### **Conclusion**:
- The correct answer is **D) 0.5**.

Would you like more details on how the Durbin-Watson statistic is calculated or how the residual plot impacts the interpretation of autocorrelation?

### **Related Questions**:

1. What is the Durbin-Watson statistic and how is it used in regression analysis?
2. How can the residual plot help in diagnosing the assumptions of a linear regression model?
3. What are the consequences of positive and negative autocorrelation in time series data?
4. How can autocorrelation in residuals be remedied in a regression analysis?
5. What are other methods to detect autocorrelation besides the Durbin-Watson test?

**Tip**: Always check the residual plots for patterns, as they can reveal important information about model assumptions like homoscedasticity and autocorrelation.

Consider the following graph of residuals versus observation number of demand observed weekly over a certain time period, where the residuals were obtained from a simple linear regression model with observation number as the explanatory variable. For the above graph, proportion of values in the list (1.50, 1.75, 1.85, 4.25, 3.25, 3.75, 2.15, 1.90) that seem reasonable values of the Durbin-Watson statistic, is nearest to: A) 0.125 B) 0.25 C) 0.375 D) 0.5 E) 0.625.

Explore the application of the Durbin-Watson statistic to assess autocorrelation in a linear regression model. This problem involves interpreting a residual plot to estimate the Durbin-Watson statistic, crucial for understanding model accuracy. Suitable for college-level students.

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