The mean and standard deviation of 40 observations are 30 and 5...

Given,

Wrong mean $= {\mu}_{1} = 30$

Wrong standard deviation $= {\sigma}_{1} = 5$

Now mean ig given by $\dfrac{∑ {x}_{i}}{40} = 30$

$⇒ ∑ {x}_{i} = 1200$

Variance is given by ${\sigma}_{1}^{2} = 25$

$⇒ \dfrac{∑ {x}_{i}^{2}}{40} - {30}^{2} = 25$

$⇒ ∑ {x}_{i}^{2} = 925 \times 40 = 37000$

New sum of mean $= ∑ {x}_{i}^{'} = 1200 - 10 - 12 = 1178$

So, new mean $= {\mu}_{1}^{'} = \dfrac{1178}{38} = 31$

Now new variance $∑ {x}_{i}^{2} = 37000 - {\left(10\right)}^{2} - {\left(12\right)}^{2} = 36756$

So, new S.D. ${\sigma}_{1}^{'} = \sqrt{\dfrac{36756}{38} - {\left(31\right)}^{2}} = \sigma$

$36756 - {\left(31\right)}^{2} \times 38 = 38{\sigma}^{2}$

$⇒ 38{\sigma}^{2} = 238$