Question

Question 15
Let $\overline{x}$ be the mean of $x_1,x_2,\dots \dots x_n$, and $\overline{y}$ be the mean of $y_1,y_2,\dots \dots y_n$. If $\overline{z}$ represents the mean of $x_1,x_2,\dots \dots x_n,y_1,y_2,\dots \dots y_n$, then $\overline{z}$ is equal to:

A) $\overline{x}+\overline{y}$
B) $\frac{\overline{x}+\overline{y}}{2}$
C) $\frac{\overline{x}+\overline{y}}{n}$
D) $\frac{\overline{x}+\overline{y}}{2n}$

Answer 1

The answer is B.
Given, ${\sum }_{i=1}^n{x}_i=n\overline{x}and{\sum }_{i=1}^n=n\overline{y}\dots \dots \left(i\right)$ $[\because \overline{x}=\frac{{\sum }_{i=1}^n{x}_i}{n}]$
Now, $\overline{z}=\frac{(x_1+x_2+\dots \dots x_n)+(y_1+y_2+\dots \dots +y_n)}{n+n}$
$=\frac{{\sum }_{i=1}^n{x}_i+{\sum }_{i=1}^n{y}_i}{2n}$
$=\frac{n\overline{x}+n\overline{y}}{2n}=\frac{\overline{x}+\overline{y}}{2}$ [from (i)]