If `n = 5,sumx = sumy = 20, sumx^2 = sumy^2 = 90 , sumxy = 76`
Find the regression equation of x on y.

Answer

To find the regression equation of \(x\) on \(y\), we need to follow these steps:

### Step 1: Identify the given values
We have the following values:
- \( n = 5 \)
- \( \sum x = 20 \)
- \( \sum y = 20 \)
- \( \sum x^2 = 90 \)
- \( \sum y^2 = 90 \)
- \( \sum xy = 76 \)

### Step 2: Calculate the regression coefficient \(b_{xy}\)
The formula for the regression coefficient \(b_{xy}\) of \(x\) on \(y\) is given by:

\[ b_{xy} = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2} \]

Substituting the values into the formula:

\[ b_{xy} = \frac{5 \cdot 76 - 20 \cdot 20}{5 \cdot 90 - (20)^2} \]

Calculating the numerator:

\[ 5 \cdot 76 = 380 \]
\[ 20 \cdot 20 = 400 \]
\[ \text{Numerator} = 380 - 400 = -20 \]

Calculating the denominator:

\[ 5 \cdot 90 = 450 \]
\[ (20)^2 = 400 \]
\[ \text{Denominator} = 450 - 400 = 50 \]

Now substituting back into the equation:

\[ b_{xy} = \frac{-20}{50} = -\frac{2}{5} \]

### Step 3: Calculate the means \( \bar{x} \) and \( \bar{y} \)
The mean of \(x\) and \(y\) is calculated as follows:

\[ \bar{x} = \frac{\sum x}{n} = \frac{20}{5} = 4 \]

\[ \bar{y} = \frac{\sum y}{n} = \frac{20}{5} = 4 \]

### Step 4: Write the regression equation of \(x\) on \(y\)
The regression equation of \(x\) on \(y\) is given by:

\[ x - \bar{x} = b_{xy} (y - \bar{y}) \]

Substituting the values we found:

\[ x - 4 = -\frac{2}{5} (y - 4) \]

### Step 5: Rearranging the equation
To rearrange this equation, we can multiply both sides by 5:

\[ 5(x - 4) = -2(y - 4) \]

Expanding both sides:

\[ 5x - 20 = -2y + 8 \]

Rearranging gives us:

\[ 5x + 2y = 28 \]

### Final Result
Thus, the regression equation of \(x\) on \(y\) is:

\[ 5x + 2y = 28 \]

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