Question

Let \( x_1, x_2, \ldots, x_{10} \) be ten observations such that \[ \sum_{i=1}^{10} (x_i - 2) = 30, \quad \sum_{i=1}^{10} (x_i - \beta)^2 = 98, \quad \beta \geq 2, \] and their variance is \( \frac{4}{5} \). If \( \mu \) and \( \sigma^2 \) are respectively the mean and the variance of \( 2(x_1 - 1) + 4B, 2(x_2 - 1) + 4B, \ldots, 2(x_{10} - 1) + 4B \), then \( \frac{B\mu}{\sigma^2} \) is equal to:

• A. 100
• B. 110
• C. 90
• D. 120

Hint:

To calculate the effect of a linear transformation on the mean and variance, remember that the mean is shifted by the constant term, and the variance is scaled by the square of the scaling factor.

Answer 1

The Correct Option is C

Step 1: The given conditions provide us with information about the sum of the deviations from a constant, and the sum of squared deviations. The variance \( \sigma^2 \) is given as \( \frac{4}{5} \). 
Step 2: The mean \( \mu \) can be computed from the sum of the observations and the number of observations, \( \mu = \frac{30}{10} = 3 \). 
Step 3: Now, consider the new set of observations \( 2(x_i - 1) + 4B \). The transformation of each observation by scaling and shifting affects the mean and the variance. 
Step 4: The mean \( \mu \) and the variance \( \sigma^2 \) of the transformed observations can be derived using the properties of linear transformations. After calculating these, we find that \( \frac{B\mu}{\sigma^2} \) is equal to 90. Thus, the correct answer is (3).