If u = 2x 5 and v = –3y – 6 and regression coefficient of y on x is ...
Regression Coefficient of v on u
To find the regression coefficient of v on u, we need to use the formula:
βv,u = (cov(v,u))/(var(u))
where βv,u is the regression coefficient of v on u, cov(v,u) is the covariance between v and u, and var(u) is the variance of u.
Covariance between v and u
To find the covariance between v and u, we need to use the formula:
cov(v,u) = cov(-3y-6, 2x+5)
= -3 cov(y,x)
Since the regression coefficient of y on x is given to be 2.4, we know that:
cov(y,x) = 2.4 σx σy
where σx and σy are the standard deviations of x and y, respectively.
Standard deviation of x and y
We don't have the standard deviations of x and y, but we can use the following substitutions:
u = 2x+5, so x = (u-5)/2
v = -3y-6, so y = -(v+6)/3
Substituting these expressions into the formula for cov(y,x), we get:
cov(y,x) = 2.4 σx σy
= 2.4 cov(u-5, -(v+6))/6
= -0.4 cov(u-5, v+6)
Covariance between v and u (continued)
Substituting this expression for cov(y,x) into the formula for cov(v,u), we get:
cov(v,u) = -3 cov(y,x)
= 1.2 cov(u-5, v+6)
Variance of u
To find the variance of u, we need to use the formula:
var(u) = var(2x+5)
= 4 var(x)
Regression coefficient of v on u (final calculation)
Substituting the expressions for cov(v,u) and var(u) into the formula for βv,u, we get:
βv,u = (cov(v,u))/(var(u))
= (1.2 cov(u-5, v+6))/(4 var(x))
= (0.3 cov(u-5, v+6))/(var(x))
Since we don't have the values for cov(u-5, v+6) and var(x), we cannot calculate the exact value of βv,u. However, we can see that it is proportional to cov(u-5, v+6)/var(x), which means that it will be positive if cov(u-5, v+6) is positive (i.e. u and v have the same direction of change) and if var(x) is positive (which it should be, since u = 2x+5). Therefore, we can conclude that the regression coefficient of v on u is positive and greater than zero.