The mean of a set of 30 observations is 75 . If each observation is multiplied by a non-zero number

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\

Professor PC Thomas Classes · Answer

(d) : Let $x_{1}, x_{2}, \ldots ., x_{30}$ be 30 observations.\ Given, $\frac{x_{1}+x_{2}+\ldots+x_{30}}{30}=75$\ Each observation is multiplied by $\lambda$ and decreased by 25 .\ $	herefore$ New observations are\ $\lambda x_{1}-25, \lambda x_{2}-25, \ldots, \lambda x_{30}-25$. Now, new mean\ $=\frac{\left(\lambda x_{1}-25ight)+\left(\lambda x_{2}-25ight)+\ldots+\left(\lambda x_{30}-25ight)}{30}$\ $=\frac{\lambda\left(x_{1}+x_{2}+\ldots+x_{30}ight)}{30}-\left(\frac{25 	imes 30}{30}ight)$\ $=\lambda(75)-25$\ According to question, $\lambda(75)-25=75$\ $\Rightarrow \lambda(75)=100 \Rightarrow \lambda=\frac{100}{75}=\frac{4}{3}$.\