Let the mean and the variance of 20 observations x _ 1 , x _ 2 , x...

Given,

Mean of observation is $15$,

So $\dfrac{∑ {x}_{i}}{20} = 15$

$⇒ ∑ {x}_{i} = 15 \times 20 = 300 ⋯ \left(i\right)$

Also variance is given as $9$

So, $\dfrac{∑ {x}_{i}^{2}}{20} - {\left(\dfrac{∑ {x}_{i}}{20}\right)}^{2} = 9 ⋯ \left(ii\right)$

$⇒ \dfrac{∑ {x}_{i}^{2}}{20} - {\left(15\right)}^{2} = 9 ⋯ \left(ii\right)$

$⇒ ∑ {x}_{i}^{2} = 234 \times 20 = 4680$

Given mean of ${\left({x}_{1} + \alpha\right)}^{2}, {\left({x}_{2} + \alpha\right)}^{2}, \ldots , {\left({x}_{20} + \alpha\right)}^{2}$ is $178$

So, $\dfrac{∑ {\left({x}_{i} + \alpha\right)}^{2}}{20} = 178 ⇒ ∑ {\left({x}_{i} + \alpha\right)}^{2} = 3560$

$⇒ ∑ {x}_{i}^{2} + 2\alpha ∑ {x}_{i} + ∑ {\alpha}^{2} = 3560$

$⇒ 4680 + 600\alpha + 20{\alpha}^{2} = 3560$

$⇒ {\alpha}^{2} + 30\alpha + 56 = 0$

$⇒ \left(\alpha + 28\right)\left(\alpha + 2\right) = 0$

$\alpha = - 2, - 28$

Square of maximum value of $\alpha$ is $4$