Master Regression Equations for CBSE Class 11 Applied Mathematics. Learn to calculate regression lines, coefficients, and predictions with solved examples.
Regression analysis is a statistical method used to estimate the relationship between a dependent variable and one or more independent variables. The regression line of Y on X is given by Y = a + bX, where b is the regression coefficient representing the slope. This line minimizes the sum of squared vertical deviations between the observed data points and the fitted line.
Step 1: Given x̄ = 10, ȳ = 20, bᵧₓ = 0.5.
Step 2: Substitute into Y - 20 = 0.5(X - 10).
Step 3: Simplify to Y = 0.5X + 15.
Answer: The regression equation is Y = 0.5X + 15.
The regression coefficient bᵧₓ indicates the change in Y for a unit change in X. It is calculated using the covariance of X and Y and the variance of X. The relationship between the correlation coefficient r and regression coefficients is r = ±√(bᵧₓ · bₓᵧ).
Step 1: Given Cov(X, Y) = 12 and σₓ² = 4.
Step 2: Apply bᵧₓ = 12 / 4.
Step 3: Calculate bᵧₓ = 3.
Answer: The regression coefficient bᵧₓ is 3.
Both regression lines of Y on X and X on Y intersect at the point (x̄, ȳ). The regression coefficients are independent of the change of origin but not of the change of scale. If r = 0, the regression lines are perpendicular to each other.
Step 1: Given lines 2X - Y = 1 and X - 2Y = -4.
Step 2: Solve the system: Y = 2X - 1 and X - 2(2X - 1) = -4.
Step 3: X - 4X + 2 = -4 ⟹ -3X = -6 ⟹ X = 2, Y = 3.
Answer: The point of intersection (x̄, ȳ) is (2, 3).