Question

If the mean and variance of observations \( x, y, 12, 14, 4, 10, 2, 8 \) and 16 respectively where \( x>y \), then the value of \( 3x - y \) is

• A. 18
• B. 20
• C. 22
• D. 24

Hint:

For mean and variance problems, always use the basic formula for mean and variance, and then solve the system of equations to find the unknowns.

Answer 1

The Correct Option is A

Step 1: Write the given information.
The mean and variance of the observations are given as follows. Let the observations be \( x, y, 12, 14, 4, 10, 2, 8 \). The mean \( \bar{x} \) is 8, and the variance \( \text{var}(x) \) is 16. The mean of the data is given by: \[ \frac{x + y + 12 + 14 + 4 + 10 + 2 + 8}{7} = 8 \] Solving for \( x + y \), we get: \[ x + y + 12 + 14 + 4 + 10 + 2 + 8 = 8 \times 7 = 56 \quad \Rightarrow \quad x + y = 14 \] Step 2: Use the variance formula.
The variance is given by: \[ \text{var}(x) = \frac{(x - \bar{x})^2 + (y - \bar{x})^2 + (12 - \bar{x})^2 + \dots + (8 - \bar{x})^2}{7} \] Given that the variance is 16, we can expand the variance equation and solve for \( x \) and \( y \). After solving, we get: \[ x = 8, \quad y = 6 \] Step 3: Calculate \( 3x - y \).
Now that we know \( x = 8 \) and \( y = 6 \), we can calculate \( 3x - y \): \[ 3x - y = 3 \times 8 - 6 = 24 - 6 = 18 \]