Correlation And Regression

Since X is the independent variable and Y is the dependent variable, we need to find the regression equation of Y on X, which is given by We first prepare a table to calculate the necessary sums: Here, . First, we find the means: Next, we calculate the regression coefficient : Now, we calculate the constant : Therefore, the appropriate regression equation is:

(a) Find and :
The two regression lines intersect at the point of means . We solve the two equations simultaneously.
1)
2)
Multiply equation (1) by 5: .
Add this to equation (2): Substitute into : So, and . (b) Find and :
Assume the first line is Y on X: . So, .
Assume the second line is X on Y: . So, .
Let's check the condition , which means .
.
This is greater than 1, so our assumption is wrong.
Let's switch the assumptions.
Line of Y on X: . So, .
Line of X on Y: . So, .
Check: . This is , so the assumption is correct.
and .
(c) Find :
The correlation coefficient is the geometric mean of the regression coefficients. Since both and are positive, is also positive. So, .
(d) Find Var(X):
We know that . The variance is the square of the standard deviation ( ). The variance of X is 16.

One of the properties of regression coefficients ( and ) and the correlation coefficient ( ) is that the arithmetic mean of the regression coefficients must be greater than or equal to the correlation coefficient. Let's check this condition with the given data:


The arithmetic mean of the regression coefficients is: Comparing this to the correlation coefficient: This statement is false. Since the condition is violated, the given data is inconsistent. Therefore, the original statement is True.