Two samples from bivariate populations have 15 observations each. The sample means of
X and Y are 25 and 18 respectively. The corresponding sum of squares of deviations from means are 136 and 148 respectively. The sum of product of deviations from respective means is 122.Obtain the regression equation of x on y.

Answer

To obtain the regression equation of X on Y, we will follow these steps:

### Step 1: Identify the given values
- Sample size (N) = 15
- Sample mean of X (\( \bar{X} \)) = 25
- Sample mean of Y (\( \bar{Y} \)) = 18
- Sum of squares of deviations from means for X (\( SS_X \)) = 136
- Sum of squares of deviations from means for Y (\( SS_Y \)) = 148
- Sum of product of deviations from respective means (\( SP \)) = 122

### Step 2: Calculate the regression coefficient of X on Y
The regression coefficient \( b_{XY} \) is calculated using the formula:
\[ b_{XY} = \frac{SP}{SS_Y} \]
Substituting the values we have:
\[ b_{XY} = \frac{122}{148} \]

### Step 3: Simplify the regression coefficient
To simplify \( \frac{122}{148} \):
\[ b_{XY} = \frac{61}{74} \]

### Step 4: Write the regression equation
The regression equation of X on Y can be expressed as:
\[ X - \bar{X} = b_{XY} (Y - \bar{Y}) \]
Substituting the known values:
\[ X - 25 = \frac{61}{74} (Y - 18) \]

### Step 5: Rearranging the equation
To rearrange the equation:
\[ X - 25 = \frac{61}{74} Y - \frac{61 \times 18}{74} \]
Calculating \( \frac{61 \times 18}{74} \):
\[ \frac{61 \times 18}{74} = \frac{1098}{74} \]
Thus, we can rewrite the equation as:
\[ X - 25 = \frac{61}{74} Y - \frac{1098}{74} \]

### Step 6: Multiply through by 74 to eliminate the fraction
Multiplying through by 74 gives:
\[ 74X - 1850 = 61Y - 1098 \]

### Step 7: Rearranging to standard form
Rearranging gives us:
\[ 74X - 61Y - 752 = 0 \]

### Final Result
The regression equation of X on Y is:
\[ 74X - 61Y - 752 = 0 \]

---