If n = 6, `sumx = 36, sumy = 60,sumxy = -67 , sumx^2 =50 , sumy^2` =106, Estimate y when x is 13.

Answer

To estimate \( y \) when \( x = 13 \), we will follow these steps:

### Step 1: Calculate the regression coefficient \( b_{yx} \)

The formula for the regression coefficient \( b_{yx} \) is given by:

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

Substituting the given values:
- \( n = 6 \)
- \( \sum{xy} = -67 \)
- \( \sum{x} = 36 \)
- \( \sum{y} = 60 \)
- \( \sum{x^2} = 50 \)

Calculating the numerator:
\[ n \cdot \sum{xy} - \sum{x} \cdot \sum{y} = 6 \cdot (-67) - 36 \cdot 60 = -402 - 2160 = -2562 \]

Calculating the denominator:
\[ n \cdot \sum{x^2} - (\sum{x})^2 = 6 \cdot 50 - 36^2 = 300 - 1296 = -996 \]

Now, substituting these values into the formula:
\[ b_{yx} = \frac{-2562}{-996} = \frac{2562}{996} \]

### Step 2: Simplify \( b_{yx} \)

To simplify \( \frac{2562}{996} \):
- Dividing both the numerator and denominator by 3:
\[ \frac{854}{332} \]
- Dividing again by 2:
\[ \frac{427}{166} \]

Thus, \( b_{yx} = \frac{427}{166} \).

### Step 3: Calculate \( \bar{x} \) and \( \bar{y} \)

Calculate the means:
\[ \bar{x} = \frac{\sum{x}}{n} = \frac{36}{6} = 6 \]
\[ \bar{y} = \frac{\sum{y}}{n} = \frac{60}{6} = 10 \]

### Step 4: 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:
\[ y - 10 = \frac{427}{166}(x - 6) \]

### Step 5: Estimate \( y \) when \( x = 13 \)

Substituting \( x = 13 \) into the regression equation:
\[ y - 10 = \frac{427}{166}(13 - 6) \]
\[ y - 10 = \frac{427}{166} \cdot 7 \]
\[ y - 10 = \frac{2989}{166} \]
\[ y = 10 + \frac{2989}{166} \]

Calculating \( y \):
\[ y = 10 + 18.00 \approx 28.006 \]

Thus, the estimated value of \( y \) when \( x = 13 \) is approximately \( 28.006 \).