Find the two equation of regression line for the given data:
`n=102, sumx=510, sumy=7140, sumx^(2)=4150, sumy^(2)=740200, sumxy=54900`
Also estimate the value of y when x = 7.

Answer

To find the two equations of regression lines for the given data, we will follow these steps:

### Step 1: Calculate the Means \( \bar{x} \) and \( \bar{y} \)

Given:
- \( n = 102 \)
- \( \sum x = 510 \)
- \( \sum y = 7140 \)

The mean values are calculated as follows:

\[ \bar{x} = \frac{\sum x}{n} = \frac{510}{102} = 5 \]

\[ \bar{y} = \frac{\sum y}{n} = \frac{7140}{102} = 70 \]

### Step 2: Calculate the Regression Coefficients \( b_{xy} \) and \( b_{yx} \)

The formulas for the regression coefficients are:

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

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

Given:
- \( \sum xy = 54900 \)
- \( \sum y^2 = 740200 \)
- \( \sum x^2 = 4150 \)

Now we can calculate \( b_{xy} \):

\[ b_{xy} = \frac{54900 - 102 \cdot 5 \cdot 70}{740200 - 102 \cdot 70^2} \]

Calculating the numerator:

\[ 54900 - 102 \cdot 5 \cdot 70 = 54900 - 35700 = 19200 \]

Calculating the denominator:

\[ 740200 - 102 \cdot 70^2 = 740200 - 102 \cdot 4900 = 740200 - 499800 = 240400 \]

Thus,

\[ b_{xy} = \frac{19200}{240400} \approx 0.08 \]

Now we calculate \( b_{yx} \):

\[ b_{yx} = \frac{54900 - 102 \cdot 5 \cdot 70}{4150 - 102 \cdot 5^2} \]

Calculating the numerator (same as before):

\[ 54900 - 35700 = 19200 \]

Calculating the denominator:

\[ 4150 - 102 \cdot 5^2 = 4150 - 2550 = 1600 \]

Thus,

\[ b_{yx} = \frac{19200}{1600} = 12 \]

### Step 3: Write the Regression Equations

The regression equations are given by:

1. \( y - \bar{y} = b_{xy}(x - \bar{x}) \)
2. \( x - \bar{x} = b_{yx}(y - \bar{y}) \)

Substituting the values:

1. For \( y \):
\[ y - 70 = 0.08(x - 5) \implies y = 0.08x + 70 - 0.4 \implies y = 0.08x + 69.6 \]

2. For \( x \):
\[ x - 5 = 12(y - 70) \implies x = 12y - 840 + 5 \implies x = 12y - 835 \]

### Step 4: Estimate the Value of \( y \) when \( x = 7 \)

Using the first regression equation:

\[ y = 0.08(7) + 69.6 = 0.56 + 69.6 = 70.16 \]

Using the second regression equation:

\[ x = 12y - 835 \implies 7 = 12y - 835 \implies 12y = 842 \implies y = \frac{842}{12} \approx 70.17 \]

### Final Results

The two regression equations are:
1. \( y = 0.08x + 69.6 \)
2. \( x = 12y - 835 \)

The estimated value of \( y \) when \( x = 7 \) is approximately \( 70.16 \).