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 then the mean and variance become 10.1 and 1.99, respectively. Then equals: [2026]

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Solution

Q.
The mean and variance of a data of 10 observations are 10 and 2, respectively. If an observation $α$ in this data is replaced by $β,$ then the mean and variance become 10.1 and 1.99, respectively. Then $α + β$ equals: [2026]

1 5

2 15

3 10

4 20

Ans.
(4)

$Let first 10 numbers are x_{1}, x_{2}, \dots, x_{9}, α$

$\Rightarrow α + \sum_{i = 1}^{9} x_{i} = 100 \Rightarrow \sum_{i = 1}^{9} x_{i} = 100 - α$

$Variance = (\frac{\sum x_{i}^{2}}{n}) - {(\frac{\sum x_{i}}{n})}^{2}$

$\Rightarrow \frac{\sum x_{i}^{2}}{n} = 98$

$\Rightarrow x_{1}^{2} + x_{2}^{2} + \dots + x_{9}^{2} + α^{2} = 1020 \Rightarrow \sum x_{i}^{2} = 1020 - α^{2}$

$In second case, let numbers are x_{1}, x_{2}, \dots, x_{9}, β$

$100 - α + β = 101 \Rightarrow α - β + 1 = 0$

$\frac{\sum x_{i}^{2} + β^{2}}{10} - {(10.1)}^{2} = 1.99$

$\Rightarrow β^{2} - α^{2} = 20$

$α = \frac{19}{2}, β = \frac{21}{2}$

$α + β = \frac{19 + 21}{2} = 20$

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