Determine the regression line of x+2y-5=0 and 3x+3y-8=0
(i) y on x and (ii) x on y, (iii) Find r using the regression coefficients
To determine the regression line for the equations x + 2y - 5 = 0 and 3x + 3y - 8 = 0, we can use the least squares method. Let's solve it step by step.
(i) Regression line of y on x:
Step 1: Rearrange the equation x + 2y - 5 = 0 to isolate y:
2y = -x + 5
y = (-1/2)x + (5/2)
So, the regression line of y on x is y = (-1/2)x + (5/2).
(ii) Regression line of x on y:
Step 1: Rearrange the equation 3x + 3y - 8 = 0 to isolate x:
3x = -3y + 8
x = (-1)y + (8/3)
So, the regression line of x on y is x = (-1)y + (8/3).
(iii) Finding r using the regression coefficients:
The coefficient of determination, r^2, is given by the square of the regression coefficient.
For the regression line of y on x, the regression coefficient is -1/2.
For the regression line of x on y, the regression coefficient is -1.
Step 1: Calculate the coefficient of determination for y on x:
r^2 = (-1/2)^2
r^2 = 1/4
r = ±√(1/4)
r = ±1/2
Step 2: Calculate the coefficient of determination for x on y:
r^2 = (-1)^2
r^2 = 1
r = ±√(1)
r = ±1
So, the correlation coefficient (r) for the regression line of y on x is ±1/2, and for the regression line of x on y, it is ±1.
To determine the regression line between x and y, we need to use the given equations to find the slope and intercept for both y on x and x on y.
(i) To find the regression line of y on x:
1. Solve the equation x + 2y - 5 = 0 for y:
y = (5 - x) / 2
2. Rearrange the equation to be in the form of y = mx + c, where m is the slope and c is the intercept:
y = (-1/2)x + (5/2)
The regression line of y on x is y = (-1/2)x + (5/2).
(ii) To find the regression line of x on y:
1. Solve the equation 3x + 3y - 8 = 0 for x:
x = (8 - 3y) / 3
2. Rearrange the equation to be in the form of x = my + c, where m is the slope and c is the intercept:
x = (-1/3)y + (8/3)
The regression line of x on y is x = (-1/3)y + (8/3).
(iii) To find the coefficient of determination (r) using the regression coefficients:
The coefficient of determination (r) is calculated using the regression coefficients of the regression line.
The regression coefficient of y on x is -1/2, and the regression coefficient of x on y is -1/3.
The coefficient of determination (r) can be calculated using the following formula:
r = sqrt(regression coefficient of x on y * regression coefficient of y on x)
Substituting the regression coefficients into the formula:
r = sqrt((-1/3) * (-1/2))
= sqrt(1/6)
= 0.408
Therefore, the coefficient of determination (r) is approximately 0.408.