Question

The mean of a set of $30$ observations is $75$. If each observation is multiplied by a non-zero number $\lambda$ and then each of them is decreased by $25$, their mean remains the same. Then $\lambda$ is equal to :

• A. $\frac{1}{3}$
• B. $\frac{2}{3}$
• C. $\frac{4}{3}$
• D. $\frac{10}{3}$

Answer 1

The correct option is C $\frac{4}{3}$

Let the mean be, $\mu =75$
$\Rightarrow \frac{\sum _{i=1}^{30}{x}_i}{30}=75\cdots \left(1\right)$

Now, $\frac{\sum _{i=1}^{30}(\lambda x_i-25)}{30}=75$
$\Rightarrow \frac{\lambda \sum _{i=1}^{30}{x}_i}{30}-25=75$
Using equation $\left(1\right),$
$\lambda \times 75-25=75\Rightarrow \lambda =\frac{4}{3}$