x and y are a pair of correlated variables . Ten observations of their values `(x_(p)y_(t))` have the following result : `sum x = 55, sum y = 55, sum x^(2) = 385, sum xy = 350 `. Predict the value of y when the value of x is 6

Answer

To solve the problem, we need to find the regression line of the correlated variables \(x\) and \(y\) and then use it to predict the value of \(y\) when \(x = 6\).

### Step-by-Step Solution:

1. **Identify the given values:**
- \( \sum x = 55 \)
- \( \sum y = 55 \)
- \( \sum x^2 = 385 \)
- \( \sum xy = 350 \)
- Number of observations \( n = 10 \)

2. **Calculate the slope \( b \) of the regression line:**
The formula for \( b \) is:
\[ b = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2} \]
Plugging in the values:
\[ b = \frac{10 \times 350 - 55 \times 55}{10 \times 385 - 55^2} \]
Calculate the numerator:
\[ 10 \times 350 = 3500 \]
\[ 55 \times 55 = 3025 \]
So, the numerator becomes:
\[ 3500 - 3025 = 475 \]
Now calculate the denominator:
\[ 10 \times 385 = 3850 \]
\[ 55^2 = 3025 \]
So, the denominator becomes:
\[ 3850 - 3025 = 825 \]
Therefore:
\[ b = \frac{475}{825} = 0.57576 \approx 0.57 \]

3. **Calculate the intercept \( a \) of the regression line:**
The formula for \( a \) is:
\[ a = \frac{\sum y \sum x^2 - \sum x \sum xy}{n \sum x^2 - (\sum x)^2} \]
Plugging in the values:
\[ a = \frac{55 \times 385 - 55 \times 350}{10 \times 385 - 55^2} \]
The denominator is already calculated as \( 825 \).

Now calculate the numerator:
\[ 55 \times 385 = 21175 \]
\[ 55 \times 350 = 19250 \]
So, the numerator becomes:
\[ 21175 - 19250 = 1925 \]
Therefore:
\[ a = \frac{1925}{825} \approx 2.33333 \approx 2.33 \]

4. **Formulate the regression equation:**
The regression line is given by:
\[ y = a + bx \]
Substituting the values of \( a \) and \( b \):
\[ y = 2.33 + 0.57x \]

5. **Predict the value of \( y \) when \( x = 6 \):**
Substitute \( x = 6 \) into the regression equation:
\[ y = 2.33 + 0.57 \times 6 \]
Calculate \( 0.57 \times 6 = 3.42 \):
\[ y = 2.33 + 3.42 = 5.75 \]

### Final Answer:
The predicted value of \( y \) when \( x = 6 \) is approximately \( 5.75 \).