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When regression lines perpendicular to each other then angle will be: Property 1 : The regression coefficients remain unchanged due to a shift of origin but change due to a shift of scale. This property states that if the original pair of variables is (x, y) and if they are changed to the pair (u, v) where Property 2 : The two lines of regression intersect at the point where x and y are the variables under consideration. Property 3 : The coefficient of correlation between two variables x and y in the simple geometric mean of the two regression coefficients. The sign of the correlation coefficient would be the common sign of the two regression coefficients. This property says that if the two regression coefficients are denoted by byx and bxy then the coefficient of correlation is given by If both the regression coefficients are negative, r would be negative and if both are positive, r would assume a positive value. Property 4 : The two lines of regression coincide i.e. become identical when r = –1 or 1 or in other words, there is a perfect negative or positive correlation between the two variables under discussion. Property 5 : The two lines of regression are perpendicular to each other when r = 0. Solved ProblemFor the variables x and y, the regression equations are given as 7x – 3y – 18 = 0 and 4x – y – 11 = 0 (i) Find the arithmetic means of x and y. (ii) Identify the regression equation of y on x. (iii) Compute the correlation coefficient between x and y. (iv) Given the variance of x is 9, find the SD of y. Solution (i) : By property, always the two lines of regression intersect at the point (mean of 'x', mean of 'y') Solving the given two regression equations, we get the point of intersection (3, 1). Hence Arithmetic mean of 'x' = 3 Arithmetic mean of 'y' = 1 Solution (ii) : Let us assume that 7x – 3y – 18 = 0 represents the regression line of y on x and 4x – y – 11 = 0 represents the regression line of x on y. Now, 7x - 3y - 18 = 0 ------> y = (-6) + (7/3)x Therefore, byx = 7/3 4x - y - 11 = 0 ------> x = 11/4 + (1/4)y Therefore, bxy = 1/4 We can get the value of 'r', using the formula given below Both byx and bxy are positive, so 'r' is also positive. Using the above formula, the value of r is 0.7638. Since r = 0.7638 which lies in the interval -1 ≤ r ≤ 1, our assumptions are correct. Thus, 7x – 3y – 18 = 0 truly represents the regression line of y on x. Solution (iii) : From solution (ii), r = 0.7638 Hence, correlation coefficient between x and y is 0.7638. Solution (iv) : Given byx = r ⋅ Sᵧ/Sₓ Variance of x = 9 ------> Sx = 3 Then, (7/3) = 0.7638 ⋅ Sy/3 Sy = 7 / 0.7638 Sᵧ = 9.1647 So, the standard deviation of y is 9.1647. Kindly mail your feedback to We always appreciate your feedback. ©All rights reserved. onlinemath4all.com When regression lines are perpendicular to each other?If the correlation coefficient r xy = 0, then the two lines of regression are perpendicular to each other.
What is the correlation coefficient for the regression line?Correlation in Linear Regression
The square of the correlation coefficient, r², is a useful value in linear regression. This value represents the fraction of the variation in one variable that may be explained by the other variable.
Is the correlation coefficient and regression same?Both variables are different. Correlation coefficient indicates the extent to which two variables move together. Regression indicates the impact of a unit change in the known variable (x) on the estimated variable (y).
When value of correlation coefficient is one the two regression lines coincide?Solution: (2) 1
If the lines of regression coincide, then the value of the correlation coefficient is 1 or -1.
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