The arithmetic mean of a set of observation is ¯ X . If each...

Let ${x}_{1}, {x}_{2}, \ldots , {x}_{n}$ be $n$ observations.

Then, $\overset{¯}{X} = \dfrac{1}{n}Σ{x}_{i}$

Let ${y}_{i} = \dfrac{{x}_{i}}{\alpha} + 10$

Then. $\dfrac{1}{n}\mathop{∑}_{i = 1}^{n}{y}_{i} = \dfrac{1}{\alpha}\left(\dfrac{1}{n}Σ{x}_{i}\right) + \dfrac{1}{n}\left(10n\right)$

$⇒ \overset{¯}{X}\text{new} = \dfrac{1}{\alpha}\overset{¯}{X} + 10 = \dfrac{\overset{¯}{X} + 10\alpha}{\alpha}$

Hence option (c) is correct.