Question

Consider 10 observations $x_1, x_2, \ldots, x_{10}$ such that \[\sum_{i=1}^{10} (x_i - \alpha) = 2 \quad \text{and} \quad \sum_{i=1}^{10} (x_i - \beta)^2 = 40,\]where $\alpha, \beta$ are positive integers. Let the mean and the variance of the observations be $\frac{6}{5}$ and $\frac{84}{25}$, respectively. The value of $\frac{\beta}{\alpha}$ is equal to:

• A. 2
• B. $\frac{3}{2}$
• C. $\frac{5}{2}$
• D. 1

Answer 1

The Correct Option is A

We are given:

$\sum_{i=1}^{10} X_i - 10A = 2 \implies \sum_{i=1}^{10} X_i = 10A + 2$.

$\sum_{i=1}^{10} X_i - 10B = 40 \implies \sum_{i=1}^{10} X_i = 10B + 40$.

Equating both expressions for $\sum_{i=1}^{10} X_i$, we get:

$10A + 2 = 10B + 40 \implies 10A - 10B = 38 \implies A - B = 3.8$.

Since A and B are integers, $A = 4$ and $B = 2$.

Thus, $B = 2$.