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The mean and standard deviation of 40 observations are 30 and 5 respectively. It was noticed that two of the observations 12 and 10 were wrongly recorded. If $σ$ is the standard deviation of the data after omitting the two wrong observations, then $19 σ^{2} - 113$ is equal to

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Detailed Solution

$\frac{\sum x_{i}}{40} = 30; \frac{\sum {x_{i}}^{2}}{40} - 30^{2} = 25 \Rightarrow \sum x_{i} = 1200; \sum {x_{i}}^{2} = 37000$

After wrong data is omitted
$\sum x_{i} = 1178; \sum {x_{i}}^{2} = 36756;$
$\Rightarrow$ new mean $\bar{x} = \frac{1178}{38} = 31$
new variance $σ^{2} = \frac{36756}{38} - 31^{2}$
$\Rightarrow 38 σ^{2} = 238$

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