If the mean and variance of the data 65, 68, 58, 44, 48, 45, 60, 60 where , are 56 and 66.2 respectively, then is equal to

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Solution

Q.
If the mean and variance of the data 65, 68, 58, 44, 48, 45, 60, $α,$ $β,$ 60 where $α > β$ , are 56 and 66.2 respectively, then $α^{2} + β^{2}$ is equal to _____ . [2024]

Ans.
(6344)

Mean = 56

$Variance = \frac{1}{n} \sum_{i = 1}^{n} x_{i}^{2} - {(\bar{x})}^{2}$

$\Rightarrow 66.2 = \frac{1}{10} [{(65)}^{2} + {(68)}^{2} + {(58)}^{2} + {(44)}^{2} + {(48)}^{2} + {(45)}^{2} + (60)$ ²+α²+β²+(60)²]-(56)²

$\Rightarrow 3202.2 \times 10 = 4225 + 4624 + 3364 + 1936 + 2304 + 2025 + 3600 + α^{2} + β^{2} + 3600$

$\Rightarrow α^{2} + β^{2} = 6344$

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