The mean and standard deviation of 15 observations were found to be...

Given: Incorrect mean, $x = 12$ and incorrect standard deviation, ${\sigma}^{'} = 3$.

$⇒ \dfrac{{Σx}_{i}}{15} = 12$

$⇒ {Σx}_{i} = 180$

Correct mean is given by,

$\mu = \dfrac{{Σx}_{i} - 10 + 12}{15}$

$⇒ \mu = \dfrac{182}{15}$

Also, ${\sigma}^{'} = \sqrt{\dfrac{∑ {x}_{i}^{2}}{15} - 144} = 3$

$⇒ \dfrac{∑ {x}_{i}^{2}}{15} = 9 + 144$

$⇒ ∑ {x}_{i}^{2} = 2295$

Correct ${Σx}_{i}^{2} = 2295 - 100 + 144 = 2339$

${\sigma}^{2}$ (correct variance) $= \dfrac{2339}{15} - \dfrac{182 \times 182}{15 \times 15}$

$⇒ 15\left(\mu + {\mu}^{2} + {\sigma}^{2}\right) = 15\left(\dfrac{182}{15} + \dfrac{182 \times 182}{15 \times 15} + \dfrac{2339}{15} - \dfrac{182 \times 182}{15 \times 15}\right)$

$⇒ 15\left(\mu + {\mu}^{2} + {\sigma}^{2}\right) = 15\left(\dfrac{182}{15} + \dfrac{2339}{15}\right)$

$⇒ 15\left(\mu + {\mu}^{2} + {\sigma}^{2}\right) = 2521$