• Software Deterioration and Entropy (Pressman):
• Physical hardware wears out due to physical friction, heat, and environmental fatigue.
• Software does not suffer physical wear-out; however, it deteriorates over time as it evolves because:
• Repeated maintenance cycles, bug patches, and new feature requests introduce architectural entropy.
• Unanticipated side effects and dependencies emerge between interacting components, increasing complexity and error rates.
• Conclusion: Multiple change requests introduce errors in component interactions (Option 3).

Correct option: (3) Multiple change requests introduce errors in component interactions.

• Diagnostics of Multicollinearity in Regression:
• Multicollinearity refers to linear dependencies among explanatory variables.
• Standard Diagnostic Measures:
• Kappa Statistic (Condition Number $\kappa$): $\kappa = \sqrt{\lambda_{\max} / \lambda_{\min}}$. Values $\gt 20$ indicate moderate to severe multicollinearity.
• Variance Inflation Factor (VIF): $\text{VIF}_j = \frac{1}{1 - R_j^2}$. Values $\gt 5$ or $10$ indicate serious collinearity.
• Eigenvalues near zero: Indicate that the design matrix $X^T X$ is near-singular.
• Incorrect Measure:
• Coefficient of determination ($R^2 \gt 0.7$): Measures the overall goodness of fit and proportion of total variation in the response variable explained by the regression model, not collinearity between predictors.
• Conclusion: Coefficient of determination > 0.7 (Option 4).

Correct option: (4) Coefficient of determination > 0.7

• Invariance of Regression Coefficients to Change of Origin:
• Let $Y$ and $X$ be the dependent and independent variables with regression coefficient:

$$b_{YX} = \frac{\text{Cov}(X, Y)}{\text{Var}(X)}$$

• If a constant $C$ is subtracted from both variables:

$$u = X - C, \quad v = Y - C$$

• The variances and covariance under linear translation remain identical:

$$\text{Var}(u) = \text{Var}(X - C) = \text{Var}(X)$$ $$\text{Cov}(u, v) = \text{Cov}(X - C, Y - C) = \text{Cov}(X, Y)$$

• The new regression coefficient is:

$$b_{vu} = \frac{\text{Cov}(u, v)}{\text{Var}(u)} = \frac{\text{Cov}(X, Y)}{\text{Var}(X)} = b_{YX}$$

• Conclusion: The regression coefficient Remains unchanged (Option 3).

Correct option: (3) Remains unchanged

• Goodness of Fit in Regression Analysis:
• The overall explanatory strength and goodness-of-fit of a linear regression model is measured by the Coefficient of Determination ($R^2$):

$$R^2 = \frac{SSR}{SST} = 1 - \frac{SSE}{SST}$$

• It quantifies the proportion of the total variation in the dependent variable explained by the regression equation, bounded between 0 and 1.
• Other Options:
• Partial correlation coefficient: Measures association between two variables while controlling for others.
• Partial regression coefficient: The slope parameter associated with a single predictor.
• Conclusion: Coefficient of determination (Option 2).

Correct option: (2) Coefficient of determination