From the following data find regression equation of y on x, `Sigma x= 55, Sigma y= 88, Sigma x^(2)= 385, Sigma y^(2) = 1114, Sigma xy= 586,n= 10`

Answer

To find the regression equation of \( y \) on \( x \) using the given data, we will follow these steps:

### Step 1: Calculate the Mean of \( x \) and \( y \)
The mean of \( x \) is calculated as:
\[ \bar{x} = \frac{\Sigma x}{n} = \frac{55}{10} = 5.5 \]

The mean of \( y \) is calculated as:
\[ \bar{y} = \frac{\Sigma y}{n} = \frac{88}{10} = 8.8 \]

### Step 2: Calculate the Mean of \( xy \)
The mean of \( xy \) is calculated as:
\[ \overline{xy} = \frac{\Sigma xy}{n} = \frac{586}{10} = 58.6 \]

### Step 3: Calculate the Covariance of \( x \) and \( y \)
The covariance \( \text{Cov}(x, y) \) is calculated using the formula:
\[ \text{Cov}(x, y) = \overline{xy} - \bar{x} \bar{y} \]
Substituting the values:
\[ \text{Cov}(x, y) = 58.6 - (5.5 \times 8.8) = 58.6 - 48.4 = 10.2 \]

### Step 4: Calculate the Variance of \( x \)
The variance \( \sigma^2_x \) is calculated using the formula:
\[ \sigma^2_x = \frac{\Sigma x^2}{n} - \bar{x}^2 \]
Substituting the values:
\[ \sigma^2_x = \frac{385}{10} - (5.5)^2 = 38.5 - 30.25 = 8.25 \]

### Step 5: Calculate the Regression Coefficient \( b_{yx} \)
The regression coefficient \( b_{yx} \) is calculated using the formula:
\[ b_{yx} = \frac{\text{Cov}(x, y)}{\sigma^2_x} \]
Substituting the values:
\[ b_{yx} = \frac{10.2}{8.25} \approx 1.236 \]

### Step 6: Write the Regression Equation
The regression equation of \( y \) on \( x \) is given by:
\[ y - \bar{y} = b_{yx}(x - \bar{x}) \]
Substituting the values:
\[ y - 8.8 = 1.236(x - 5.5) \]
Expanding this gives:
\[ y - 8.8 = 1.236x - 6.798 \]
Thus, rearranging gives:
\[ y = 1.236x + (8.8 - 6.798) \]
Calculating the constant:
\[ y = 1.236x + 2.002 \approx 1.236x + 2 \]

### Final Regression Equation
The regression equation of \( y \) on \( x \) is:
\[ y = 1.236x + 2 \]

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