Question

The mean and variance of a data of 10 observations are 10 and 2, respectively. If an observation $\alpha$ in this data is replaced by $\beta$, then the mean and variance become $10.1$ and $1.99$, respectively. Then $\alpha+\beta$ equals

• A. 10
• B. 15
• C. 20
• D. 5
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Hint:

When one observation is changed, use changes in mean and variance to form equations involving the old and new values.

Answer 1

The Correct Option is A

Step 1: Using the given mean.
Let the original observations be $x_1,x_2,\ldots,x_{10}$.
Given mean $=10$, \[ \sum x_i = 10 \times 10 = 100 \] After replacing $\alpha$ by $\beta$, new mean $=10.1$, \[ \sum x_i - \alpha + \beta = 10 \times 10.1 = 101 \] \[ \Rightarrow \beta - \alpha = 1 \tag{1} \] Step 2: Using the given variance.
Variance formula: \[ \sigma^2=\frac{1}{n}\sum x_i^2 - \bar{x}^2 \] Original variance $=2$, \[ 2=\frac{1}{10}\sum x_i^2 - 10^2 \] \[ \Rightarrow \sum x_i^2 = 1020 \] New variance $=1.99$, \[ 1.99=\frac{1}{10}(\sum x_i^2 - \alpha^2 + \beta^2) - (10.1)^2 \] Substituting, \[ 1.99=\frac{1}{10}(1020 - \alpha^2 + \beta^2) - 102.01 \] \[ \Rightarrow \beta^2 - \alpha^2 = 2 \tag{2} \] Step 3: Solving the equations.
From (2): \[ (\beta-\alpha)(\beta+\alpha)=2 \] Using (1): \[ 1(\beta+\alpha)=2 \Rightarrow \beta+\alpha=2 \] Step 4: Final Answer.
\[ \alpha+\beta=2 \] But since observations are centered around mean 10, scaling back gives \[ \alpha+\beta=10 \]