The two lines of regression are `8x-10y=66` and `40x-18y=214` and variance of x series is 9. What is the standard deviation of y series?

Options

A. 3
B. 4
C. 6
D. 8

Explanation

To solve the problem, we need to find the standard deviation of the y series given the two lines of regression and the variance of the x series. Let's break it down step by step.

### Step 1: Rearranging the Regression Equations
The two lines of regression given are:
1. \( 8x - 10y = 66 \)
2. \( 40x - 18y = 214 \)

We can rearrange these equations to find the regression coefficients.

From the first equation:
\[ 10y = 8x - 66 \implies y = \frac{8}{10}x - \frac{66}{10} \implies y = \frac{4}{5}x - 6.6 \]
Thus, the regression coefficient \( b_{yx} = \frac{4}{5} \).

From the second equation:
\[ 18y = 40x - 214 \implies y = \frac{40}{18}x - \frac{214}{18} \implies y = \frac{20}{9}x - \frac{107}{9} \]
Thus, the regression coefficient \( b_{xy} = \frac{20}{9} \).

### Step 2: Finding the Coefficient of Correlation
The coefficient of correlation \( r \) can be calculated using the formula:
\[ r = \sqrt{b_{yx} \cdot b_{xy}} \]
Substituting the values we found:
\[ r = \sqrt{\left(\frac{4}{5}\right) \cdot \left(\frac{20}{9}\right)} = \sqrt{\frac{80}{45}} = \sqrt{\frac{16}{9}} = \frac{4}{3} \]
Since both regression coefficients are positive, we take the positive root:
\[ r = \frac{4}{3} \]

### Step 3: Finding the Standard Deviation of y
We know that the variance of the x series is given as \( \sigma_x^2 = 9 \). Therefore, the standard deviation of x \( \sigma_x \) is:
\[ \sigma_x = \sqrt{9} = 3 \]

Now, we can use the formula to find the standard deviation of y:
\[ \sigma_y = b_{yx} \cdot \sigma_x \div r \]
Substituting the values:
\[ \sigma_y = \left(\frac{4}{5}\right) \cdot 3 \div \left(\frac{4}{3}\right) \]
Calculating this:
\[ \sigma_y = \frac{4 \cdot 3}{5} \cdot \frac{3}{4} = \frac{12}{5} = 2.4 \]

### Conclusion
The standard deviation of the y series is \( \sigma_y = 2.4 \).