(A) 15

(B) 10

(C) 5

(D) 20

Correct Answer: 20

Explanation

Original number of observations $n = 10$.

Original mean $=10$:

$$ \sum x = 10 \times 10 = 100 $$

Original variance $=2$:

$$ \frac{1}{10}\sum x^2 - 10^2 = 2 $$

$$ \sum x^2 = 10(100 + 2) = 1020 $$

After replacing $\alpha$ by $\beta$, new mean is $10.1$:

$$ \sum x - \alpha + \beta = 10 \times 10.1 = 101 $$

$$ \beta - \alpha = 1 \quad (1) $$

New variance is $1.99$:

$$ \frac{1}{10}(\sum x^2 - \alpha^2 + \beta^2) - (10.1)^2 = 1.99 $$

$$ \sum x^2 - \alpha^2 + \beta^2 = 10(102.01 + 1.99) = 1040 $$

$$ 1020 - \alpha^2 + \beta^2 = 1040 $$

$$ \beta^2 - \alpha^2 = 20 $$

$$ (\beta - \alpha)(\beta + \alpha) = 20 $$

Using (1):

$$ 1 \cdot (\alpha + \beta) = 20 $$

$$ \alpha + \beta = \boxed{20} $$

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