The two lines of regression meet at

Options

A. `(overset(-)x,overset(-)y)`
B. `(sigmax, sigmay)`
C. `(sigma x^2, sigma y^2)`
D. None of these

Answer

To find the point where the two lines of regression meet, we can follow these steps:

### Step 1: Understand the Regression Lines
The two lines of regression are typically represented as:
1. The regression line of Y on X (denoted as Y = a + bX)
2. The regression line of X on Y (denoted as X = c + dY)

### Step 2: Set the Two Equations Equal
To find the point where these two lines intersect, we need to set the equations equal to each other. This means we will solve the equations:
\[ a + bX = c + dY \]

### Step 3: Express One Variable in Terms of the Other
From the first equation (Y on X), we can express Y in terms of X:
\[ Y = a + bX \]

From the second equation (X on Y), we can express X in terms of Y:
\[ X = c + dY \]

### Step 4: Substitute and Solve
Now, we can substitute one equation into the other. Let's substitute the expression for Y into the equation for X:
\[ X = c + d(a + bX) \]

### Step 5: Rearranging the Equation
Rearranging the equation gives us:
\[ X = c + da + dbX \]
\[ X - dbX = c + da \]
\[ X(1 - db) = c + da \]

### Step 6: Solve for X
Now, we can solve for X:
\[ X = \frac{c + da}{1 - db} \]

### Step 7: Substitute Back to Find Y
Now that we have X, we can substitute it back into the equation for Y:
\[ Y = a + b\left(\frac{c + da}{1 - db}\right) \]

### Step 8: Final Expression for Y
This gives us the final expression for Y:
\[ Y = a + \frac{b(c + da)}{1 - db} \]

### Conclusion
Thus, the two lines of regression meet at the point \((X, Y)\) where:
- \(X = \frac{c + da}{1 - db}\)
- \(Y = a + \frac{b(c + da)}{1 - db}\)