Given that the regression equation of y on x is `y = a + 1/(m)x`, find the value of m when `r = 0.5 , sigma_(x)^2=1/4sigma_(y)^2` .

Answer

To solve the problem step by step, we will start with the given regression equation and the provided values.

### Step 1: Understand the regression equation
The regression equation of y on x is given as:
\[ y = a + \frac{1}{m} x \]

### Step 2: Use the correlation coefficient
We know that the correlation coefficient \( r \) is given as \( r = 0.5 \).

### Step 3: Relate standard deviations
We are also given that:
\[ \sigma_x^2 = \frac{1}{4} \sigma_y^2 \]

This implies:
\[ \frac{\sigma_x^2}{\sigma_y^2} = \frac{1}{4} \]

### Step 4: Find the ratio of standard deviations
Taking the square root of both sides gives:
\[ \frac{\sigma_x}{\sigma_y} = \frac{1}{2} \]

### Step 5: Use the formula for the regression coefficient
The regression coefficient \( p_{yx} \) (the slope of the regression line) can be expressed as:
\[ p_{yx} = r \cdot \frac{\sigma_y}{\sigma_x} \]

### Step 6: Substitute the known values
Substituting the known values into the equation:
\[ p_{yx} = 0.5 \cdot \frac{\sigma_y}{\sigma_x} \]
Since we have \( \frac{\sigma_x}{\sigma_y} = \frac{1}{2} \), we can find \( \frac{\sigma_y}{\sigma_x} \):
\[ \frac{\sigma_y}{\sigma_x} = 2 \]

### Step 7: Calculate \( p_{yx} \)
Now substituting \( \frac{\sigma_y}{\sigma_x} \) back into the equation for \( p_{yx} \):
\[ p_{yx} = 0.5 \cdot 2 = 1 \]

### Step 8: Relate \( p_{yx} \) to \( m \)
From the regression equation, we know that:
\[ p_{yx} = \frac{1}{m} \]
Since we found \( p_{yx} = 1 \), we can set up the equation:
\[ \frac{1}{m} = 1 \]

### Step 9: Solve for \( m \)
To find \( m \), we rearrange the equation:
\[ m = 1 \]

### Conclusion
Thus, the value of \( m \) is:
\[ m = 1 \]

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