Master Regression Coefficients for CBSE Class 11 Applied Mathematics. Learn properties, formulas, and step-by-step solutions for board exam success.
Regression coefficients, denoted by bxy and byx, measure the average change in one variable for a unit change in the other. The coefficient bxy represents the slope of the regression line of y on x, while byx represents the slope of the regression line of x on y. These coefficients are essential for predicting values in bivariate data sets.
Step 1: Given r = 0.8, σx = 2, σy = 3.
Step 2: Calculate bxy = 0.8 · (3 / 2) = 1.2.
Step 3: Calculate byx = 0.8 · (2 / 3) = 0.533.
Answer: bxy = 1.2, byx = 0.533
Regression coefficients possess unique properties: they share the same sign as the correlation coefficient r. Furthermore, the arithmetic mean of the two regression coefficients is always greater than or equal to the correlation coefficient. If one coefficient is greater than 1, the other must be less than 1.
Step 1: Given bxy = 1.6 and byx = 0.4.
Step 2: Calculate arithmetic mean = (1.6 + 0.4) / 2 = 1.0.
Step 3: Since r = √(bxy · byx) = √(1.6 · 0.4) = √0.64 = 0.8.
Answer: 1.0 ≥ 0.8, property verified.
The correlation coefficient r is the geometric mean of the two regression coefficients. This relationship allows us to determine the strength and direction of the linear relationship between two variables. It is important to note that regression coefficients are independent of change of origin but not of scale.
Step 1: Given bxy = 0.9 and byx = 0.4.
Step 2: r = √(0.9 · 0.4) = √0.36.
Step 3: r = 0.6.
Answer: r = 0.6