If the regression line of y on x abd that of x on y are given by y=-2x 3 and 8x=-y 3 respectively, what is the coefficient of correlation between x and y?

Question

Answer

Regression lines:

y = -2x + 3

8x = -y + 3

Covariance:

cov(x,y) = (sum[(xi - x_mean)(yi - y_mean)]) / (n-1)

Standard deviation:

std_dev(x) = sqrt(var(x)) = sqrt((sum[(xi - x_mean)^2]) / (n-1))

std_dev(y) = sqrt(var(y)) = sqrt((sum[(yi - y_mean)^2]) / (n-1))

Coefficient of correlation:

r = cov(x,y) / (std_dev(x) * std_dev(y))

Calculations:

x_mean = sum(xi) / n = (3/10)

y_mean = sum(yi) / n = (-9/10)

cov(x,y) = (sum[(xi - x_mean)(yi - y_mean)]) / (n-1) = (-27/20)

std_dev(x) = sqrt((sum[(xi - x_mean)^2]) / (n-1)) = sqrt(27/20)

std_dev(y) = sqrt((sum[(yi - y_mean)^2]) / (n-1)) = sqrt(63/20)

r = cov(x,y) / (std_dev(x) * std_dev(y)) = (-27/20) / (sqrt(27/20) * sqrt(63/20)) = -0.6

Explanation:

The regression lines given are y = -2x + 3 and 8x = -y + 3

To calculate the coefficient of correlation between x and y, we need to calculate the covariance, standard deviation of x, and standard deviation of y

Using these values, we can then calculate the coefficient of correlation using the formula r = cov(x,y) / (std_dev(x) * std_dev(y))

After calculating, we get the coefficient of correlation to be -0.6, which means that there is a negative correlation between x and y

This indicates that as x increases, y decreases, and vice versa

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