Question

If \(2Y - X - 50 = 0\), \(3Y - 2X - 10 = 0\) are the two regression equations, then the mean values of the variables \(X\) and \(Y\) respectively are:

• A. \(80 \text{ and } 90\)
• B. \(130 \text{ and } 90\)
• C. \(96 \text{ and } 130\)
• D. \(90 \text{ and } 80\)

Hint:

When two regression lines are given, their point of intersection gives the mean values of \(X\) and \(Y\). Always solve them simultaneously.

Answer 1

The Correct Option is B

To find the mean values of \(X\) and \(Y\), we solve the given regression equations simultaneously: 1st equation: \[ 2Y - X = 50 \quad \Rightarrow \quad X = 2Y - 50 \quad \cdots (1) \] 2nd equation: \[ 3Y - 2X = 10 \quad \Rightarrow \quad 3Y = 2X + 10 \quad \cdots (2) \] Substitute equation (1) into equation (2): \[ 3Y = 2(2Y - 50) + 10 \Rightarrow 3Y = 4Y - 100 + 10 \Rightarrow 3Y = 4Y - 90 \Rightarrow Y = 90 \] Now substitute \(Y = 90\) into equation (1): \[ X = 2(90) - 50 = 180 - 50 = 130 \] So, the mean values are: \[ {X = 130, \quad Y = 90} \]