If the relationship between two variables x and y in given by 2x 3y 4 = 0, then the value of the correlation coefficient between x and y is:?

Question

If the relationship between two variables x and y in given by 2x   3y   4 = 0, then the value of the correlation coefficient between x and y is:?

Answer

Correlation Coefficient between x and y

Given:

2x + 3y + 4 = 0

Calculating Correlation Coefficient

To find the correlation coefficient between x and y, we need to determine the covariance and standard deviation of both variables.

Covariance

The covariance between x and y can be calculated using the following formula:

cov(x,y) = E[(x - E(x))(y - E(y))]

Where E(x) and E(y) represent the mean of x and y, respectively. We can rewrite the given equation as:

2x = -3y - 4

x = (-3/2)y - 2

Now, we can compute the means:

E(x) = E((-3/2)y - 2)

E(x) = (-3/2)E(y) - 2

Substituting the value of E(x) in the formula for covariance:

cov(x,y) = E[(x - (-3/2)E(y) - 2)(y - E(y))]

cov(x,y) = (-3/2)E[(y - E(y))^2]

Standard Deviation

The standard deviation of x and y can be calculated using the following formulas:

SD(x) = sqrt(E[(x - E(x))^2])

SD(y) = sqrt(E[(y - E(y))^2])

Substituting the value of E(x) in the formula for SD(x):

SD(x) = sqrt(E[((-3/2)y - 2 - (-3/2)E(y) - 2)^2])

SD(x) = sqrt((9/4)E[(y - E(y))^2])

Substituting the value of E(y) in the formula for SD(y):

SD(y) = sqrt(E[(y - (-4/3))^2])

Correlation Coefficient

Using the formulas for covariance and standard deviation, we can compute the correlation coefficient:

corr(x,y) = cov(x,y) / (SD(x) * SD(y))

corr(x,y) = (-3/2) / (3/2 * sqrt(E[(y - E(y))^2]))

Therefore, the correlation coefficient between x and y is:

corr(x,y) = -1/sqrt(3)