Regression Analysis: Simple and Multiple Linear Regression, Least Squares Principle, Properties and R-Squared

Analysis of regression properties under transformation:

• Incorrect: Adding a constant does not increase the slope.
• Incorrect: Division by 35 occurs only if variables are scaled by multiplying by 35.
• Incorrect: Subtraction does not affect the slope.
• Correct: Regression coefficients ($b_{yx} = \frac{\text{Cov}(X, Y)}{\text{Var}(X)}$) are strictly independent of the change of origin. Adding any constant (here 35) to all values of $X$ and $Y$ shifts the mean but leaves deviations $(X - \bar{X})$ and $(Y - \bar{Y})$ completely unchanged. Hence, the regression coefficient remains exactly the same as the original.

Analysis of linear line slopes:
In the equation $Y = a + bX$, $b$ represents the slope $\frac{dY}{dX}$.

• Incorrect: $b \ne 0$ produces an inclined line.
• Incorrect: $b = 1$ gives an angle of $45^\circ$ with the horizontal.
• Correct: A line is completely horizontal (parallel to the X-axis) when its slope is zero ($b = 0$). In this case, $Y = a$, meaning $Y$ remains constant regardless of $X$.
• Incorrect: $a = b$ specifies a relationship between the intercept and slope, not horizontality.

Analysis of simple linear regression parameters:

• Incorrect: A regression line requires both intercept and slope, not just one.
• Correct: The simple linear regression model is formulated as $Y = \beta_0 + \beta_1 X + \varepsilon$. The two unknown regression constants to be estimated via the method of least squares are the intercept parameter $\beta_0$ (or $a$) and the slope parameter $\beta_1$ (or $b$).
• Incorrect: Multiple regression involves 3 or more parameters.
• Incorrect: 4 is incorrect for a bivariate single-predictor model.

Relationship between regression slopes and correlation coefficient:

• $1/4$ is the reciprocal slope of the equation, not necessarily the correlation coefficient.
• Correlation coefficient $r$ cannot exceed $+1$, so it cannot be $4$.
• In the regression of $X$ on $Y$, $b_{xy} = +4$. A fundamental statistical theorem states that the regression coefficients $b_{yx}$, $b_{xy}$, and the Pearson correlation coefficient $r$ must all share the exact same mathematical sign. Since $b_{xy} \gt 0$, $r$ must be strictly positive ($r \gt 0$).
• $r$ cannot exceed $1$ by definition ($-1 \le r \le +1$).