If is given the `sigma_(x)^(2)=9,r=0.6` and regression equation of Y on X is `4x-5y+33=0` find Var (Y).

Answer

To solve the problem step by step, we will use the given information and apply the necessary formulas.

### Step 1: Understand the given information
We are given:
- Variance of X, denoted as \( \sigma_x^2 = 9 \)
- Correlation coefficient, \( r = 0.6 \)
- Regression equation of Y on X: \( 4x - 5y + 33 = 0 \)

### Step 2: Rearrange the regression equation
We need to express the regression equation in the slope-intercept form \( y = mx + c \).
Starting from:
\[ 4x - 5y + 33 = 0 \]
Rearranging gives:
\[ 5y = 4x + 33 \]
\[ y = \frac{4}{5}x + \frac{33}{5} \]
Here, the slope \( b_{yx} = \frac{4}{5} \).

### Step 3: Use the formula for the slope of the regression line
The formula relating the slope of the regression line to the correlation coefficient and the standard deviations is:
\[ b_{yx} = r \frac{\sigma_y}{\sigma_x} \]
Where:
- \( b_{yx} \) is the slope of the regression line of Y on X
- \( r \) is the correlation coefficient
- \( \sigma_y \) is the standard deviation of Y
- \( \sigma_x \) is the standard deviation of X

### Step 4: Find the standard deviation of X
Since we know the variance of X:
\[ \sigma_x^2 = 9 \]
Taking the square root gives:
\[ \sigma_x = \sqrt{9} = 3 \]

### Step 5: Substitute the known values into the slope formula
Now we can substitute the known values into the slope formula:
\[ \frac{4}{5} = 0.6 \frac{\sigma_y}{3} \]

### Step 6: Solve for \( \sigma_y \)
To isolate \( \sigma_y \), we first multiply both sides by 3:
\[ 3 \cdot \frac{4}{5} = 0.6 \sigma_y \]
Calculating the left side:
\[ \frac{12}{5} = 0.6 \sigma_y \]

Now, convert \( 0.6 \) to a fraction:
\[ 0.6 = \frac{6}{10} = \frac{3}{5} \]
Thus, we have:
\[ \frac{12}{5} = \frac{3}{5} \sigma_y \]
Now, multiply both sides by \( \frac{5}{3} \):
\[ \sigma_y = \frac{12}{5} \cdot \frac{5}{3} = \frac{12}{3} = 4 \]

### Step 7: Find the variance of Y
The variance of Y is given by:
\[ \text{Var}(Y) = \sigma_y^2 \]
Substituting the value we found:
\[ \text{Var}(Y) = 4^2 = 16 \]

### Final Answer
Thus, the variance of Y is:
\[ \text{Var}(Y) = 16 \]

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