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Solution

Let observations be $x_1,x_2,\ldots,x_9,\alpha$ Mean $=10 \Rightarrow \sum x_i =100$ $\sum_{i=1}^{9}x_i =100-\alpha$ Variance $\frac{\sum x_i^2}{10}-10^2=2$ $\Rightarrow \sum x_i^2=1020$ $\sum_{i=1}^{9}x_i^2=1020-\alpha^2$ New mean $\frac{100-\alpha+\beta}{10}=10.1$ $\Rightarrow \beta-\alpha=1$ …(1) New variance $\frac{1020-\alpha^2+\beta^2}{10}-(10.1)^2=1.99$ $\Rightarrow \beta^2-\alpha^2=20$ $(\beta-\alpha)(\beta+\alpha)=20$ Using (1): $1(\alpha+\beta)=20$ $\Rightarrow \alpha+\beta=20$