Interpreting Bivariate Linear Regression Statements

We need to evaluate each statement regarding simple bivariate linear regression.

Statement Analysis

• (a) The b coefficient is a slope of the regression line.

  In a simple linear regression model, typically represented as $Y = a + bX + \epsilon$, the coefficient 'b' indeed represents the slope of the regression line. This indicates the rate of change in the dependent variable (Y) for a one-unit change in the independent variable (X).

  Statement (a) is TRUE.
• (b) The b coefficient denotes change in Y, when X changes by 1 unit.

  This is the direct interpretation of the slope coefficient 'b'. It quantifies the expected increase or decrease in Y associated with a one-unit increase in X, holding other factors constant (though in simple regression, there are no other predictors).

  Statement (b) is TRUE.
• (c) The errors of prediction have a mean and variance of zero.

  A key assumption in linear regression is that the error term (or residuals) has a mean of zero ($E(\epsilon) = 0$). However, the variance of the errors ($\text{Var}(\epsilon) = \sigma^2$) is assumed to be constant and positive, not zero. Zero variance implies perfect prediction, which is generally not the case.

  Statement (c) is FALSE.
• (d) If the b coefficient is significant, the product moment correlation between X and Y is also significant.

  In simple linear regression (with only one predictor variable), the statistical significance of the slope coefficient ('b') is directly linked to the significance of the product moment correlation coefficient ('r') between X and Y. If 'b' is significantly different from zero, it implies a significant linear relationship, which means 'r' will also be statistically significant.

  Statement (d) is TRUE.

Conclusion

Based on the analysis, statements (a), (b), and (d) are true, while statement (c) is false.

Therefore, the correct option includes statements (a), (b), and (d).