Problem 1a: Complete The ANOVA Table Based On 20 Observation

Problem 1a Complete The Following Anova Table Based On 20 Observati

Complete the following ANOVA table based on 20 observations for the regression equation. Determine if the overall regression is significant by filling in the missing values in the table:

• Source: Regression, Error, Total
• DF (degrees of freedom)
• SS (Sum of Squares)
• MS (Mean Squares)
• F-statistic

Given data:

• Regression SS = 350
• Error DF = ?
• Total SS = 500

Section A: Completing the ANOVA Table

First, note that the total degrees of freedom (DF) is always equal to the total number of observations minus 1. With 20 observations, total DF is 19. The regression degrees of freedom depend on the number of predictors; assuming a simple linear regression (one predictor), regression DF is 1. The error degrees of freedom is the total DF minus the regression DF: 19 - 1 = 18.

Next, individual MS values are computed as SS divided by their respective DF:

• Regression MS = SS (regression) / DF (regression) = 350 / 1 = 350
• Total MS is not directly calculated; the total SS is 500, but total MS is not necessary here for significance testing.

Calculate Error SS: Error SS = Total SS - Regression SS = 500 - 350 = 150.

Where Error DF is 18 (from above), Error MS = Error SS / Error DF = 150 / 18 ≈ 8.33.

Compute F-statistic: F = Regression MS / Error MS = 350 / 8.33 ≈ 42.0.

This high F-value suggests the regression model is statistically significant.

Section B: Sequential Sums of Squares and ANOVA Output

Suppose the sequential SS due to regression are given as 300, 250, 340, 325 for successive variables. The partially filled output includes F-values and p-values. To fill in missing values:

• Source: Variables in Model
• DF: 1 for each variable added sequentially
• Partial SS: as provided
• F-value and Pr>F: calculated using the formula: F = Partial SS / Error MS (from the full model); p-value derived from F-distribution

Given the partial sums of squares and their F-values, the significant F-value of 0.0042 corresponds to a p-value of around 0.04, indicating significance at the 5% level.

Conclusion

The completed ANOVA table indicates a highly significant relationship between the predictor and response variables, with an F-value of approximately 42, confirming that the regression model explains a significant portion of the variability in the response variable.

References

• Kutner, M. H., Nachtsheim, C. J., Neter, J., & Li, W. (2004). Applied Linear Statistical Models (5th ed.). McGraw-Hill.
• Minitab Inc. (2020). ANOVA: Analysis of Variance. Retrieved from https://support.minitab.com
• Neter, J., Wasserman, W., & Kutner, M. H. (1990). Applied Linear Regression Models. Irwin.
• Sheskin, D. J. (2004). Handbook of Parametric and Nonparametric Statistics. Chapman & Hall/CRC.
• Montgomery, D. C. (2017). Design and Analysis of Experiments. Wiley.
• Osborne, J. W. (2013). Best Practices in Data Cleaning. Practical Assessment, Research & Evaluation, 18(1), 1-21.
• Verbeke, G., & Molenberghs, G. (2000). Linear Mixed Models for Longitudinal Data. Springer.
• Cook, R. D., & Weisberg, S. (1999). Applied Regression Including Computing and Graphics. Wiley.
• Field, A. (2013). Discovering Statistics Using SPSS. Sage Publications.
• Zimmerman, D. W. (2004). A Note on the Calculation of Power in Multiple Regression. The American Statistician, 58(2), 94-97.